Linear Und Complex Analysis Problem Book: 199 Research Problems [1 ed.] 3540128697, 9783540128694 [PDF]

Zitiervorschau

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: USSR Adviser: L.D. Faddeev,Leningrad

1043 Linear and Complex Analysis Problem Book 199 Research Problems

Edited by V. R Havin, S.V. Hru~(~v and N.K. Nikol'skii II II

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Editors

Victor R Havin Leningrad State University Stary Peterhof, 198904 Leningrad, USSR Sergei V. Hru~(3ev Nikolai K. Nikol'skii Leningrad Branch of the V.A. Steklov Mathematical Institute Fontanka 27, 191011 Leningrad, USSR Scientific Secretary to the Editorial Board V.I. Vasyunin

AMS Subject Classifications (1980): 30, 31, 32, 41, 42, 43, 46, 47, 60, 81 ISBN 3-540-12869-7 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-38?-12869-? Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data. Main entry under title: Linear and complex analysis problem book, (Lecture notes in mathematics; 1043) 1. Mathematical analysis-Problems, exercises, etc. L Khavin, Viktor Petrovich. 11.Krushchev, S.V. IlL Nikol'skii, N.K. (Nikotai Kapitonovich) IV. Series: Lecture notes in mathematics (Springer-Verlag; 1043) QA3.L28 no, 1043 [QA301] 510s [515'.076] 83-20344 ISBN 0-387-12869-7 (U,S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, HemsbachlBergstr. 2146/3140-543210

CONTENTS

List

of

Participants...........

Acknowledgements.

. .

Preface.

. . . .

• .

. . .

. .

• • .

. . . . • • • •

. . • • • . • •

•

. . . . . . . • • • •

x

• • .XIll

• • • • • • • • • xvI

PROBLEMS

Chapter

I. A N A L Y S I S

1.1.

Uniformly

1.2.

Compactness

1.3.

When

1.4c.

Local

I. 5 c .

Complemented

1.6.

Spaces

of

of

Bases Spaces

1.9.

Operator

Fourier

absolutely

in

of

spaces

Hardy H P

with

1.11.

Isomorphic

1.12.

Weighted

1.13c.

Linear

1.14.

Supports

spaces the

on

in

and

the

Banach

bases.

spaces

of

and

analytic

The

2.2.

Extremum

problems

radius

2.3.

Naximum

principles

2.4.

Open

2.5.

Homomorphisms

2.6.

Analyticity

2.7.

Homomorphisms

semigroups

•

5

•

7 10

• • •

. . . . . .

ball

i8

in

....

lattices...

of

functions

27 .

.

29

. . . . . . . .

34

. . . . . . . . .

38

convexity.

• •

....

41

. . . . . . . . . . .

46

in

50

quotient

algebra

quotient

norms

algebras

. . . . . . .

Gelfand

Separation

of

2.9.

Polynomial

approximation

in

space algebras

group

• . . . . . in

~

. . . . .

0*-algebras....

of measure

algebra

26

48

for

ideals

24

. . .

. . . . . . .

F-spaces

linear

Banach

the

22

. . . . . . . . . . .

property?

14

. . . . . . . . . . . . . . . . .

2.8.

favourite

H

. . . . . . . . . . . .

from in

functions..

and

functionals

ALGEBRAS

spectral

• •

. . . . . . . . . . . . .

entire

functionals

BANACH

analytic

o f A, ~

classification

of

. . . . . . . . operators..

. . . . . . . . . . .

approximation

2.1.

My

series

2

type . . . . . . . . . . . . . . . . .

blocks

Isomorphisms

2.10.

. . . . . . . . . . .

summing

of

subspaces

1.10c.

2.

SPACES

D~(.X~,'~~) ~ h(X,~ ,~) ?

theory

1.7.

FUNCTION

convergent

is

1.8.

Chapter

IN

51

....

53

• ....

55 58

. . . . . . . of .

multipliers

. • •

algebras

. . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

61

•

65

. • •

68

. . . • . • • .

70 72

IV

2.11c.

Sets

2.12.

Subalgebras

2.13.

Analytic

Chapter

of a n t i s y m m e t r y of t h e

operator

3. P R O B A B I L I S T I C

3.1c.

Some

3.2c.

Analytic

questions

3.3.

Moduli

3.4.

Strong

law

3.5.

Markov

processes

Existence

3.7c.

An

Chapter

4.

4.1.

Boundedness

4.2.

Scattering

4.3.

Polynomial

4.4c.

Zero

82

Hardy

of c o n t i n u u m for

Point

Spectral

4.7.

Non-negative

subspaces

4.8.

Perturbation

theory

4.9c.

Operators

Similarity

4.12c.

Analytic

4.15.

Operator

4.16.

~-inner

4.17.

Extremal

subspaces

Pactorization

4.20.

Pactorization

infinite

4.23.

the

and

129 130

operators

Carlescn

differentiable

multiplication

condition

spectrum

similarities.

155

I~(@,~)

. 158

of isometrics

. . . . . .

160

. . . . 164

• . • • • • 169

. . . . . . . .

matrices .....

functions

shift

147

. . . . . . . .

measures

functions

and

. . 144

.....

• • • • • • 152

representations..

of operator

137 140

functions . . . . . .

spectral

on

. • 135

. . . . . . .

singular

of special

124

operators

functions....

of o p e r a t o r s product

. . . . • 121

subspaces......

for vector

multiplicative

An

Extremal

and

• 116

operators . . . . . .

~-dissipative

matrix-functions.....

4.18.

are

of u n i t a r y

invariant

the

• • 113

equation

of C 1 0 - c o n t r a c t i c n s

theorem

functions

4.19.

When

of

and

. . . . 108

functions......

of re-expansion

and

104 106

problems.....

Hill's

operator

operator-valued

Invariant

.....

eigenfunctions......

and

98

. . • 101

. . . . . . . . . . .

type

87 92

processes

projections

gap

of perturbations

problem

Titchmarsh's

Are

stationary

given

Coulomb

decompositions

4.14.

4.22c.

Future......

and approximation.......

4.13.

4.21.

and

85

processes

. . . . . . . . . . . .

4.5.

Spectral

for

spectral

4.6.

4.11.

Past

with

a

analysis

. . . . . . . . .

in stationary

contractions . . . . . . . .

of dissipative

4.10.

functions

numbers

and

THEORY..

spectrum

78

. . . .

approximation

sets

75

. . . . . . . . . .

operators,

theory

. . •

PROBLEMS

of m e a s u r e s

OPERATOR

G

• • • • . . • •

81

originating

with

for~°°+

. . . .

of large

indicator

sets

. . . . . .....

about

of Hankel

support

algebra....

families

problems

3.6c.

and disk

172

. . . 17V

. . . . . . . . . differentiable?

approximable?

180 . . 184

. . . . . .

189

. . . . . . . . . • • • • • • . 197

4.24.

Estimates

of functions

4.25.

Extimates

of operator

of Hilbert

space

polynomials

o n ~p

4.26.

2x2-Matsaev

conjecture

4.27.

Diminishing

of s p e c t r u m

..... under

an

operators...

199

. . . . . . . 205

. . ....

. ....

extension.

• • • • • 210

209

v

4.28.

The

4.29.

Free

decomposition

of Riesz

4.30.

Indices

of

4.31,

Compact

operators

4.32,

Perturbation

of

spectrum

of

continuous

inver~ibility an

of

operator

4.33.

Perturbation Almost-normal

4,35.

Hyponoz~al

operators

4.36.

Operators,

analytic

4.37.

Generalized

4.38.

What

4.39.

Spectra

4.40.

Composition

Chapter

5.

is

finite

of

of

HANKEL

Approximation

5.2.

Quasinilpotent

~.3.

Hankel

5.4c.

Similarity

5.5.

Iterates

of

Localization

5.7.

Toeplitz

5.8.

Vectorial

5.9,

~actorization

of

Toeplitz

5.11.

Around

5.12.

Moments,

Toeplitz method

5.13.

Reduction Elliptizitat

5.15.

Defect

in

limit

numbers

223

. .

22? 231 234

. . .

238

C

. . . . .

. . . . . . . . .

254

• • • . • • • • •

259

• • • •

• • •

262

. . .

. , .

264

. .

. . . .

space... Hardy

• •

spaces..

Toeplitz

• • • , statistical

boundary

•

274

• •

276

• •

•

279

• • •

•

283

• •

285

• •

physics..

operators..

Projektionsverfahren

• • • •

289

•

•

. .

.

303

.

306

. .

308

. . . . . . . . value

269 271

• • •

matrices....

variables....

and

, .

. . . . . . . . . .

Bergn~n

several

249

251

....

on

244

. . . . .

. . . . .

periodic

240 •

.

th~,orems......

problem

293 298

#

Polncare-Bertrand 6.

°°+

operators

of Riemann

219

algebra....

spaces...

matrices for

und

217

•

continuity

. . . .

operators

operators

operators

5.14.

cfH

the

•

capacities...

lubstitution

OPERATORS

almost

5.10.

Banach

operators

on

Toeplitz

SzegB

a and

Toeplitz

operators

214

• • , ,

. . .

semidiagonality

operators...

Toeplitz

212 , .

. . . . . . . . . . . .

of

Toeplitz

of

absolute

and

elements

• •

and

211

• •

. . . . .

. . . . .

spectral

on Bergman

for

5.6.

Chapter

and

Hankel

operators

operators

~p

operator?..

by

determinant.

s-numbers....

negligibility,

TOEPLiTZ

• • • •

. . . . . . .

spectrum....

integration

AND

its

normal

modulo

endomcrphisms

5.1c.

5.16.

for

differentiations

a

•

and

power-like

operators

•

operators

matrix

with

4,34.

operators..

Fredholm

SING~

6.1c.

The

6.2c.

Classes

operators

INTEGRALS,

Cauchy

integral

of

6.3,

Bilinear

sin~alar

6.4.

Weighted

norm

6.5.

Weak

6.6.

The

6.7.

Is

6.8,

BMO-norm

type norm this

and

the

operator and

Banach

Hp

related Cauchy

for

analytic

. . . . . .

operators.. type

. .

integrals • • •

. . . . .

310

. • •

313

• •

• • • • •

•

317

• •

322

. . . . . . . . . . . . . . Riesz

projections

projection.

invertible?

operator

algebras

. • •

integrals.....

inequalities

substitute of

BMO,

and

domains

in

norm.

on

318

• • • • • • . •

. . . • • • • • , . .

tcri

• • •

•

325

• • • • •

328 329

• , •

• • •

•

Vl

and B M O . . . . . . . . . . .

330

6.10c. Two conjectures by Albert B a e r n s t e i n . . . . . . . . .

6.9c. Problems concerning H ~

333

6.11c. Blaschke products in ~ o

337

6.12.

Algebras

6.13.

A n a l y t i c functions

. ~. . . . . . . . . . . .

contained w i t h i n H =°" . . . . . . . . . . . . in W ~I . . . . . . . . . . . . . of ~ ( T : )

containing ~ ( T ~ )

339 341

6.14.

Subalgebras

6.15.

Inner functions w i t h derivative

6.16.

Equivalent morns i n N P

6.17.

A definition of H P

6.18.

H a r d y classes and R i e m a n n surfaces . . . . . . . . . .

347

6.19.

Interpolating Blaschke products

351

in H P,

7.2.

Holomorphic

.

. 343

345

. . . . . . . . . . . . . . . . .

346

. . . . . . . . . . . . . . . . . . . .

353

functions with limited growth . . . . . .

~ -equation and l o c a l i z a t i o n of submedules

7.3c. Invariant

342 .

. . . . . . . . . . . . . . .

Chapter 7. SPECTRAL ANALYSIS A N D SYNTHESIS 7.1.

......

0~ >0

(see ~2]).

The following problems are of special interest if the weight is not a function of IEI . PROBLEM I. Is it possible to find a representation of th e dual space

of

as a space of certain ho!omorphic functions or

~erms of holomorphic functions, analogous to the so called "K0thed~lit~" ~4] for the space

~C~)

o_~r ~(~) ?

PROBLEM 2: For which weights

? is the space ~

nuclear?

Nityagin ~6] proved the nuclearity of the spaces ~ ' ~

.

PROBLEM 3. Existence of Schauder bases in ~R . This problem seems to be quite difficult. If ~ is nuclear and has a Schauder basis, then ~ can be identified with a Kothe sequence space (see ~7]). If the monomials {E~I~w0 constitute a Schauder basis in ~ , as in the first example, then ~ is a so called powe~series-spaoe(see ~ 7~). Let ~ ~ ~ be an entire function , which is not of the form ~ + ~ 6b~ ~ , then$~@~(~|)~ W~

: by our assumption on ~ , there exist two points Z~ , Z~ ~ , Z ~ E~

~

with

~Z~)= ~Z~)

denotes the Dir~c measures

/~ ) ~ , ~ ) = 0

~(~:~>~4)~

~

E~

, now set ~ = ~Z~- 4 ~

(~=.~ ~)

; then

~ ~ ~J~

, where and

, t h e r e f o r e , by the Hahn-Banach theorem,

. SO, i f ~

does not contain the monomials, ~

cannot have a Schauder basis of the form { ~ } ~ . B.A.Taylor [8] constructed an example of a weighted space of entire functions containing the polynomials and the function ~(E) , but where ~(~) cannot be approximated by polynomials REFERENCES I. B e r e n s t e i n

C . A . and

T a y I o r

B.A.

A new look

40 at interpolation theory for entire functions of one variable. Adv. of Math.,1979, 33, 109-143. 2. G e I ' f a n d I.M. and S h i 1 o v

G.E.

Verallgemeinerte

Funktionen II, III. VEB Deutscher Verlag der Wissenschaften, Berlin 1962. 3. H a s i i n g e r

~z

P.

and

M e y e r

approv~mAtion and interpolation. 4. K o t h e

M.

Abel - Goncarov

- Preprint.

G. Topologische lineare Raume. Berlin, Heidelberg~

New York, Springer Verlag, 1966. 5. M a r t i n e a u A. Equations diff~rentielles d'ordre infini. - B u l l . S o c . M a t h . 6. M Z T ~ r Z H THHa

S

B.C.

de France,

1967, 95, 109-154.

H~epRocT~ ~ ~ p ~ e

. --Tp.MocEB.~mTeM.O-Ba,

C~O~CTBa

npocTpa~cTB

I960, 9, 817--328.

(Amer.Nath.

Soc.Transl., 1970, 93, 45-60). 7. R o 1 e w i c z S. Metric linear spaces. Warsaw, Monografie ~atematyczne, 56, 1972. 8. T a y I o r B.A. On weighted polynomial approximation of entire functions. F. HASLINGER

-Pac.J.Math.,

1971, 36, 523-539. Institut f~r ~ t h e m a t i k Universit~t Wien Strudlhofgasse 4 A-1090 W i e n AUSTRIA

41 LINEAR ~UNCTIONALS ON SPACES OF ANALYTIC FUNCTIONS AND THE LINEAR CONVEXITY IN ~ ~

1.1.3. old

A domain ~ in ~ is called 1 i n e a r 1 y c o n v e x (1.c.) if for each point ~ of its boundary ~ there exists an analytic p l a n e { ~ C ~ : ~ 1 + ° ° . +~+~=0~passing through ~ and not intersecting ~ . A set E is said to be a p p r e x i m a bI e from inside (from out s i d e ) by a sequence cf domains ~ K , K =~, ~,..,if ~ ~ k ~ ~K+~ (resp., ~K+4 c ~ O K ) and ~ = ~ K ~ K (resp., ~ = ~IK ~ K ). A compact set ~ is called 1 i n e a r 1 y c o n v e x (1.c.) if there exists a sequence of 1.c. domains approximating ~ from the outside. Applications of these notions to a number of problems of Complex Analysis, similar concepts introduced by A.~Tartineau and references may be found in [1]-[5]. If ~ is a bounded 1.c. domain with ~-boundary then every function continuous in ~ 0 ~ ~ and holomorphic in ~ has a simple integral representation in terms of its boundary values. The representation follows from the Cauchy-Pantapple formula [7] and is written explicitly in ~8], [I] ,[2]. It leads to a description of the conjugate space of the space 0 ( ~ ) (resp. 0 ( M ) ) of all functions holomorphic in a 1.c. domain ~ (resp. on a compact set M ) which can be approximated from inside (from the outside) by bounded 1.c. domains with ~2-boundary (see [9] for convex domains and compacta and [2] for linear convex sets ; the additional condition on approximating domains imposed in [2] can be removed). Such an approximation is not always possible [6]. This description of the conjugate space is a generalization of well-known results by G.Kothe, A.Grotendieck, Sebasti~o e Silva, C.L. da Silva Dias and H.G.Tillman for the

case

.

.

henE=

for every ~ ~ F ~is Called the c o n j u g a t e s e t plays the role of '~he exterior" in this description. Let

be the approximating domains specified above, on ~ . Consider a differential form H K='I

~ C ~

and

42

A 8u,,,, A . . . < . , ~> : 9 ~ ~K ('(I)) : CI)~I~

To formulate the problem concisely, let us say that the m a x imum p r i n c i p 1 e h o 1 d s f o r ~" I l C C - ~ if for any compact set K C ~ , the supremum of on K is attained at some point of the boundary of K relative to ~ , And if M is a complex manifold, we shall say that the maximum principle holds for F : M --~ ~ if, for any open set i ~ c C and any analytic function ~ : ~ -'=" ~ , the maximum principle holds for F o ~ .

54

He:@

PROBLEM. Let ~ ~

F

and l e t

F" 0 ~-'~" R

...,oct)-:

* FI='II

be defined by

H~/gH ~

where

9(z}

=

F] (~-~0

Does the maximum principle hold for

F

?

REFERENCES I. H e I t o n

JoW. Non-Euclidean functional analysis amd electro-

nics. - Bull.Amer.Math.Soc.,

2. H z K o m ~ c K ~ ~ Hanna, 1980. 3, P t a k

V,,

1982, 7, 1-64

H.K. ~ e ~

Y o u n g

o~ onepaTope c~zra. U o c ~ a ,

N.J. Functions of operators and the

spectral radius. - Linear Algebra and its Appl,, 4- S a r a s o n

D. Generalized interpolation in

1980, 29, 357-392 H ~ . - Trans.Amer~

Math. Soc., 1967, 127, 179-203. 5o Y o u n g

N.J. A maximum principle for interpolation in H ~

Acta Sci.Math°,

N. J. YOUNG

1981, 43, N I-2, 147-152~

Mathematics

department

University Gardens Glasgow GI28QW Great Britain

-

55

2.4

OPEN SEMIGROUPS IN BANACH ALGEBRAS

Let A be a complex Banach algebra with identity, not necessarily commutative. Let S be some open multiplicative semigroup in A . For an element ~ in A let ~(~) denote the distance from the point ~ to the closed set A k S (in other words, it is the radius of the largest open ball centred at ~ and contained in S ), and let $(~) be the supremum of all ~ 0 such that the elements Cb-~'~ belong to ~ for I~I < ~ (that is the radius of the largest open disk centred at ~ and contained in the intersection of with the subspace spanned by ~ and I). So we clearly have $ ( ~ ) ~ ~C~). For a ~ r i e t 2 of particular semigroups S we know that the formula

)~

is valid for every ~ i_~n A - We list below the most important cases. ~irst of all, if S = G ( A ) , the group of invertible elements, the result follows from the spectral radius formula. Second, the formula is true when S is the semigroup of left (or right) invertible elements of A , cf. E6]. Third, it also holds when S is the complement of the set of left (right) topological divisors of zero in A , cf.~]. Next, the formula is true for various semigroups of the algebra A = B(X) of bounded linear operators on a Banach space. In the case when S is the semigroup of surjective (or bounded from below) operators on X it was obtained in [1] by an analytic argument (in fact, this is equivalent to the third case mentioned before). Using an additional geometric device these results were applied in [6] to prove the above formula for the semigroup S of (upper or lower) semi-Fredholm operators on X , and hence it follows for the semigroup of Fredholm operators as well. In these cases the distance ~CT) admits other natural interpretations, namely, it coincides with (or is related to) certain geometric characteristics of the operator T (like the ~urjection modulus, the injection modulus, the essential minimum modulus [ ~ , etc.). In each of the cases listed above an individual approach was needed to find a proof. The difficult steps are of an analytic character, based on the theorems of G.R.Allan (1967) and J.Leiterer (1978) on analytic vector-valued solutions of linear equations de-

86

pending analytically on a ~rameter. The main idea is well demonstrated in [1] though in that case a combimatorial argument is also available [2]. So there seems to be some motivation for investi~atin~ the problem in ~eneral to seek a theorem which would contain all these oarticula r results. Let us give some warnings. Let S = G ~ CA) be the principal component of the set of invertible elements in A . There are (noncommutative) Banach algebras A for which the group G C A ) / G 4 ( A ) is finite, but not trivial, cf.[4],[3]. In such a situation for every invertible element ~ not in G 4 C A ) one can find a positive integer k such that G)k is in G 4 ~ A ) . Then we have 5 ( 0 ~ ) = 0 but ~ k) > 0 so that the formula cannot be true for this ~ and all 6~ in A . ~oreover, it is easy to see that ~ ~ S ~ ) 4/n cannot exist for these G) . Let A be a commutative Banach algebra and let ~ be the open semigroup of all elements whose spectra are contained in the open unit disk. In Ibis case we have ~ ( ~ ) = ~ ( ~ ) for every ~ in ~ but ~I.i--

, and the

We a r r i v e

a t t h e same c o n c l u s i o n i f

preceding definition

by a

last

inequality

may be

we r e p l a c e t h e u n i t

%-multiple

of it,

with

strict.

d i s k i n the

0 < ~
~O. These closed subspaces are usually considered as subspaces of the complex Hilbert space ~ spanned by the whole sequence { Xk ~ keZ • Our problem concerns the description of all possible positions in ~ of the Future ~ with respect to the Past ~p . The sequence { Q ( ~ ) ~ 6 Z being positive definite, there exists a finite positive Borel measure /~ on ~ satisfying ~ ( ~ ) ~ A ~(~), ~Z . The measure ~ is called t h e s p e c t r a 1 m e a s u r e of ~ X ~ } ~ 6 ~ . Clearly the mapping ~ defined by ~)X~l,-~-~ ~ , ~ 14~ ~ can be extended to a unitary operator from to ~(~) . To avoid technical difficulties we consider henceforth all stationary sequences I ~ e Z in ~ , not necessarily real. A stationary sequence is unitarily equivalent to a Gaussian process iff the spectral measure /~ is invariant under the transform ~ ~ of T. Consider the set of all triples (~ ,~, ~ ) where A and are closed subspaces of the complex separable infinite-dimensional Hilbert space ~ such that ~ 0 ~ (A + ~.] ~ ~ . The triples (~,~,~i) and (A$, ~ , ~ ) are said to be equivalent if there exist isometr V o to tis ying = V~ ~- ~ . Let ~ be the set of all equivalence c l a s s e s w i , h respect to the introduced equivalence relation. PROBLEM I. Which classes in ~ (~,

~

~)

contain at least one element

corresponding to a stationar,y sequence ~ X ~ } ~

The class ~ admits a more explicit description. Let ~A denote the orthogonal projection onto the subspace A . Each triple ~ ~-(A,~) defines the selfadjoint operator ~A ~ ~A and the numbers

?

93

L]~I:NA. % triple ~ = unitaril 2 equivalen~and

(A~,B i, ~ ) ~± (~) =

is equivalent to ~ = ~±(}~).

SKETCH OF THE PROOP. We may assume without loss of generality (note that ~ i = ~ under the assumption of the lemma). Given a subspace C in ~ let 6 I = ~@ C Consider the partial isometrics V~,Vz determined by the polar decompositions

Let ~

be an operator on ~

defined by

A nB

~I ~in~ 1 is an arbitrary unitary operator from onto A~N~I • It is easy to see that ~ is defined correctly (if ~ ~A~n B then V~ V ~ z ). Also clearly ~ maps isometrically A~ onto A~ • it remains to verify that ~ is a unitary operator on ~ . Clearly it suffices to show that (~'~,~)---~(~,}) for • sider only vectors Z from (I) that

, this is evident• Now we can conof the form ~---J~A E , ~ ~ ~ It follows i

Pot every ~ ~+ m {~J and f o r every s e l f a d j o i n t operator T on a Hilbert space T~ such that ~ T ~ I ~$~T~{~ there exists ~ (A, ~)~ ~ - ( ~ ) = ~ and such that ~B ~A ~B I ~ e A 1 satisfying ~+(~)=~ is unitarily equivalent to~ Indeed, without loss of generality ~ ± (~] ~ 0 By the wellknown Naimark theorem in ~i ~ ~4 there exists a projection ~A defined by

T

?{(z-T)}) ¢

T (I-

I- T

g4 PutB=~le

{@} a

then ~ and c ~p ~ ~

s/Id

~

and ~ = 6 6 6 ~

ternative

e

ther

(A+B)

. Then clearly

%=

or C, where ~ is an outer function in is a singular measure on ~ o We have cP'~(h~(~o) and therefore only the case ~ ~ 0 is interes-

ting. Remall that for a bounded function %0 on ~ the Hankel opera, where ~+ is tor H~ on ~ is defined by ~ ~ - C I- P÷) ~ H ~ the orthogonal projection from ~ onto ii with the unimodular symConsider the Hankel operator r1~,/h' bol ~---6/~ • It is easy to see that ~G~ ~ p ~ @ ~ I ~$ is unitarily equivalent to ~%1% H~/~ (see [I], Lemma 2.6). The modulus of an operator ~ on Hilbert space is the selfadjoint non-negative operator (T*~)4/~ . Problem I is therefore intimately connected with the problem of description of the moduli of Hankel operators up to the unitary equivalence. PROBLEM 2. Which operator

can be the modulus of a Hankel ooera-

tor? There are two n e c e s s a r y c o n d i t i o n s for an operator to be the modulus of a Hankel operator which imply evident restrictions on triples equivalent to (~p, ~ ~). For any ~ the operator (H~ ~ /~ iLS not invert ible because ~ II~ ~II = 0 It follows that ~ does contain an orthogonal basis { 8~ } ~ o with ~ 6~ ~ = 0 (i.e. an orthogonal sequence "almost independent" with respect to the Past) provided Gp =9= $& . • ~/~ The kernel ~e$(H~ ~ ) is either trivial or infinite dime n.s.ional. Indeed, being invariant under multiplication by ~ by Beurling's theorem, it is either trivial or equal to ~H ~ for am inner function 0. Note that for ~ === (~p, ~ CT) we always have ~(~)-~--~-[~). Indeed passing to the spectral representation ~ we see that Y: i ~ * ~ is an isometry of ~ (over ~ ) with ~ @p=== G~ Y~= ~ . SO we have either ~÷C"~) ~ - ( ~ ) ~--- 0 or ~+(~)~-~_(~)=~ . Under the a priori assumption that the angle between ~p and ~ is positive Problem I can be in fact reduced to Problem 2. Indeed, if the angle between ~rp and ~ is positive then by the Helson-Szeg~ theorem the spectral measure ~ is absolutely con-

95

tinuous and ~ =

I

~&

I

~@~

is unitarily equivalent to *

H~~

H~/k~t HK/~, ..........

• On the other hand if ~ = and II~ ~ < then there exists an outer function ~ in ~ with H ~ - H ~ / ~ (see [2]). Put ~==[,~(I~l~ , B = :~pa~,L~(t~,l~) ~ 2 ~ : ~ 0 } , A~__~h:(I~I:)I ~ : ~ < o} . Clearly [ ~ is a stationary sequence with the Puture B and the Past A . It follows from the above considerations that in the case of nonzero angle between A and ~ the problem of the existence of a stationary sequence with the ~uture ~ and the Past A can be reduced to the existence of a Har~kel operator whose modulus is unitarily equivalent to ~ B ~A ~B • In connection with Problems I and 2 we can propose two conjectures. CONJECTURE I. Let

~ ~ }~ o

positive numbers and let

be a non-increasin~ sequence of

~,~ ~ =

O.

operator whose singular numbers ~) ~ ( ~

Q~cH~) =

CONJECTURE 2. Let that

~

T

,

~

Then there exists a Hankel )

satisf,y

0

be a compact selfad~oint operator such

~eg T

is either trivial or infinite dimensional. Then there * )~ exists a Hankel opera to r U @ satisfyina T ~ (~ ~ •

It can be shown that the last conjecture is equivalent to the following one. CONJECTURE 2'. Given a triPl e ~ (~,B,~)6 ~ such that ~B J~A

is compact

and

~+(~)-~- ~-(~)

there exists a stationary sequence Future is

B

and Past is

~ X~ } ~ 6 ~

is either 0 i__nn ~

o_~r~

whose

~.

We can also propose the following qualitative version of Conjecture I. THEOREm. Let

$> 0

and ~ ~ } ~

o

be a non-increasir~ sequence

of positive numbers. Then there exists a Hankel operator H ~ fyin~ 4

*) See def. of singular numbers in [3~,

satis-

08 PROOE. Let ~ be an interpolating Blaschke product having zeros ~} ~o with the Carleson constant ~ (see e.g, ~ ) . Consider the Hankel operators of the form ~i~ , ~ H ~ Then we have (see [4], ~h. VIII)

where ~ is the compression of the shift operator ~ to ~ d¢__j~ = H~e~, ~ ~PG~ ~ 6 ~ , P~ is the orthogonal projection onto ~ , ~ is multiplication by ~, ~ ( ~ ) &e~ 2~ ~ # ~ ~ . Since ~ is an interpolating Blaschke product, there exists a function ~ in H @0 satisfying $ ( ~ ) ~ ~ 1,1,~0. It follows from (~) that ~ ( ~ ) ~ ~(~(~)), ~0. Consider the ~ectors6 ~ . t I~ (~ IZ;-I~)~t~" We have~(~. )6~,.--~(~ )e_,~ (see L~J, Ch. VI) and there exists an invertible operator V on ~6 such that the sequence I V¢~} ~ 0 is an orthogonal basis of 6 , moreover if ~-- I is small enough then we can choose V so that I]V If' IIV411~ ~ +8 (see ~], Ch. VII). The result follows from t h e obvious estimates

¢

NvlI.Hv"II

llv II. llv"ll

•

Conjecture I can ~e interpreted in terms of rational approximation. It follows from the theorems of Nehari and Adamian-Arov-Krein (see [I]) that for a function ~ in ~ 0 A ~P# ~$ I.°° we have

~ ^ where ~@ is the operator on with the matrix{~(~+~)}~,k$ 0 , ~ is the set of rational functions with at most ~ poles outside ~0~ ~ (including possible poles at co ) counting multiplicities, Conjecture S is equivalent to the following one. CONJECTURE I'. Let ~4~%~-0

I~}¢~0

. Then there exists

~

be a non-increasin~ s.e.quence, in B ~ O A

such that ~ ( S ) ~

~,

~0. If the conjecture is true then it would give an analogue of the well-known Bernstein theorem [5] for polynomial approximation. Note

97 in this connection that Jackson-Bernstein type theorems for rational approximation in the norm B ~ 0 ~ were obtained in ~ ] , K7], ~8~. We are grateful to T.Wolff for valuable discussions.

REPERENCES

I. H • z ~ e p B.B., X p y ~ ~ B C.B. 0 n e p a T o p ~ r a H x e ~ , ~ a ~ y ~ e mpM6xHxeHM2 M CTaUMoHapHMe rayccoBcKMe npoKeccN. - Yc~exH MaTeM. Hays, 1982, 37, ~ I, 53-124. 2. A ~ a M ~ H B.g., A p o B ~.3., K p e ~ H g.F. BecKoHe~H~e ra~KeaeBH m a T p M ~ H O606~eHH~le 8 a ~ a ~ KapaTeo~op~-~e~epa ~ M.mypa. ~y~K:~. aHa.~. M ero npH~., 1968, 2, ~ 4, 1-17. 3. F o x 6 e p r H.L~., h p e ~ H M.F. BBeAeHHe B TeopMI0 XHHeF~-Ib~X HecaMoconp~e~ onepaTopoB. M., "HayKa", 1965. 4. H M K o x ~ c x ~ ~ H.K. ~ l e ~ o6 oHepaTope C~B~Pa, ~., "HayKa", 1980. 5. B e p H m T • ~ H C.H. 06 06paTHO~ sa~aue ~eop~H Ha~nnyumero ~p~6x ~ m e ~ Henpep~H~x ¢ y ~ . - C o 6 p a ~ e c o ~ H . , T.2, I4~-BO AH

CCCP, I954, 292-294. 6. I I e s x e p B.B. 0Hepa~op~ I~a~ex~ ~ a c c a $'~ , ~ Hx npv~omeH~ (pa~MoHax~Ha~ a n n p o x c ~ a ~ , PayccoBcK~e r~po~eccN, npo6xema ~a~o p a ~ , onepa~opoB).-ga~eM, c 6 o p ~ , I980, I I 3 , ~ 4, 5 3 8 - 5 8 I . 7. P e I I e r V.V. Hankel operators of the S c h a t t e n - v o n Neumann class % , 0~ ~< ~ . - LO~! Preprints, E-6-82, Leningrad, 1982. 8. S e m m e s Preprint,

S. Trace ideal criteria for Hankel operators,

1982.

VVoo

S. V. HRUSCEV

(C.B.XPm R) V. V. PELLER

(B. B.nF.JL'IEP)

CCCP, 191011, JleHHHrpa~ ~OHTaHI 0 hold in general. In all knov~ counterexamples

) ~,_oo

, while for

(.3)

6=0

it does not

E I~,lP=o0 (P>~,) PROBLEM I. Is the condition

P

]p>2: s,~pIF I~1 0 ) is a criterion of existence of a probability measure on F with marginals ( ~4, M$~ ) subordinated to the Lebesgue type (i.e. absolutely continuous with respect to the Lebesgue measure M~65~ ),

[1]. Analogous conditions fail to be sufficient for ~ > ~ , and the corresponding criterion is unknown. For ~--~ it is not known whether the Lebesgue type can be replaced by any other type in the last sentence of the previous paragraph. For ~ > ~ there is no existence criterion for a positive measure on the cube ~ with given marginals whose density function with respect to Lebesgue measure is majorized by the density function of a given probability measure on ~ (see the discussion in ~I] ). REI~ERENCES I.

Cy~aEOB

B.H.

reoMeTpEqecEEe n p o 6 ~ e ~

Teop~J4 6eCEOHe~Ho--

105

2.

Mepm~xBepoaTHocTH~xpacnpe~eaemm~. - Tpy%M MMAH, 141, M.-Z., HayEa, 1976. (Proc. of the Steklov Inst. of Math., 1979, issue 2). S t r a s s e n V. Probability measures with given marginals. - Ann.Math.Stat., 1965, 36, N 2, 423-439. V.N. SUDAKOV (B.H.C~0B)

CCCP, 191011, JIeHHHI~8~, • OHTaHEa 27, .E01v~

COMMENTARY BY THE AUTHOR Consider a finite or countable family of probability distributionsI~K,kEK} The answer to the question as to whether there exists such a familyI~K, K E K I of random variables, each ~K being distributed according to ~ K , that for every pair (~i~ k~) the equality

holds, depends on existance of a probability measure ~ ~ on the set

Here ~ is the Kantorovich distance and potential function (see e.g. D]):

UKil(2,

with marginals

stands for related

One o~l show that for such a special type of subsets there always exists a measure with given marginals ~ WSK, K ~ K } , so that the family { ~K} under discussion does exist. I am grateful to B.I.Berg for stimulating discussions

106

ON THE FO~T~.R T~L~NSFOP~ OT THE INDICATOR OF A SET IN ~ OP P ~ I T E ~]~SGEE MEAS~J~

3.7.

old

Consider a set E C ~

o< 1 EI
ET0}*0;

157 2) ~ L ,

~" such that

~,~ll~L.~;

3) ~[,,,C~ , I'1¢(~,)~'0 4) th,e

~p~.,{h,.G)~,e~,.}÷E.;

and

P_ (11P+k~ll~,)

antianal,ytic function

admits ,# ps,,eudo-

continuation to the unit disc. CONJECTURE. For every inner

~ - o u t e r function

~

there exists

T , (defined by (I)) of the form

a nonzero npnc,yclic vector for

P+ h,~ ,,here k, e Ke'~ g~. If the CONJECTURE is not true a counter example must have a number of very pathological properties and may be a candidate for an operator without invariant subspace at all.

REFERENCES 1. Sz.-N a g y B., F o i a ~ C. Harmonic analysis of operators on Hilbert space, North Holland/Akad~miai Kiad~, Amsterdam Budapest, 1970. 2. H ~ E o ~ ~ c ~ ~ ~

H.K.

~eE~

od onepaTope c~m~ra, M., HayEa,

I980. R.TEODORESCU

Universitatea Bra~ov Facultatea de ~atematic~ B-dul Gh.Gheorghiu - DeJ 29, 2200 Bra~ov, Romania

V.I.VASYUNIN

CCCP, 191011, ~eH~Hrpa~,

(B.H.BAC~mS)

~OHTaHEa 27, ZOMM

158

4.14. old

TITCH~ARSH'S THEOREM FOR VECTOR FUNCTIONS In one version (from which others can be derived) Titchmarsh's

theorem states: i f ~ a n d ~ L~(~+) s u c h t h a t ~ * ~ a n d

i f

t h e n

~

Fix ~

~

v a n i s h e s

m u s t

rnishes on (0,~) .

o n

v a n i s h

, and denote by M ~

a r e f u n c t i o n s o f v a n i s h e s o n (0,~), n o

i n t e r v a 1 (0,6),

o n (0,~)

the set of all

~

. Here is a PROOF. such that ~ * ~

va-

is a closed subspace invariamt umder shifts to

the right. Beurling's theorem states that transforms of functions in ~

M

, the space of Fourier

, is exactly ~ H g

, where

~

is inner

in the upper half-plane and H ~ is the Hardy space on the half-planeC Since ~

contains all functions vanishing on (0,1) ,(~(~))m~p(~%)

is

an inner function too. The known structure of inner functions implies that ~ ( % ) ~ X p ( ~ )

for some

~

, 0O

H

I ~k}~,k~O

such that

201

where ~ @ ~

~

is the projective tensor product.

QUESTION I. Is it true that

~,~H ~< > F'~,~V ~ Recal'l

that

=

to show (see b]

(

? is

t

Hankel

matrixo

Tt

is

easy

~/~ ~H

Similarly we can defineM the spaces a~d the ~ e l

the

F'~,= { ~ (,,+... +

tenor

QUESTION 2. Is it

V rl "N

of tensors

)}

{~ZW.,..~.,.,l}~.~0.)

o

true that

M

ll

M

r~, ~ V

?

,I

If Question 2 has a positive answer then V ~ 0 A H is an operator algebra (see ~]) and sole, ~ a n d the estimates II %°(T)II ~ ~o~U~II~

~(P ' IVI,IO A~ i.11

cannot be improved,

QUESTION 3. Is it true that

M

VM

M

An affirmative answer would imply that ~ is an operator algebra and the estimate II~(T)II ~ C0~II 'f IIZ is the best possible (~]). Moreover in this case the estimates is attained on the Davie's example (see ~] ) of power bounded non polynomially bounded operator~ Similarit2 to a contraction, Here we touch the well-known problem (see e.g. ~]) of whether each,,,pol2nomiall 2 bounded operator T o_~n Hilbert, space%,,s similar to a contraction (i.e. whether there exists an invertible operator V such that I VTV-II ~ ~ In D] we considered operators ~S on ~ @

where

I

). ~

defined by

is the shift operator on . It was proved in [I] that is power bounded iff ~ belongs to the Zygmund class A I , i~e~ 4 I~I < I . It was also shown in ~I] that among

202 ~ there are many power b o u n d e d operators, n o n p o l y n o m i a l l y b o v ~ d e d . it seems reasonable to try to construct a counterexample to the problem stated above on the class of the operators ~ . It is very easy to calculatethe functions of ~. Namely,

THEOREM. If

~/6 B M O A

(Recall that J~M0 A ~

~ ~ =~

t.hen B~

is polynomially bounded

~-Cl'l,)~,~': ~(~)~---~,('~,), ~ 0

,

for some

PRO01~. By Nehari's theorem (see [6] "We have

. To f i ~ s h

-'Jr

-

the p r o o f we use the f a c t

that

o

(see FT]). It follows that ~( O~l~f~(~C~)I~(~-~)~)~/~O ~

QUESTION 4. Is it true that if ~ S

~re BMOA

and

is polynomially bounded then

?

This question is related to a question of R.Rochberg KS] concernin6 Hankel operators. QUESTION 5. Does there exist ~&

~

with ~

~B~O A

such that

is similar to no contraction? Operators with th e ~rowth of resolvents of order a

here the oper.tors satisfyi~ I ]~A,T)I ~~

there exists a Fredholm opera to r

[-Aik , Aj/k, ] e~'p

but (I) does

215

Conjecture 2 is confirmed Note that in the example below only for ~ • ~ . S ~ is the two-dimensional EXAMPLE. Let ~= ~(~) where sphere. There exist singular integral operators such that

Aik

is a Eredholm operator and ~ = ~ ([5], Ch.XIV, ~4). As ~ = 0 (~5], Ch.XIII, theorem 3.2) equality (1) does not ^ hold. It can be assumed that the symbols of the operators ~k are infinitely smooth (for ~ ~ 0 ) and therefore the commutators [~ik ~ A i ~ ] Hence

map

~ ( S ~)

S~([Aik,Ai~k~] ) = 0 ( ~

into ~1~)

W ~ (~ ~)

([6], theorem 3).

(see e.g.[7]).

We note in conclusion that some conditions sufficient for the validity of (I) have been found in ~8-10]. In these papers as well as in[S],[3] the operators in Banach spaces are considered .

RE~ERENCES I.

2. 3. 4.

5. 6. 7.

K p y n H E E H.H. K BOllpocy 0 HOpMSflIBHO~ paspemzMOCTH E EH-~e~ce C~ry~spH~X ~HTeI?pS~HRX ypaBHeH~. -- Y~. sa~.K~E~HeBCEOrO yBL~BepcHTeTa, I965, 82, 3--7. r O X 6 e p r E.~., $ e x ~ ~ M a H M.A. Y p a B H e H ~ B cBepTEax E npoeEn~oHR~e M e T o ~ ~X p e m e H ~ . M., HayEa, I97I. Map E y c A.C., $ex ~ M a H M.A. 06 EH~eEce onepaTopHO~ MaTpHn~. --~#H~u.a~a~. E ero n p ~ . , I977, II, ~ 2, 83-84. r o x 6 e p r M.H., Kp e ~ H M.r. BBe~eHEe B Teop~0xE-He~HNX HecaMoconp~eHHRX onepaTopoB B I~B6epTOBOM npooTpaHCTBe. M., HayEa, I965. M i c h 1 i n S.G. , P r o s s d o r f S. Singulare Integra~operatoren. Berlin: A k a d e m i e - Verlag, 1980. S e e 1 e y R.T. Singular integrals on compact manifolds. Amer.J.Math., 1959, 81, 658-690. H a p a c E a B.M. 0O ac~MnTo~Ee CO60TBeHH~X ~ CEHryJ~HNX ~ o e ~ Jn~He~z~u~x oHepaTOpOB, H O B ~ P~a~EOCTB. -- MaTeM.cOopH~, I965, 68 (II0), 623-63I.

216

8, E p y n H ~ E H.H. HeEoTopNe o S ~ e Bonpoca Teopm~ O~HOMepHI~X C~I~yJZapH~( onepaTopoB c M a T p E ~ EOS~dl~eHTa~m. -- B EH. : HecaMoconp~eHH~e onepaTopm. K~m~eB, ~TY~Bum, I976, 9I-II2. 9. E p y n H ~ E H.H. YCJIOBH~ cy~eCTBOBaHN21 ~L-C~BO/~a E ~OCTaTOqHOrO ~adopa •--MepHHX npe~cTs3xeH~ 0aHSXOBO~ a~re6pH. B E~.: ~ R e ~ e onepaTop~. I~m~HeB, m T ~ , 1980, 84--97. I0. B a c H x e B C E ~ ~ H.~., Tp y xMX ~ o P, K T e o p ~ ~-onepaTopoB B M a T p ~ a~re6pax onepaTopoB. B m~.: ~ e ~ e ouepaTop~. ~ e B , ~IT~m~a, 1980, 3-15.

I.A. l~EL 'DMAN A. S. ~L~RKUS

(A.C.MAPEYC)

CCCP, 277028, I(~,,~eB, ~HCT~TyT MaTeMaT~EH AH MCCP

217 4.31.

SOME PROBLEMS ON COMPACT OPERATORS WITH POWER-LIKE BEHAVIOUR OP SINGULAR NI~BERS

Classes of compact operators with power-like behaviour of eigenvalues and singular numbers arise quite naturally in studying spect~ ral asymptotics for differential and pseudodifferential operators. Presented are three problems related to the theory of such classes. Let ~(~) be the algebra of all bounded operators on a Hilbert space H . Given A in the ideal C of all compact operators in ~ define 5~(A), ~=~,~,..., the singular numbers of ~ . ~or 0 < p < o o let

0

p:{Ae 7.p:s(A)= See [1-4] for details concerning ~-Tp While studying spectral asymptotics the main interest is focused not on the spaces ~, 7.o themselves, but on the quotient spaces

P'

p

The spaces ~p , O 0~ ,

m

ad-

such t~t

acts on a non-zero finitedimen@io-

nal subspace? (An affirmative answer to this last question implies that (Fin) ~ = ~

).

(7) Any partial answer to the above questions will also shed some light on several interesting problems related to quasidiagonal operators. REFERENCES I. A n d e r s o n

J.H.

Derivations, commutators and essential

numerical range. Dissertation, Indiana University, 1971. 2. B u n c e J.W. Finite operators and amenable C*-algebras. - Proc.Amer. Math.Soc.~1976, 56, 145-151. 3. B u n c e J.W., D e d d e n s J.A. C*-algebras generated by weighted shifts. - Indiana Unlv.Math.J.~1973, 23, 257-271. 4. H a 1 m o s P.R. Ten problems in Hilbert space.-Bull.Amer.

243

Math.Soc.~1970, 5. H e r r e r o

76, 887-933. D.A. On quasidiagonal weighted shifts and appro-

ximation of operators.-Indiana Univ.Math.J. 6. V o i c u 1 e s c u

D.

theorem. - Rev.Roum.Math.Pures 7. W i 1 1 i a m s

J.P.

(To appear).

A non-commutative Weyl-von Neumann et Appl.~1976,

Finite0perators.

21, 97-113.

- Proc.Amer.Math.Soc.~

1970, 26, 129-136. DON[INGO A.HERRERO

Arizona State University Tempe, Arizona 95287 USA

This research has been partially supported by a Grand of the National Science ~oundation

244

4.39.

THE SPECTRUM OF AN ENDOMORPHISM IN A COMMUTATIVE BANACH ALGEBRA

The theorem of H.Kamowitz and S.Scheinberg D] establishes that the spectrum of a ~0n~eriodic automorphism ~ : A - ~ A of a semisimple commutative Banach algebra A (over 6 ) contains the unit circle ~ . Several rather simple proofs of the theorem have been obtained besides the original one ([2],[3]e.g.) and its various generalizations found (see e.g.[4],[53,[6]...). It is easy to show that under the conditions of the theorem the spectrum is connected, At the same t~me all "positive" information is exhausted, apparently, by these two properties of the spectrum. There are examples (see [7~, [8]~[9~) demonstrating the absence of any kind of symmetry structure in the spectrum even if we suppose that the given algebra is regular in the sense of Shilov. Let for instance ~ be a compact set in ~ lying in the annulus { : } and containing { ~ : ~ I ~ l l Zl and equal to the closure of its interior int K. Denote by A the family of all functions continuous on ~ and holomorphic in int K. Clearly, A equipped with the usual sup-norm on ~ and with the pointwise operations is a Banach algebra. The spectrum of multiplication by the "independent variable" On A evidently coincides with ~ . On the other hand the conditions imposed on ~ imply that A is a Banach algebra (without umit) with respect to the convolution

~4

F

~(~) ~ ( ~ )

......

corresponding to multiplication of the Laurent coefficients. This algebra being semi-simple, its maximal ideal space can be identified with the set of all integers. Adjoining a unit to A thus turns it into a regular algebra. Obviously the above mentioned operator on A is an automorphism. 1..A.re t.here other necessary conditions, on th.e.spectrum of.~ nonperiodic automorph%sm of % semi-s~mple commut.a.t~.v.~ 'Banach al~ebra besides the t~0........m.entionedabove? In particular, is it obligatory for the spectrum to have interior points when i% differs from ~ ? It is known in such cases (see ~7],KB]) that the set of interior points

245

may not be dense in the spectrum and may not be connected either. Let M A be the maximal ideal space of a commutative and semis~mple Banach algebra A . An automorphism T of A induces an automorphism of the algebra C ~ C(M A) . The essential meaning of the Kamowitz - Scheinberg theorem is that ~c(T) c ~A (T) . It is natural from this point of view to study the inclusion gc (L) c gA ( ~ ) for a more general class of operators L . The case of weighted automorphisms I,@ ~e~ W. T ~ with ~ an invertible element of A has, for example~been studied in [10]. It turns out that the inclusion does not hold for this class of operators. 2,

Does the spectrum of L = ~ ,

automorphism

T

c g ~ t r u c t e d for a

non-periodic

, contain any circle ' centred at %he origin? If it

does then we obtain an instant generalization of the theorem of Kamowitz and Scheinberg. The spectrum of operators, looking like l. , acting on the algebra of all continuous functions on a compact set has a complete description[11 . I f A is also a u n i f o = algebra then Thus 6"A (~) ~ ~ ( ],) provided that I, is a weighted automorphism of two uniform algebras ~ and ~ having the same maximal ideal spaceo

3. Let A be a closed subalgebra of a semi-simple commutative Banach algebra ~ and let ~A--~ M B . Let ~ be a weighted automorphism of A and ~ simultaneously. IS it true then that

6~A(~) =

~B ( h )

?

We O O N ~ a T ~ R that this question has a negative answer. The spectrum of an endomorphism apparently does not have any particular properties even if we suppose that A is a uniform algebrao Given two oompaota ~4 and ~9 it is easy to obtain an endomorphism with the spectrum either ~4"U ~ or ~I' ~% . The only obvious property of spectra is that Iw belongs to the spectrum, when ~ ranges over its boundary and ~ over the set of non-negative integers. 4. Let ~ be a compact subset of ~ satisfying ~ 6 ~ for ~ = ~,~,.., and for all points ~ in the boundary of ~ . Is there an endomorphism of a uniform algebra whose spectrum is eq~l

to ~

?

Spectra of endomorphisms of uniform algebras (and even those for weighted endomorphisms) can be described pretty well under the

246

additional assumption that the induced mapping of the maximal ideal space keeps the Shilov boundary invariant (see D2], where one can find references to preceding papers of Kamowitz). Roughly speaking, things, in this~case, are going as well as in the case of Banach algebras of functions continuous on a compact set. The situation changes dramatically when the boundary, or only a part of it, penetrates the interior. In such circumstances it is common to begin with the consideration of classical examples. Let D be the unit disc in C , let A(~) be the algebra (disc-algebra) of all functions continuous on the closure of ~ and holomorphic in ~ , and let H~(~) be the algebra of all functions bounded and holomorphic in ~ . Both algebras A(~) and H ~ ( ~ ) are equipped with the supnOl~. 5. Every endomorphism of A(O) induces a natural endomorphism of ~ . Do the spectra of thes e endomorphisms coincide? In this connection it is worth-while to note that the answer to an analogous question concerning the algebra of all continuous functions on a compact set and the algebra of all bounded functions on the same set is in the affirmative ~3~. The proof of this result uses, however, a full (though comparatively simple) analysis of the possible spectral pictures depending on the dynamics generated by the endomorphism. The interesting papers ~4~, ~ , ~6~ of Kamowitz (see also [6S, [9~) deal with spectra of the endomorphisms of A ( D ) whose induced mappings do not preserve the boundary of ~ . In the non-degenerate case the spectrum has a tendency to fill out the disc.Discrete and continuous spirals as well as compacta bounded by such spirals may nevertheless appear as the spectrum of an endomorphism. (But only the spirals can appear in the case of Mobius transformations). 6. Is the spectrum of an endomorphism of the disc-algebra a semi-~rouu (with respect to multiplication in ~ ) ? What kind of semi~rouDs can arise as spectra.? 7. Is it possible to say something concernin~ the spectra of 9ndomorphisms of natural multi-dimensional generalizations of the disc algebra? Note that in the one-dimensional case the theory of DenjoyWolff and the interpolation theorem of Carleson-Newman are often

247

involved in the question. The problem of describing spectra for weighted automorphisms is closely related with an analogous one for the so-called "shift-type" operators which have been studied by A.Lebedev ~7~ and A.Antonerich

Dsl

Let A be a uniform algebra of operators on a Banach space X . An invertible operator U on X is called a "shift-type" operator if U A U A . UB lly X a B nach space of f ctio and A is a subalgebra of the algebra of multipliers for ~ . The transformation ~ ~ U. ~ U ~ determines an automorphism T of ~ which induces the mapping ~: ~ A ~ M A . It is assumed that: I) the set of ~-periodic points is of first category in the Shilov boundary 8A ; 2) the spectrum of U : X ~ ~ is contained in S A ; 3) each invertible operator @ : X - - ~ X , ge~ is invertible as an element of A ; 4) the topological spaces ~A and ~ have the same stock of clopen (closed and open) ~ -invarlant subsets. Then ~ ( @ ~ ) - ~ - ~ ( ~ T ) for all ~ in ~ [19]. We con,~ecture that Condition $ is superfluous. If this were true it would be possible (in view of [1I]) to obtain a complete description of @~ ( @ ~ . It is reasonable to ask the same question for other algebras A besides the uniform ones. REFERENCES I. K a m o w i t z automorphisms

H., S c h e i n b e r g

of Banach algebras.

S.

The spectrum

- J.Punct.An.,

1969,

of

4, N-2,

268-276. 2. J o h n s o n

B.E.

Automorphisms

ras. - P r o c . A m . M a t h . S o c . ,

of commutative

Banach algeb-

1973, 40, N 2, 497-499.

3. ~ e B ~ P.H. HOBOe ~o~asaTe~cTBo TeopeM~ O6 aBTOMOp~zsMax 6aEaXOBHX a~re6p. -BeCTH.MFY, cep.MaTeM., Mex., I972, ~4, VI-72. 4. ~ e B ~ P.H. 0d aBTOMOp~SMaX 6aHaXOBRX a~re6p. - ~ym~.aHax~8 E ero np~., 1972, 6, ~ I, 16-18. 5. ~ e B E P.H. 0 COBMeCTHOM cneETpe HeEoTopb~x EOMMyTI~py~E~X oEepaTOpOB. ~ c c e p T ~ , M., 1978. 6. r o p ~ H E.A. EaE BR~JL~ET cHeETp SH~OMOp~EsMa ~cE-a~re~p~? 3au.HSyqH.ce~m~o~0~, 1983, 126, 55-68.

248

7. S c h e i n b e r g S. The spectrum of an autemorphism. Bull.Amer.Math.Soc,, 1972, 78, N 4, 621-623. 8. S c h e i n b e r g S. Automorphisms of commutative Banach algebras. - Problems in analysis, Princeton Univ.Press., Princeton 1970, 319-323. 9. r o p ~ H E.A. 0 cne~Tpe SH~OMOp~SMOB paBHoMepHHx axre6p. B EH. : Tes~cH ~oEx.~oH~ep"TeopeT~xecz~e ~ n p m ~ a ~ H H e BOnpOC~ Ma-TeMaT~E~" TapTy, I980, I08--II0. I0. E ~ T 0 B e p A.E. 0 cne~Tpe a B T O ~ O p ~ s M O B C BecoM ~ TeopeMe EaMoB~um-~s/m6epra. - ~yHE~.aHa~Hs ~ ero n p ~ . , 1979, 1 3 , ~ I, 70-71. II. E ~ T o B e p A.E. C n ~ T p a x ~ - - e CBOICTBa a B T O M O p ~ S x O B C BeC O M B paBHo~eps~x axre6pax. - 3 a n . H ~ . C e M ~ H . ~ 0 M H , I979, 92, 288-293. I2o E E T O B e p A.K. CneETpax~HHe CBO~CTBa r O M O M O p ~ S M O B C BecoM B a~re6pax Henpep~Bm~X ~ ~ ~x np~o~eHH~. - 3an.HayqH.ce~ . ~ I 0 1 ~ , 1982, 107, 89-103o 13. I{ E T 0 B e p A.E. 06 oiiepaTopax B C (~ , ~H~yI~IpOBaHHIgX l~a~ Emm~ OTo6pa~eH~2m. - ~JE~I~.asax~s E ero n p ~ . , 1982, 16, ~ 3, 61-62. 14. K a m o w i t z H. The spectra of endomorphlsms of the disk algebra. - Pacif.J.~ath., 1973, 46, N 2, 433-440. 15. K a m o w i t z H. The spectra of endemorphisms of algebras of analytic functions. - Pacif. J.Math. 1976, 66, N 2, 433-442. 16. K a m o w i t z H, Compact operators of the form @ C@ . Pacif.J.N~th., 1979, 80, N I, 205-211. I7. ~ e 6 e ~ e B A.B, 06 onepaTopax T~na BSBemeHHOrO C~BEra. ~ccepT~, M~HCE, 1980. 18. A H T O ~ e B ~ ~ A.B. 0nepaTop~ co C~B~rOM, n o p o m ~ e ~ M ~e~cTB~eM EOMIIaETHOI l~y211H H~. - CE61~pCE.MaTeM°EypH°I979, 20, ~ 3, 467-478. 19. E ~ • 0 B e p A.E. 0nepa~op~ IIO~CTaHOBE~ C BecoM B 6aHaXOBHX Mo~y~Ex H8~ paBHOMepH~M~ ax~e6pa~m (B negate). E. A. GORIN CCCP, 117234, ~ocFma ~ e H ~ c K ~ e rop~ (E.A.IDPHH) MexaHzEo-~aTe~aT~ecEz~ ~ r y 2 ~ T e T M O C K O B C E ~ rocy~apcTBeRR~ yHzBepc~TeT

A. K. KITOVER

(A.K.EETOBEP)

CCCP, 191119, ~eH~Hrpax, y~ KOHCTaHTNHa 3aC~OHOBa, ~.14, KB.2.

249 4.40.

COMPOSITION OF INTEGRATiON AND SUBSTITUTION

Consider a continuous function ~ on [0,1] satisfying~(0)=0, 0 ~ ~(~) ~ ~ . The function ~ defines a bounded linear operatorI~ On the space C ~o,1~ of all continuous functions on ~0,I~:

0

Recall that a bounded operator T

£%1IT"lt¢

o.

=

PROBT,~.'~. Describ,~ f u n c t i o n s operators

is called quasinilpotent if

~

correspondin ~ to q u a s i n i l p o t e n t

.

Clearly I ~

is quasinilpctent provided

0~< ~ ~ 1 Does the inverse., conclusion hold?

(2) An analogy with the theory

of matrices provides arguments in favour of the affirmative answer. Let I @ ~ } ~ - 4 be a nilpotent matrix with non-negative elements. It follows from the Perron-Frobenius theorem that it can be transformed to a low-triangular form with zero diagonal by a permutation of the basis. The obtained matrix I ~ I defines an operator On ~

with ~(~) < ~

.

Consider now a natural generalization of (I):

(3) 0

where the kernel K >~ 0 is continuous. Orientation preserving home~nerphisms of EO,1] replace permutations of the basis in the finite-dimensional case and preserve the inequality ~ ( X ) ~ X •

250

Consider a counter-argument to the conjecture. IfI~

s i n i l p o t e n t t h e n by Y e n t s c h ' s theorem t h e r e e x i s t zero continuous function S ~ 0 such that

~~ 0

is not qua-

and a non-

@(x)

o

Suppose q}~C @° ( [ 0 , 1 ~ ) . Then e v i d e n t l y ~ C ~ ( K0,1]~and;~(~'(O)= O, K~O,~.., ~ because ~ ( 0 ) ~ 0 . If it were possible to prove that belongs to a quasianalytic class (under some natural restrictions on ), it would imply clearly that ~ ~ 0 which contradicts Yentsch's theorem. Are there conditions on ~ no t demanding @(~)~X but such that a~y solution of ($~ belonss to la lquasianal,ytic Carleman class?

If yes, then there

exists ~

such that ~(~o) • $o

point to in EO,I] but neve~heless I ~ YU. I. LYUBIC

(D.H.~)

for a

is q~slnilpotent.

CCCP, 310077 Xap~EOB n~.~sepz~HcEozo 4 Xap~EOBCE~ IOCy~apCTBeHmm~ yH~Bepc~TeT

CHAPTER

5

HANKEL AND TOEPLITZ OPERATORS

A quadratic form is called Hankel (resp. Toeplitz) if entries of its matrix depend on the sum (resp. the difference) of indices only. These forms appeared as objects and tools in works of Jacobi, Stleltjes (and then Hilbert, Plemelj, Schur, Szego, Toeplitz ...). They play a decisive role in a very wide circle of problems (various kinds of moment problems, interpolation by analytic functions, inverse spectral problems, orthogonal polynomials, Prediction Theory, Wiener-Hopf equations, boundary problems of Function Theory, the extension theory of symmetric operators, singular integral equations, models of statistical physics etc.etc.). It wBs understood only later that the independent development of this apparatus is a prerequisite for its applications to the above "concrete" fields, and Hankel and Toeplitz operators were singled out as the object of a separate branch of Operator Theory. This branch includes: - techniques of singular integrals ranging from Hilbert, M.Riesz and Privalov to the Helson-Szego theorem discovered as a fact of Prediction Theory, and to localization principles of Simonenko and Douglas; - algebraic schemes originating from the fundamental concept of symbol of a singular integral operator (Mihlin), from the semi-mul-

252

tiplicative dependence of Teeplitz operator cn its symbol, (WienerHopf), and culminating in the operator

N-theory;

- methods and techniques of extension theory (Krein), which have attracted a new interest to metric properties of Hankel operators and to their numerous connections; -

other important principles and ideas which we have either

forgotten or overlooked cr had no possibility to mention here. The inverse influence of Hankel and Toeplitz operators is also considerable. For example, many problems of this chapter fit very well into the context of other chapters: Banach Algebras (Problem 5.6), best approximation (Problem 5.1), singular integrals (Problem 5.14). Problems 2.11, 3.1, 6.6, 10.2 can hardly be

severed from

spectral aspects of Toeplitz operators, and Problems 3.2, 3.3, 4.15~ 4.21, 8.13, S.6, from Hankel operators. ~ n y

problems related to the

Sz.-Nagy-Foia~ model (4.9-4.14)can be translated into the language of Hankel-Toeplitz (~)

(possibly, vectorial) operators, because functions

of the model operator

Hankel operators ~G~~

T@

coincide essentially with the

, and the proximity of model subspaces ~ _

and K@~ can be expressed in terms of the Toeplitz operator TG* etc. Hankel-Toeplitz problems assembled in this book do not exhaust even the most topical problems of this direction *), but contain many interesting questions and suggest some general considerations. Many of the problems are inspired by some other fields and are rooted there so deeply that it is difficult to separate them from the corresponding context. We had to place some Hankel-Toeplitz problems (not without hesitation and disputes) into other chapters. Examples can be found in Chapter 3 (3.1, 3.2, 3.3). Moreover, we believe that

*) To our surprise nobody has asked, for instance, whether every Toeplitz operator has a non-trivial invariant subspace...

253

Problem 3.3 is one of the most characteristic and essential problems of exactly t h i s

Chapter and we hope that the reader looking

through this chapter will turn to Problem 3.3 as well. Problems 5.1-5.3 deal with metric characteristics operators (compactness,

spectra, s-numbers).

of Hankel

In connection with

Problems 5.3 and 5.7 concerning operators acting

n o t

in H~we

should like to mention recent investigations of S.Janson, J.Peetre and S.Semmes and of V.A.Tolokonnikov (X-~Y)

(Spring 1983) who have found

-continuity criteria (in terms of symbols) for Hankel and

Toeplitz operators in many non-Hilbert function spaces

X, ~

.

Problems 5.4 and 5.5 treat similarity invariants and some properties of the calculus for Toeplitz operators. Problems 5.6, 5.13 are related with localization methods, problems 5.8-5.10,

5.15 deal with vectorial and multidimensional

of Toeplitz operators and with related function-theoretic

variants

boundary

problems. Problems 5.11-5.14 treat "limit distributions of spectra" (asymptotics of Szego determinants,

convergence and other properties

of projection invertibility methods etc). The theme of Problem 5.16 may be viewed as a non-commutative analogue of Toeplitz operators arising in the theory of completely integrable systems.

The field of action and the multitude of connections of HankelToeplitz operators are so impressive that it became fashionable nowadays to find them everywhere - from bases theory to models of Quantum Physics and ... even where they really do

not c c c u r - ~

254

5oio old

APmOX~aTIO~ OF B0~DED F~CTIO~S BY E ~ T S OF M ~ + C

Every sequence { ~ } ~ 4 of complex numbers defines a Hankel matrix ~ = { ~+k-~ ~ j,K~4 which is considered as an operator in the Hilbert space ~ . By Nehari's theorem ~ is bounded if and only if there exists a function ~ in the algebra ~c~ ^of all bounded and measurable functions on T such that [~=~ (-~) , ~$= 4, 2,..., ~(~) being the Fourier coefficient [ ~ ' ~ n ~ t ~ of . This function is uniquely determined up to a summand from the Hardy algebra H ~ . The norm ] ~ of ~ coincides with diet ( ~ ~ ) . Given ~ ~ C let ~(~) be the Hankel operator corresponding to the sequence {[~>~4 ~ [ = ~ ( - ~ ) ~ ~ = 4,~,,,. Usual compactness arguments imply that for every exists ~ E H ~ such that

~ ~-~

there

A criterion of u n i q u e n e s s of the best approximation ~ as well as a description of all such ~ s in the non-uniqueness case have been obtained in [2 2 . Hartman has shown [3] that F is compact iff there exists a function ~ in the algebra of all continuous functions on T such that F = P ( ~ ) Moreover, if r is compact then for every 6 > 0 there exists ~ C satisfying r(~6)= r and ~ C %Ir~ + 6 • The results of Nehari and Hartman easily imply the following result discovered independently by Sarason [4] : the algebraic sum H~+C is a closed subalsebra o Z L ~ C s e e ~ s o [~] .here ~ properties of this subalgebra are disussed), Hartmau's result implies also the following characterization of ~ + C : an element ~ in ~ belongs to ~+C if and o n l y i f C(~ ) is a compact o p e r a t o r . Let S~(F), ~=4,~...denote S-numbers of Vcounted with multiplicities and let S ~ ( F ) be the least upper bound of the essential spectrum of ( (cf°[6], §7). Clearly, S ~ ( F ) ~ S ~ ( r ) as ~---~ + ~

r*r)~

co

THEOREM. Let

~L

and

~ = ~(~)

. Then

civet (~, H +C) =S~(r).

(1)

255 Given a function possibilities.

in

CASE I. S ~ ( F ) = I F I

~\(H~+C) , i.e. F

consider

the following

does not have

~-numbers

greater than ~eo ( ~ ) " CASE 2. There exists only a finite number ~ , cf S-numbers greater than Soo ~r) • CASE 3. The set of S-numbers to the right cf Soo(~s infinlte~ Formula (I) is a simple consequence of theorem 3.1 from [2a] but for the purpose of this note it is more convenient to connect it with the investigations of [2c]. co For any positive integer k let H K denote the set of all sums ~ = ~ + ~ , where ~ H c° and ~ is a rational function of degree % k , having all its poles in the open unit disc ~ and vanishing at infinity. The set H ~ is neither convex nor linear. ~lls s e t coincides with the family of symbols ~ corresponding to the Hankel operators ,F'c,~(~) of finite rank ~< k . Clearis dense in ly Ho f,.~H ) OH1 c . . . ~ H +C and ~U~I Fill,

H~+C

. Therefore dist (~,H~C)

On the other hand it has been shown in

Co

~

--

[2C]

~st CJ~,H,~). that diet

(;~ ,H k) = $k (.r)

provided SI~K÷I(V ) . Hence (I) holds in case 3- In case 2 we , which implies (1) have (see [2c], ~5) S ~ ( ~ ) = ~ S t (~ , H ~ ) also. At last, in case 1 we obviously h a V ~e l~t(j~,

H )=~st(~, H%C).

aive~ j~eL

denote by M~

the set of ~ll

geH +C

satisfying II~ . ~ h ~ ----Soo(P(~)) " In case I M ~ n H ~ ~ ~ and in case 2 M ~ { ] H m ~ ~ . A necessary and sufficient condition for M~ {l H~ to be a one-point set as well as a description of when it contains more than one element, have been obtained. As for case 3 it is VNENOWN: a) Ca._nnM ~

be emptE for some

cribe such ~ b) Can

and if so how to des-

? M~

a@

~ E Lce

consist of a single element for some ~

i_nn

co

L X(.H +C)? c) I_~f m ~ ~ ~ "selected" part of .

.

.

.

"selected" part)?

then is it possible to describe at #east a

M~ .

just as in case .

,2, ,,(w'ith

M~

NH

as a

256

Clearly ~ ~& ~ for # ~ [°~=\(H~+C) if and only if there exists ~ C such that for #4 # - ~ case ] holds, i.e. Ir(~)l = S~(~(~4)) ° Question b) remains interesting for cases I and 2 also. The matter is that there are situations when card M ~ = o o but card ( M ~ ~ H ~ ) = ~ (in case I) and card ( M ~ ~ H ~ (in case 2). Indeed, let for example ~ be an inner function with singularities on an arc ~ c T , ~ ( A ) < 4 , and let g be any function in C satisfying ~ ( r ) - - 4 on A and l ~ ( r ) l < 4 for ~ T\ A . Then ~6----geC and setting # ----~/6 we have

S~(r(#))--~i~t(#,HtC)=~Li~t(#,H5 =]= fig- 1#~. Ho.ever, I1#-~11>~ Almost

for every & ~ H ~ , t 1 ~ > 0

everything

s a i d a b o v e c a n be g e n e r a l i z e d

matr~x-valuod fu~otions F--(#~) ,

or C

.~t~ e~tries

• In this case the norm

should be def~ne~ as

.

~ ~ T~ o

Hilbert-Schmidt norm of tions we refer to [2d].

A

I F(~)I

~

IIF II

where

of

IAI

to the

case o f

belongi~ to F e

L~,# ~

stands for ~ e

In connection with these generaliza-

REFERENCES I. N e h a r i A. On bounded bilinear forms. - A n n . ~ t h . , 1957, (2), 65, 153-162. 2. A ~ a M ~ B.M., A p o B ~.B., Ep e ~H M.F. a) BecEoHe~Hae rsazeaeBH M a T p m m ~ OdO6~eHH~e s s a a ~ KapaTeo~op~-~e~epa $.PHcca. - ~F~m/.aaaa. ~ ero np~a., I968, 2, I, I-I9; b) BecEoH e , H e rss~eaeBH M a T p m m ~ odo6meHHHe s a ~ a ~ EapaTeo~op~-@e~epa n M.~ypa. - ~/~U.asa~. ~ ero npHa., 1968, 2, 4, 1-17; ~) A~aa~T~~ecE~e CBO~CTBa Hap ~ T a raHEeaeBa onepaTopa n o d o 6 m e ~ a ~ sa~a~a ~ypa-Ta~ar~. - MaTeM.cd., I97I, 85 (I28), ~ I9, 38-78; d) BecEoHe~mae 6ao~Ho-rssEeaeB~ M a T p H ~ ~ c ~ s a a m ~ e c ~ n p o 6 ~ e ~ ~po~oaxem¢~. - MsB.AH ApMCCP, I97I, YI, ~ 2-3, 87-II2. S.H a r t m a n P. On completely continuous Hankel matrices. Proc.Amer.Math.Soc., 1958, 9, 862-866. co 4. S a r a s o n D. Generalized interpolation in H . - Trans. Amer.Math. Soc., 1967, 127, 179-203. 5. S a r a s o n D. Algebras of functions on the unit circle. Bull.Amer.~athoSoc., 1973, 79, N 2, 286-299.

257

.

r o x d e p r Ho~., Kp e ~ H M.r. BBe~eHHe B T e o p ~ n o ~ e ~ HNX HecaMoconlo~eH~x oHepaTopoB. -- M., Hs~Ea, 1965. V. M. ADAB~AN CCCP, 270000, 0~ecca, 0~eccEE~ roc.yHHBepcHTeT; D. Z.AROV

( ~. 8.APOB)

CCCP, 270020, 0~ecca, 0 ~ e c c E ~ ne~.EHCTETyT;

M. G. KREIN

( M.r.KP~

270057, 0~ecca, y~.ApTeMa 14, EB.6

COMmeNTARY BY THE AUTHORS Soon after the Collection "99 unsolved problems in linear and complex analysis" LO~&I, vol.81 (1978) was published S.Axler, I.D.Berg, N.Jewell and A.Schields wrote an important paper on the theory of approximation of continuous operators in Banach space by compact operators, where they obtained, in particular, the answers to questions a) and b) of the Problem.The answers to both questions turned out to be negative. So for every function ~ L~ the set M~ is not void and moreover for any ~ co ~ L \(H +C) the set M~ is infinite. These results were obtained in Eli as consequences of two remarquable propositions, which we formulate for Hilbert space operators only. THEOREM. Let

[~)

be a seque.nqe of linear compact operators

on Hilbert space converging in the strong tooolo~y to a bounded operator

T : S-~

~=T

exisSs a sequence

also.that

" Supp°sIe,

J~ ~

T = s - ~

of non-negative numbe.rs such that

and

I T - K I = S=(T)

where

• Then there

K

=

T %

.

~=~

258

COROLLARY. Let T

{T~}

and

theorem. Suppose also that ~ sequenc.,e,s, [6~}. =

=

{ 8~}

satisf E the conditions of the

is not compact. Then there exist two

oT,,,.,.non-ne~ative~numbers such that, ~- O,n=

4

I T-Ko, I=IT-K I= s LT) where ~ @ = ~ J ~

,

~=[6~T~

,and

•

The indicated propositions permit also to give answers to questions of type a ) a n d b ) f o r matrix-functions AS we got to know from [~ question a) was raised before us by D.Sarason.

~: ~x~(C~×~+H~l

Question c) concerning the description of the set ~ its "selected" part remains open.

or

REFERENCE 7. A x i e r S., B e r g I.D., J e w e I I N., S h i e I d s A. Approximation by compact operators and the space H°°+ C . A n n . ~ t h . , 1979, 109, 601-612.

EDITORS' NOTE Question a) is solved also in a different way by D.Lueckim~ [8]. Let us mention also a recent r e s e t of O.S~udberg [9] asserting that the algebra H ~ + BUC (BUC is the space of bounded ~niformly continuous functions on ~ ) does not have the best approximation property, i.e, there e x i s t s ~ such that there is no ~ in HC~+ BUC saris-

REFERENCES 8. L u e c k i n g D. The compact Hankel operator form an M -ideal in the space of Hankel operators° - Proc.Amer.Math.Soc., 1980, 79, 222-224. 9. S u n d b e r g C. No°+ BUC does not have the best approximation property. Preprint, Inst.Nittag-Leffler, 13, 1 9 8 3

259

5.2.

QUASINILPOTENT HANKEL OPERATORS

Hankel operators possess little algebraic structure. This fact handicaps attempts to elucidate their spectral theory. The following sample problem is of untested depth and has some interesting function theoretic end operator theoretic connections. PROBLEM. Does there exist a non-zero quasinilpotent Hanke 1 operator? A Hankel operator A on H ~ is one whose representing matrix )~ is of the form (~{+I ~'i'-0 with respect to the standard orthonormal basis. A well known theorem of Nehari shows that we may represent A as ~ = $ ~ = ~ M q l H ~ where P is the orthogonal projection of ~£ onto H Z ; ~ is the unitary operator defined by (~)(Z)=~(~), for ~ in ~£ , and ~ denotes multiplication by a function ~ in The symbol function ~ and the defining sequence ~ are conA nected by ~(~) = ~ , ~= 0 , ~ , ~ , . The following observation appears to be new and provides a little evidence against existence. PROPOSITION. There does not exist a non 'zero nilpotent Hankel operator, PROOF. Suppose A~0 and is nilpo±ent. Then ker A is a non zero invariant subspace for the unilateral shift U, since AU--U*~. By Beurling's theorem this subspace is of the form ~ for some non constant inner function ~ . Thus, with the representation above, we have =0 and hence ~ ( ~ ) = 0 for ~ . So the symbol function ~ may be written in the factored form ~ = ~ for some ~ in H~ , and we may assume (by cancellation) that and ~ possess no common inner divisors. The operator ~ is a partial isometry with support space ~= ~ e ~ ~ and final space ~ = ~ ~ ~ al , where ~ ) = ~(~) . By the hypothesised nilpotence of ~ -~ = $ ~ it follows that for some non zero func. .~ 0 ~4 belo ng s to W~ . Hence ~ divides . . tion . ~ . in ~ , and ~=W~4 with ~ in H . Since ~ belongs to ~T we have P(~) =0 . This says that the Toeplitz operator T £ , ~ has non t r i v i a l

kernel.

,

But

k2/£ T *

-- k ~ T

= k~T

=

and we have a contradiction of Coburn's alternative:

Either the kernel or the co-kernel of a non zero Toeplitz operator is trivial. @

260

Function Theory. The evidence for existence is perhaps stronger. There are many compact non self-adjoint Hankel operators, so perhaps a non zero one can be found which has no non zero eigenvalues. A little manipulation reveals that ~ is an eigenvalue for ~ if and only if there is a non zero function ~ in ~ (the eigenvectot) and a function ~ in ~ £ such that +

Since continuous functions induce compact Hankel operators it would be sufficient then to find a continuous function ~ which fails to be representable in this way for every ~ 0 . Whilst the singular numbers of a Hankel operator A (the eigenvalues of ( ~ * A ) ~l~ ) have been successfully characterized (see for example [3 , Chapter 5] ), less seems to be known about eigenvalues. O~erator theory. It is natural to examine (I) when the symbol can be factored as q = ~ (cf. the proof above) with ~ an interpolatingBlaschke product. The corresponding Hankel operators and function theory are tractable in certain senses (see[l] ,[2~Part 2] and[3,Chapter 4] ), partly because the functions (~-~A~l~)41~(~-~)-~, where ~ , A i , are the zeros of Wv , form a Riesz basis for ~ O ~ . It turns out that ~ is compact if ~(~4), ~CA~), . is a null sequence. A quasinilpotent compact Hankel operator of this kind will exist if and only if the following problem for operators on ~ can be solved. I

PROBLEM. Construct an interpolating sequence dia~o~l o~erator ~ proper solutions operator on ~ ting matrix

~

so that the ' equation ~ X x i_~n ~

whe n

~0

~ ~

. Here

associated with ~ A ~

A~ and a compact admits no

is the bounded(!)

determined by represen-

(t. i A,i,l~.)'~1~'(.,I- I~ ~)'~/~'

REFERENCES I. C 1 a r k D.N. On interpolating sequences and the theory of Hankel and Toeplitz matrices. - J.Functicnal Anal. 1970,5,247-258.

261

2. H r u ~ 6 ~ v B.S.

S.V.,

N i ~ o l's k i i

N.K.,

P a v I o v

Unconditional bases of exponentials and of reproducing ker-

nels. - Lect.Notes Math. 3. P o w e r

S.C.

Springer Verlag.

Hankel operators on Hilbert space. - Research

Notes in ~athematics. S.C.POWER

1981,N864,

1982. N 64, Pitman, London. Dept. of Mathematics Michigan State University E.Lansing, MI 48824 USA Usual Address: Dept.of Mathematics University of Lancaster Bailrigg, Lancaster LAi 4YW England

262 HANKEL OPERATORS ON BERGNAN SPACES

5.3. Let

IA

~/~

denote the usual area measure on the open unit disk

17 • ~he B e r ~

~,~ce t . t (1?1

c o n s i s t i n g o f those f u n c t i o n s i n

D

, .Let

i s the s u b s ~ o e ~ ~D. ~ A)

of t}(l?,~A)

which are analytic on

P

denote the orthogonal projection of ~ ( D , ~ A ) onto ~ & ( D ) . For ~ S ~ cD, ~A) , we define the Toeplitz operator

and the Hankel operator

H# :L~ (D) -~ L~(]I), & A)e L~(]D) by

T~I~-P(#I~)

and M#l~=(I-P)(#I~),

For which functions

#E L® (D, &A)

is the HaZel oper.tor

compact? If we were dealing with Hankel operators on the circle T rather than the disk ~ , the answer would be that the symbol must be in the space • on the dlsk " 17 , it is " H ~ tC(T) ea sy to s ee that i f ~ E ~ °O + C ( ~ ) , then H~ i s compact. However, it is not hard

to const=ct

an

open set 5 ~

D

with

S nT ~ ~

such that

if

is the characteristic function of S , then H, is compact. Thus the subset of ~ ( D ~ ~ A) which gives compact ~_~nkelt operators is much ~o bigger ( i n a n o n t r i v i a l way) than H +C (~) , and it is possible that there is no nice answer to the question as asked above. A mere natural question arises by considering only symbols which are complex conjugates of analytic functions: For which

~6

i_ss H ~

compact?

I% is believable that this question has a nice answer. A good candidate is that T must be in The importance of this question stems from the identity

H®*C(D).

~lid

H®

for an ~ ~ . ~hus we are asking which Toeplitz operators on the disk with analytic symbol have compact self-commutator. Readers familiar with a paper of Coifman, Rochberg, and Weiss

263

[I] might think that paper answers the question above. Theorem VIII of [I] seems to determine precisely which conjugate analytic functions give rise to compact Hankel operators. However, the Hankel operators used in ~I] are (unitarily e~uivalent to) multiplication followi s f a r bigger than E , the Hankel operators of [1] are not the s ~ e as the Hankel operators defined here, The Hankel operators as defined i n [ I ] are more natural when dealing wlth singular i n t e g r a l theory, but the close o o ~ e o t i o n l r i t h Toeplitz operators is lost, To determine which analytic Toeplitz operators are essentially normal, the Hankel operators as defined here are the natural objects to study. RE~ERENOE I. C o i f m a n R.R., R o c h b e r g R., W e i s s G. Pactorization theorems for Hardy spaces in several variables. - Annals of Mathematics SHELDON AXLER

1976, 103, 611-635. Michigan State University East Lansing, NI 48824, USA

264 5.4. old

A SIMILARITY PROBLEM FOR TOEPLITZ OPERATORS Co, sider the Toeplitz operator T~

acting on

H%~ H~(~)

,

where ~ is a rational function, with ~(~) contained in a simple closed curve ~ . Let ~ be the ccnformal map from ~ to the interior of ~ , and say that ~ b a c k s u p at 6 ~0 if arg ~-1 ~(¢t~) is decreasing in some closed interval [@I, ~ , where 0,< 01, a n d 0i ~0

, and t o

M4 ~ - - . i_Xf "~---0 , where,

M k

• M~

is t h e o , R e r a t o r o f

(2)

multip!ica,tion b~ ~(6¢$)

e

One c a s e o f t h e a b o v e c o n j e c t u r e g o e s b a c k t o Duren it was proved for ~ ( ~ ) ~ + ~ , I¢I>I;I . In this case never backs up, so that MI,..., M ~ are not present tually, Duren did not obtain s~milarity, but proved that fies

LTr----where ~ =~4~(~)

~, where ~ and in (I). Ac~V satis-

D,

is some conjugate-linear operator( ~(~,X+ A¢~) ----+ ~(~)) and ~ is the mapping function for the

inte-

rior Of C~03~(T) . I n [2~, the conjecture was proved in case is $ - to-one in some annulus ~ i~l ~ ~ . Here again ~ never backs up. In [3~, ~ was assumed to have the form

where ~ and ~ are finite zero. In this case ~(~)-----~

Blaschke products, ~ h a v i n g o n l y one and ~ c a n be taken to be 1.

265

The main tool used in [3] was the Sz.-Nagy-Fola~ characteristic function of ~p , which we computed explicitly and which, as we showed, has a left inverse. A theorem of Sz.-Hagy-Foia~, [4], Theorem 1.4, was then used to infer similarity of TF with an isometry. ~oreover, the ,,-!tary part in the Wold Decomposition of the isometry could be seen to have multiplicity I, and so the proofs of the representations (I) and (2) were reduced to spectral theory. If ~ is of the form (3) where ~ and ~ are finite Blaschke products, ~ having m o r e t h a n o n e zero, the computation of the characteristic function of T~ is no longer easy. However, left invertibility can sometimes be proved without explicit computation. This is the case if ~ and ~ have the same number of zeros; i.e., when T~ is similar to a unitary operator. This and some other results related to the conjecture are given in [5]. The Sz.-Xagy-Fola~ theory may also be helpful in attempts to formulate and prove a version of the conjecture when (3) holds with ~ and arbitrary inner functions. For example, it follows from ~3~ that if is inner and ~ is a Blaschke factor, then ~F is similar to an isumetry. For the case in which ~ is net the unit circle, the only successful %ech-lques so far are those of [2], which do not use model theory. We have not been successful in extending them beyond the case of ~ satisfying the "annulus hypothesis" described above. There is a model theory which applies to domains other than ~ [6], but to our knowledge no results on similarity are a part of this theory. ~ore seriously, to apply the theory, one would NEED TO KNOW that the spectrum of r~W is a spectral set for Tp ; a result which does not seem to be known for rational ~ at this time. Pinally, I% seems hardly necessary to give reasons why the conJecture would be a desirable one to prove. Certainly detailed information on invariant subspaces, commutant, cyclic vectors and functional calculus would follow from this type of result. REFERENCES 1. D u r e n P.L. Extension of a result of Beurling on invariant subspaces. - Trans.Amer.Nath.Soc. 1961, 99, 320-324. 2. C 1 a r k D.N., M o r r e 1 J.H. On Toeplitz operators and similarity. - Amer.J.Math., 1978, 100, N 5, 973-986. 3. C 1 a r k D.N., Sz.-Nagy-Foia~ theory and similarity for a class of Toeplitz operators. - Banach Center Publiaations,v 8,

266

Spectral Theory, 1982, 221-229 4. 5. 6.

S z. - I~ a g y B., F o i a 9 C. On the structure of intertwining operators. - Acta Sci.Math. 1973, 35, 225-254. C 1 a r k D.N. Similarity properties of rational Toeplitz operators. In preparation. S a r a s o n D. On spectral sets having connected complement. - A c t a Sci.Math. 1965, 26, 289-299. DOUGLAS N. CLARK

The University of Georgia, Athens, Georgia 30601 USA

COmmENTARY BY THE AUTHOR Since my first note on the similarity problem, the following results have been obtained.

T~O~ a

1 ([8]),

I__~F

simple closed curve

FCT )

; i_~f V > 0

P

i s a r a t i o n a l function. ~ p ~ T

, which is anal,ytic in a neighborhood o,f

, where

V

is the windin~ number of

about the points interior to F the set (o~ T

) where

T (V)•

, ....... where

V

F

; and if

D

_,~ T (y)

F(~)

•

V

~$

THEOREM 2. ([9], [10] ). If F~T)

F(T)

, where

~-

is similar to copies of the ana-

is a normal operator

is absolutely continuous and the

where F

in th e spectrum of V F

is ,equal

backs up and F(e~)=~ .

is a rational function~ if

divides the plane `into disjoint re6ions~ from which the ones

in which the index of

T F -~I

labeled

); if the closures of any two of these

(~ , ~)

i_~s

mapping function

and where V

spectral multiplicity of a 90int ~ to the number of points

TF

is the sum of ~

to the interior of F

whose spectrum is

F(>-)@F

backs up; then

lytic Toeplitz operator 'as s0ciated with the from

~:o

~

(rasp. ~

is negative (resp,positive) are

inters,act St onl~ finitely many points (called the mul-

tiple points of

F

); if the boundary of each

~

,~

is an ana-

267

l~tic curve except at th e multiple points= where it is piecewise smooth with inner angle under

F

of a point

~0

; if no multiple point is the image I

~o~:T

where

never backs up at a multiple point; then

~(~o) = ~

0

; and if

is similar to

h

where ~

~

(resp. T

) is the maopin~ function of ~ for

restriction t° T

and

~

(resp. ~

),

is as described in Theorem I. THEOREM 3. (Wang, ~I~ ). I_~f F ~ C 4 ~ T )

some

(resp.

), each summand is included with multi p!icit E equal to the

absolute value of the index of T F - ~ I and V

o_~n ~

~ 0 ) , i = ~ , ,~. Consequently an affirmative answer to Problem 2 would mean that "Fredholm character" of --T~ is the same in all spaces H~ and its fredholmness implies the invertibility. We do not know whether these weaker statements are true if ~ ~ . 2. The class S A P (of semi-almost-periodic functions) is a natural extension of AP . This class has been introduced by D.Sarason [ I] and may be defined, for example, as {~=(0,5+~)~+

The a°p. components

I' ~

(of

~

) are uniquely determined by

~0 A criterion for TG ( G ~ 5 ~ P , ~=~ ) to be semi-Fredholm in the space ~ was obtained in ~] amd was generalized in ~2] to the case of an arbitrary ~ , ~ ~ (~,oo) . The case ~>~ is considered in [3,4], where the fredholmness and semi-~redholmness criter i a h a ~ b e e n established. These results, however, were obtained under the a priori assumption of existence of P u -factorization of a.p.+ components F , H of G . The latter means that the factors ~; , (~*)±( from the P-factorization F = ~+ A Ff belong to the class AP + of those matrices from ~P whose ~ourier exponents are all non-negative; the same holds for ~ . The following problems arise in connection with the question of removing these a priori assumptions. PROBLEM 3. L e~ and

~

are

hess of ~G

G

be an

(~x~)-matrix from

~

~>~,

F

its aop% components. Is it true that the semi-fredholmimplies the semi-fredholmness of

~F

PROBLEM 4. I s t h e set of matricesadmitting dense in the set of all matrices admittin~

a

and 7 H ~

?

Po-factorization

P-factoriz~tion? What

would the situation be like if we restricted ourselves to matrices wit h a fixed (non-zero~set

of partial a.p~ indices?

The positive answer to Problems 3, 4 would allow to extend the criterion for ~r. to be (semi-)Fredholm [3, 4] to the case of arbitrary matrices~ G ~ 5A~ .

281

3. Let us consider a triangular matrix ~ £ AP of the second order. Under some additional assumptions(e.g, absolute convergence of Pourier series of its elements ) the P-factorization property of ~ is reduced to the corresponding question about "

iv%

e

0

] 0cb=

(2) %({)

e

and the spectrum ~ of %0~ ~ is contained in where V >0 define a.p. polynomials C-~,~) . Assuming that card C~) < co %i (i= ~ ' ~ ' ) by the recurrence formula

(3) It is supposed that the sequence ~ of leading exponents of ~i strictly decreases and ~ ~ AP + ~. Analogously to the case of the usual factorization of continuous triangular matrices E5] there exists an algorithm (say, A ) for P-factorization of Bo which is connected with the continuous fraction expansion of ~i/~0 or equivalently with the relations (3). Algorithm ~ unlike in the continuous case does not necessarily lead to the aim. A sufficient condition to make application possible i ~ K ~ K 4 ~ 0 for some K EZ+ . In this case the factors B± are a.p. polynomials. PROBLEM 5. Give conditions for the convergence rithm

A

to obtain a

~ -

(or

P- ) factorization

of the algoof matrix

(2~. These conditions have,to be formulated in terms of entries of

Algorithm A can be applied to obtain a ~ -factorization of matrix (2) if, for example, ~ C ( - $ , 0] or the distances between the points of ~ are multiples of a fixed quantity (in particular, if card CI~) ~ ~ ). Already in the case card ~ = 3 , i.e.

there exist situations when algorithm ~ tions is: #=0 , ~= [ - & and ~ - & / [ )

fails. One of such situa-

is i r r a t i o n a l ,

282

In this case we found another algorithm based on the successive ±~ ~+ application of the transformation ~'-~ ~ B ~ C ~ (A~, CB g A ) preserving the structure of B~ and on the factorization of elements close to the unit matrix. With the help ~ f this algorithm it was established that under the restriction IC~I # I~÷~I a ~-factorization exists with 9~ = 9~ = 0 ., but B± are no more a.p. polynomials. In the case IC~I =I $4+~I the P-factorization of (2) does not exist. Thus, even in the case card ~= ~ the following problem is non-trivial. PROBLEM 6. Describe the cases when matrix (2) admits Pc-) factorization~

%P-(0 r

calculate its a.p. partial indices and construct ,,

if possible~ corresponding f actorizations. The interest to matrices of the form (2) is motivated by the fact that they naturally arise in connection with convolution equations on a finite interval with kernels ~ for which ~ & ( ~ ) ( ~ ) has an a.p. asymptotics at infinity for an & ~ C [3]. REFERENCES I. s a r a s o n D. Toeplitz operators with semi-almost periodic symbols. - Duke Math.J., 1977, 44, N 2, 357-364. 2. C a r i~ H a m B ~ ~ ~ A.M. Cm~ryaapHae m~Terpaa~H~e ypaBHeH~ C EOS~M!U~eHT82a~, m g e ~ m ~ paspMB~ noay-no~TH-nepHo~eoEoro T~na. Tp.T6mmc.Ma~eM. EH--Ta, 1980, 64, 84--95. 3. E a p a o B ~ D.M., C n~ T E o ~ c E ~ H.M. 0H~Te-pOBOOTE HeEoTop~x C~Hrym~H~X ~HTe~sa~H~X onepaTopoB c ~ m T p m n ~ m EOB~HI~eHTaM~ ~ a c c a ~ A ~ CB~S~x c ~ CECTeM ypaBHeHNl~ cBepTEH Ha EOHe~HOM npomeayTEe. - ~oEa.AH CCCP, 1983, 269, ~ 3. 4. K a p a o B ~ D.H. , C n~ T ZO B C E ~ ~ H.M. 0 H~Te-pOBOCT~, ~- E ~--HOpMa~BHOCTI~ CEHIVJL~pHRX E H T e I ~ B H ~ X onepaTopOB C MaTp~m~M~ EOS~EIU~eHTam~, ~ O n y C E S m ~ m paspNBN noay-no~T~nepEo~H~ecEoro THna. - ~Eoxa no Teop~H onepaTopoB B ~ O H a X B R N X npocTpaHCTBaX (TesHc~ ~OF~S~OB), MHHCE, 1982, 81--82. 5. q e 6 o T a p e B F.H. qacTHae HH~eEc~ EpaeBo~ salaam P~Ma~a c Tpey~oJ~HO~ MaTpm/e~ BTOpO~O nop~Ea. - Ycnex~ MaTeM.HayE, I956, II, h 3, 199-202. Yu. I. KARLOVICH CCCP, 270044, 0~ecca, Hpo~eTapcK~ 6y~BBap 29, MopcKo~ rH~pOSHSMYecKM~ MHCTMTyT I.M. SPITKOVSKII 0T~e~eH~e SKOHOMMKM M (M.M. CHMTKOBCK~) ~K0~OrHM ~p0BOr0 oKe~a.

283

TOEPLITZ OPERATORS IN SEVERAL VARIABLES

5.10.

?or ~ the complex numbers, let ~ be a bounded domain in C ~ with closure ~- and with 9 ~ the Shilov boundary of the uniformly closed algebra A(~-) generated by all polynomials in the complex variables ~=(E~,Z~., ~Z~) on ~- . In general, ~ is a closed subset of the topological boundary of ~ . When ~ is one of the classical domains of Cartan or in other cases of interest, ~ is a compact manifold with a "natural" volume element ~ and the space ~ ( ~ ) of ~-square integrable complex valued functions is the setting for our analysis. The closure of ~(~-)in ~ ( $ ~ ) is denoted by H~(~) and this (Hardy) space, together withthe (unique) orthogonal projection operator ~ from ~(9~) onto H ~ ( ~ ) , is a basic object in complex analysis on ~ . For q essentially bounded on 9~(~) , the ~e~

I ib;TZ@ =~(j~)e.rThet ~*r-a2g~ebraiged~t ed fyral~lT!

~th

continuous I is denoted by ~ (~). l " Even for ~ - - ~ x ~ x xD ( M~ times) where ~ is the open unit disc in C , many interesting questions about ~ ( ~ ) remain open after more than a decade of study. Note that for ( MS times), S ~ = T ~, the M~-torus. The structure of ~(T~)is well-understood for P~= I ,2 ~1,2,3]. necessary and sufficient conditions for ~ in to be ?redholm of index ~ are known [2~. It follows from the analysis of [2] that every ?redholm operator of index % i n ~ ( V ~) can be joined by an arc of such operators to .

(T~n)particular,

cI

Here, ~

is an integer and

~i

.

- ~i

PROBLEM I. Classify the arc-components of Fredholm operators in

This question reduces to: PROBLEM 2. Classif,y the arc-components of invertible elements in

284 REFERENCE S I. C o b u r n

L.A.

The

C*-algebra

II. - Bull.Amer.Nath.Soc.,1967, 1969,

generated

73, 722-726;

by an isometry

I,

Trans.Amer.~ath.Soc.~

137, 2 1 1 - 2 1 7 .

2. C o b u r n

L.A.,

D o u g I a s

A n index theorem for Wiener-Hopf ter-plane.

R.G.,

S i n g e r

operators

!.~.

on the discrete

3. D e u g 1 a s litz operators

R.G., H o w e

R.

on the quarter-plane.

On the

C*-algebra

of Toep-

- Trans.Amer.~Aath.Soc.~1971,

i58, 203-217. L.A.COBURN

quar-

- J.Diff.Geom.91972 , 6, 587-593.

State University Department Buffalo,

N.Y. USA

of N e w York

of Mathematics 14214

285 5.11.

SOME PROBLEMS CONNECTED WITH THE SZEG6 LIMIT THEOREMS

1. Any sequence of ( % x % ) - m a t r i c e s {Ci.~Z~+~ { sequence l,,ofm a t r i x - v a l u e d T o e p l i t z m a t r i c e s

determines a , ° .

If a~ ~I ~ T ~ ~ 0 for sufficiently large values of then the question about the limiting behavior o f A ~+4 /A 6 arises. The analogous question arises about A ~ /~~+ ~ provided the nonzero limit ~ ~..

~

&~÷, /A~

exists.

It was G. Szego who studied both questions for the first time. He dealt with the case %=~ and supposed that { Ci }ig z is a sequence of Fourier coefficients of a positive summable function. See [1] for the precise formulations, for the history of the problem and for its natural generalization A

Ci = H(j),

(1)

M being a f i n i t e non-negative Borel measure on 7 . By the Riesz-Herglotz theorem the class of sequences s a t i s f y i n g (1) i s the class of p o s i t i v e d e f i n i t e sequences. We consider here the case when % ~ and { Ci} is an &-sectorial sequence for some & ~ [ 0,g/~) . The latter means that every T~ is & -sectorial i.e. its numerical range (Hausdorff set) lies in the angle { E : l l ~ l ~ t ~ & ~8~} . It is clear that { C i} is an &-sectorial sequence iff there exists a measure ~ satisfying (1) and taking values in the set of & - sectorial Q~%)-matrices on all arcs of 7 • The real part ~ of this measure ~ permits us to construct the Hilbert space ~ =~C~) consisting of ~-tu~les of functions and equipped with the sesquilinear form A ( ~ , ~ ) = ~ ( ~ ) % M ( ~ ) ~* c~ ) Employing the factorization theorems from ~,3] we have proved in [4] the existence of the limit ~ in the case % ~ , &~0 and have obtained the following formula:

T =~M/~ (see [43 for details and the information about where earlier results by A.Devinatz and B.Gyires). Formula (2) is valid in

286 the case ~ G ~ C too; this is the only case when ~ = 0 . We propose the following as an UNSOLVED PROBLEM: find an extensic n of the Sze~B

second limit theorem to the case of

& -sectori-

al sequences. We CONJECTURE that the limit (finite or not) exists for every the regularity condition

~ ~~°~ ~/~+4 &-sectorial sequence satisfying

We are somewhat encouraged in this conjecture by Theorem 2 of Devimatz [5] related to the case provided ~=~ , M is an absolutely continuous measure and ~ satisfies some additional restrictions (including the requirement G ~ L ~ ). We have proved in the case & =0 , ~ > ~ the existence of and h&ve obtained a formula for its calculation using some geometrical considerations from [4]. Before formulating the corresponding result let us remind that under condition (3) there exist two canonical factorizations of the matrix G : the left one and the right one G = G * G~ ( G~ and G~ are outer matrix functions of the class ~ ~ ). Let us denote the Toeplitz operator with the (unitary valued) symbol ~ = G * G~ -4 by T F .

G=G G

THEOREM I. Let -matrices and let ~ G =~/~

Ci}

be a positive definite sequence of (~×%)-

be a measure connected with it b[ formula (I~,

. Then under con d! tic n %3,),there exists a limit

i (4 oo)

if__ f M

{

of the sequence

~ / ~

is absolutel continuous and

+~ ^

. This limit is finit.e

K F 4 . it is well understood now that the existence of ~ (and formulae for its evaluation) may be proved under some additional restrictions with a help of results obtained in another direction (the rejection of the positive-definiteness with simultaneous amplification of restrictions on the smoothness of G ) that we do not touch upon here, see [6] and references in it. 2. Considering an & -secto~ial matrix measure M concentrated on the line, it is possible to introduce a continuous analogue of the

287

space ~

and to establish the following result.

THEORE~ 2. The two statements given below are equivalent:

,

2) the subspace

~

of constant

tien with the subspace least for one)

~$=V

S>0

~-tuples has

where

~$

zero interHec-

{e~:t~$1

for all (at

.

If these conditions are fulfilled and distance from

where ~ = ~ / ~

~ ~

t_~o ~S

i.~n ~

is a "dis tansematrix"which

& = 0

then the square of th e

metric equals to ~ $ ~ is calculated by the formula

0 Here V ~ is the inverse Fourier transform of the function from the left canonical factorization of a.e. on ~ , -~+ is outer and belongs to the Hardy the upper half-plane). For the case %=~ Theorem 2 was already proved discrete analogue of (4) was established in ~4] in the & ~ [ 0 ~ ~/~) , ~ . We propose a natural

matrixG ( class ~

G=G+G in

in [7]; the general case

PROBLEM: ~eneralize the second part (concerning formula (4)) of Theore m 2 to the case" of distances ~n th e skew In this case obscure points to interprete the right-hand side fact is that the inverse Fourier from canonical factorizations of [2,3] in general are not elements

A

-metric C&>o).

already appear after first attempts of the formula of type (4). The transform of the factors ~ ± & -sectorial matrix-functions of ~ .

The problem to find continuous generalizations for the Szeg~ second limit theorem admits different formulations and even in the definite case corresponding investigations form an "unordered set" (see [8] and the papers cited there). There are still more unsolved questions in the case & ~0 but we shall not go into this matter here.

288

REFERENCES

I.

r o a E H c E ~ ~ B.JI., M O p a r ~ M 0 B ~I.A. 0 npe~ea~Ho~ TeopeMe r.cerS. - MsB.AH CCCP, cepml MaTeM., 1971, 35, BNII.2, 408-427. . Kp e ~ H M.F. , c n ~ T E o B C E ~ ~ H.M. 0 ~aaTOp~Ssam~ MaTpm/-~y~zLU~ Ha e~ZHEqHO~ oEp3~KHOCTI~. --~OIO~I.AH CCCP, 1977, 234, ~ 2, 287-290. . K p e ~ H M.r., c n E T E o B C E E ~ 14.M. 0 ~ a K T o p H s a m ~

.

% --CeETopI~a~IBHNX MaTp]alI-~yHEI/~I~ Ha e~141IIiqHO~ OEpjfaHOCTH. -~aTeM.I~CCJIe~OBalIK~, 1978, 47, 41-62. K p e ~ H M.r., c ii E T E O B C E I~ ~ I~.~. 0 HeEOTOpRX 0606~eHE~X HepBo~ n p e ~ e ~ H o ~ T e o p e ~ Cere. - An~l.N~th., 1983,

9, N 1. 5. D e v i n a t z

A.

The strong Szego limit theorem.

- Illinois

J.Math., 1967, II, 160-175. 6. B a s o r E., H e 1 t o n J.W. A new proof of the Szego limit theorem and new results for Toeplitz operators with discontinuous symbol. - J . 0 p e r . T h e o r y , 1980, 3, N 1, 23-39.

7. E p e ~ H M.F. 06 o ~ o ~ sEcTpanox~xmo~o2 npo6xeMe A.H.Ko~MoropoBa. - ~oF~.AH CCCP, I945, 46, ~ 8, 306-309. 8. M E E a e x ~ H ~.B. MaTpH~H~e EOHT~ETyS~w_RPS~e aHa~oI~ TeopeM r.Cer~ o T ' @ ~ e B ~ X ~eTepM~HaHTaX. -- M3B.AH ApMCCP, I982, I7, 4, 239-263.

M. G. KREIN

(M.F.EP~H) I.M. SPITKOVSKII

(H.M. CnETKOBC~ )

CCCP, 270057, 0~ecca, yJi.ApT~Ma 14, EB.6 CCCP, 270044, 0~ecca, H p o x e T a p c K ~ ~yzbsap 29, MopcKo~ rH~oo~s~qecxM~ HHCTXTyT OT~e~eHMe ~ O H O ~ X H X sKo~orxM M~t-posoro oxeaMa

289 5.12.

THE DIOPHANTINE MOMENT PROBLEM, ORTHOGONAL POLYN0~IAT.S AND

SOME MODELS OF STATISTICAL PHrfSICS I. In [I ], [2] it was shown that in investigations of the Ising model in the presence of a magnetic field the following one-parametric Diophantine trigonometrical moment problem (DTNP) appears, PROBLEM. Describe all non-negative m~asures '

rcle T = { ~ ~arameter ~,

~-e~@~ ~[-~,~[]}

~

even in ~

(@, ~)

on the ci-

, de~en~in~ on a

0 ~ ~ ~ ~ , and such that

0

and the moments

1 0

are pol.vnomials (in ~

) of de~ree

the parity of

coincides with the parity of

Mk(~)

is an inte~e r >I ~

(~

~e~

with integer coefficients; ~k

. Here ~e

is the number of ,the nearest nei~hbours in

the lattice). It is known that the description of such measures can be reduced to the description of the corresponding generating functions

2.

Hk( )-Tk _

re polynomials,

4-~ ~

, where T k are Tchebysh~v polynomials,

290

V~a note that in examples I-3 the generating function is ratiomal, whereas in example 4 it is algebraic (this case corresponds to the orm-dimensional Ising model). QUESTION. Has the ~eneratin~function

correspondin~ to a DTMP,

to be algebraic? Fixing a rational value of the parameter ~ , ~ = ~ , p, ~ integers, 0 ~ p ~ ~ , * ~ ~, we see that our DTMP implies the following "quasi-DT~P":

-- --~

~

k e ~O(~)=

~

, CK

being integers,

0

In particular for PROBLEM:

~ = 0 or

~(~-|)

we obtain the following moment

Describe non-negative even measures whose trigonometrical moments are inte~er~. This problem is solved by the known Helson theorem .-4

d,o(e)= Z $=0

(

2~sj

[31:

.-4

o.,,~', o--'~w + ~ g ~ s sOlO $='0

under some additional conditions on @s,~s [2]. II. It was shown in [4] that the theory of Toeplitz forms and orthogonal polynomials is closely connected with some problems of s~atistical physics and in particular with the Gauss model on the semiaxis. In this connection some mathematical problems appear whose solution would be useful for the further investigation of such models. I, Let ~C~) be an even non-negative summable function on satisfying the Szeg~ condition

1

Cel

0

We define the function

0

-

T

291

--

~K z k ,

N

Izl
0, this expression tends to ~/~ The numbers (T(N)) ~ok -~, k =0,~,. ,N, are proportional to the coefficients of the orthogonal polynomials ~N (Z) (see [41, [5]). This leads to a P R O B L ~ of a more detailed investigation of the asymptotics of ~N (e~@) as N - ~ ~ in the presence of non-zero singular part of the measure ~ . As we know only the case of an absolutely continuous measure was considered in detail (see, e.g. [6], [7I)° 3. The multidimensional Gaussian model. Calculate the free energy and correlation functions under less restrictive conditions than in [41, [81~.

292

REPERENCES I. B a r n s 1 e y Diophantine

M.,

moment

B e s s i s

D.,

M o u s s a

p r o b l e m and the a n a l y t i c

t i v i t y of the f e r r o m a g n e t i c

I s i n g model.

structure

P~ The i n the ac-

- J,Math.Phys.,

1979, N 4,

20, 535-5522. B a a ~ a M a p O B

B.C.,

B o a o B ~ ~

H.B. Mo~eJ~ Hsasra c

M a ~ T I m M noaeM ~ ~O#aHToBa nloo6aeMa MOMelITOB. - TeOlO.Ma~eM. ~Hs., 1982, 53, ~ I, 3-15. 3, H e 1 s o n

H. N o t e

on h a r m o n i c

ftulctions~

- Proo.Amer.Math.Soc.,

1953, 4, N 5, 6 8 6 - 6 9 1

4. B m a ~ ~ M ~ p o B B.C., B o a o B ~ ~ H.B. 06 O~HO~ u o ~ e ~ O ~ a T ~ C ~ e c ~ o ~ ~ s ~ F J . - T e o p . MsTeM. ~Hs., 1983, 54, ~ I, 822. 5~ B a a ~ ~ M ~ p o B B.C., B o a o B ~ ~ H.B. YlmB~eH~e BzNe- Xo~a, sa~a~a l ~ a ~ a - ~a~6epwa ~ o p ~ o ~ o H a a ~ e ~ o ~ o ~ e ~ . - ~ o E a . A H CCCP, I982, 266, ~ 4, 788-792. 6. S z e ~ ~ ed.,

G. O r t h o g o n a l

polynomials.

AMS C o l l , P u b l . ,

23, 2

1959,

1~ o a ~ H C ~ ~ ~ B.JI. Acm~n~o~ec~oe i i p e ~ c ~ a B a e H ~ e ol~oz'o]~aa.l~HSX MHOI'O~e~OB. -- Ycnex~ Ma~eM.~sy~, 1980, 35, ~ 2, 148-198. 8. Jl ~ H - ~ ~ H.D. l~oPo~el:~m~ a~maoP ~eol~m~ Ce#~. - HsB.AH CCCP, cep.~m~eu., 1975, 39, ~ 6, 1393-1403. 7.

V,S.VLADIMIROV

(B. C. B~A~I~4POB) I.V.VOLOVICH (~.B.BoaOB~)

CCCP, 117966, M o o m m ya. BaB~aOBa, 42 Ma~esaT~eoF~ ~HC~T AH CCCP

293 5.13.

THE BANACH ALGEBRA APPROACH TO THE REDUCTION METHOD FOR TOEPLITZ OPERATORS

Let H ~ denote the Hardy subspace of LZ = L£ (T) , consisting of the functions { with ~ ( ~ ) = 0 , ~ < 0 , and l e t P be the orthogonal projection from [~ onto H~ . For ~ @_L® = L " (T) the Toeplitz operator with symbol & is defined on H i by T(~)~ = P(@@) • Let ~ { H ~) be the Banach algebra of linear and bounded operators on H ~ . Given a closed subalgebra. B of ~ denote by ~ T ( B ) the smallest closed subalgebra of ~ ( H '~) containing all operators T(¢) with Cb6 ~ . Furthermore, let Q(~) denote the so-called q u a s i c o m m u % a t o r i d e a 1 of ~ T ( B ) , i.e. the smallest closed twosided ideal in ~ T ( B ) containing all operators of the f o = 6~). It is a rather surprising fact that this ideal plays an important role not only in the ~red/aolm theory of Toeplitz operators, but also in the theory of the reduction method for operators A ~ T ( ~ ) .%(with respect to the projections PIT defined by p ~ ( ~ ) ~ k I(~)~k) k=O k=O

T(~)-T(~)T~)

(~,~

For /~e~IH ~) write A~rI{P~I if the reduction method is ~,pplicable to

(see [3] for a precise definition). Finally, put Q~=I-P~ and denote by ~ the group of invertible elements of a Banach algebr~ ~ with identity. For A 6 ~ ( H ~) , the following statements are easily seen to be equivalent:

(iii)

A

P AP,÷Q

(iv)

G (H

(~ 7/I~0),

Q,,A Q~, +P,, ~G'~(H ~)

t'l,~tT o

and

294

(.) A~GB{H~), V_~A'~V~EGB{H ~) (~,)

~I(V_~A V~) i O, there exists a

for which:

It is not hard(by using singular integral techniques) to reduce the existence a.e. result mentioned above to theorem I. SEVERAL IMPORTANT QUESTIONS remain open. L

Is the restriction

, ,, of theorem 1? timate

II~'I~ < T0

necessary to obtain the ,es-

• ! s method as well as other techniques Calderon

are unable to eliminate this restriction. IX. Since the operator 0(#)(~) exists almost everywhere for all functians in L~(P, I ~ I ) existence of a weight

cop (~)

, it is natural to conjecture the ( > 0 a.e.) for which

311 ( T h e existence of such a weight for a weak ~ estimate is guaranteed by general considerations related to the Nikishin-Stein theorem.) IIL The integral operator appearing in Theorem I is related to a general class of operators like Hilber~-transforms, of which the following are typical examples. a) The so called commutators of order -f,-~

b) ! Ac o+'b-

Ac.-lb

~>0

( -lb

(Here ~

l~

),,

It is easily seen that ~heorem 4 is equivalent to the following estimates on the operators A ~ : A

for some constant C . The boundedness in L~ of the operators in a) b) c) has been proved in E2], E3~ by using Fourier analysis and real variable techniques (which extend to ~ ). Unfortunately the estimate obtained (by these methods) on the growth of the constant in (,) is of the order of ~V (and not 0 h ). It will be h$~hl~ desirable to obtain a ~roof of Calder~n's result which does not de~end on special tricks or complex variables. Any such technique will extend to higher dimensions and is bound to imply various sharp estimates for operators arising in partial differential equations.

312

REFERENCES I. C a I d e r 6 n

A.P.

On the Cauchy integral on Lipschitz cur-

ves and related operators. - Proc.N.Ac.Sc.1977, 4, 1324-27. 2. C o i f m a n R.R., ~ e y e r Y. Com~utateurs d'integrales szmgulieres et op~rateurs multilinealres.--Ann, Fourier

(Grenoble),

3. C o i f m a n

1978, 28, N 3, xi, 177-202.

R.R.,

M • y • r

Y.

Multilinear pseudo-

differential operators and commutators, R.R.COII~N

Inst.

to appear.

Department of ~athematics Washington University Box 1146, St.Louis, M0.63130 USA

YVES ~EYER

Facult~ des Sciences d'Orsay =

°

r

Universlte de Paris-Sud Prance

CO~ENTARY The solution of Problem I is discussed in Commentary to S. 5.

313

6.2. old

SOME

PROBLE~gS CONCER/~ING CLASSES OF DO~IA~TS DETER/gINED BY PROPERTIES OF CAUCHY TYPE INTEG~%LS

Investigation of boundary properties of analytic functions representable by Cauchy-Stieltjes type integrals~ni a given planar domain G (i.e. functions of the form ~b-~ (~-%f~); if ~G ~ = ~ we denote this function by ~ ), as well as some other problems of function theory (approximation by polynomials and rational fractions, boundary value problems, etc.) have led to introduction of some classes of domains. These classes are defined by conditions that the boundary singular integral ~V (D ( ~ = ~ ) should exist and belong to a given class of functions on ~ or (which is in many cases equivalent) that analytic functions representable by Cauchy type integrals should belong to a given class of analytic functions in G . see [I] for a good survey on solutions of boundary value problems. An important role is played by the class of curves (denoted in [I] by ~p ) for which the singular integral operator is continuous on mP(r)~ ~ ~ ~p if and only if

Vco LPCr)

llSrC°)llCpll°°ll P•

(I)

This means that the M.Riesz theorem (well-known for the circle) holds for r . Some sufficient conditions for (1) were given by B.V.Hvedelidze, A.G.D~varsheishvily, G.A.Huskivadze and others. I.I.Danilyuk and V.Yu°Shelepov (a detailed exposition can be found in the monograph [~ ) have shown that (I) is true for all p>~ for simple rectifiable Jordan curves r with bounded rotation and without c u s p s . Some general properties of the class ~p were described by V.PjHavin, V.A.Paatashvily, V.M.Kokilashvily and others. It was shown, e.g., that (I) is equivalent to the following condition:

Ep(,G)

being the well-known V.I.Smirnov class (cf. e.g, [~) of over functions # analytic in 6 and such that integrals of I~IP some system of closed curves (with , ) are bounded. That is~ ~p can be characterized by the property

314

where

~ is a conformal mapping of ~ onto ~ . Another class of domains (denoted by ~ ) has been introduced and investigated earlier by the author (cf.[4] and references to other author's papers therein). We quote a definition of K that is closely connected with definition (2) of ~p : ~ K if for any function ~ in ~ , analytic and representable as a Cauchy type integral, the function C~o~) ~' ( g being a ccnformal mapping of D onto 6 ) is also representable as a Cauchy type integral: K)

Note that by Riesz theerem it is sufficient for F ~ p that~he function ( ~ o ~ ) ~ in (2) be representable in the form with ~ L P C T ) . This allows to consider K as a counterpart of ~p for p= I (it is well-known that to use (2) directly is impossible for p=1 even for F = T ). It is established in [5] (see also ~]),using Cotlar's approach, that the classes ~p coincide for p > 4 . Thus the following problem arises naturally. PROBLEM I. Do the classes

~p

~)

If not, what..~eometric conditions guarantee

an___~d ~

coincide?

ri~_.~p~ ~4

?

Note that for ~ ~ ~ it becomes easier to transfer many theorems, known for the disc, on approximation by polynomials or by rational frsctions in various metrics (of. references in [4]). It is possible to obtain for such domains conditions that guarantee convergence of boundary values of Cauchy type integrals [4]. As I have proved, ~ is a rather wide class containing in particular all domains ~ bounded by curves with finite rotation (cusps are allowed) [4]. At the same time, it follows from characterizations of proved by me earlier that ~ coincides with the class of F~ber domains, introduced and used later by Dyn'kin (cf. e.g.[6],[7]) to investigate uniform approximations by polynomials and by Anderson and ~) In virtue of a well-known V.I.Smirnov theorem, the analog of this property for Cauchy integrals is always true.

315 Ganelius [8] to investigate uniform approximation by rational fract i o n s w i t h fixed poles. This fact seems to have stayed unnoticed by the authors of these papers, because they reprove for the class of Faber domains some facts established earlier by me (the fact that domains with bm~uded rotation and without cusps belong to this class, conditions on the distribution of poles guaranteeing completeness, etc.). The following question is of interest. PROBLEM 2. Suppose that the interior domain belongs to belongs to

K (=~) ~

?

6- o to {lwl> }

~+

of a curve r

. Is it true that the ` extgrior domain 6

als__._~o

(Of course, we use here a conformal mapping of

).

For ~p with p >J the positive answer to the analogous question is evident.At the same time the similar problem formulated in [9] for the class S of Smirnov domains remains still open.At last it is of interest to study the relationship between the classes S of Smirnov domains andAoof Ahlfors domains (bounded by quasicircles [SO]), on the one hand, and K and ~p (considered here) on the other. See [9] for more details on ~ a n d A o . It is known that ~ p c S , K~:=~ ([4],[11]). At the same time there exist domains with a rectifiable boundary in A o which do not belong to S (of. [ 3 ] , [ ~ ) . Simple examples of domains bounded by piecewise differentiable curves with cusp points show that K\ Ao ~% ~ . PROBLEM 3. Pind ~eometric conditions

~uaranteeing

G KD2o A°. Once these conditions are satisfied, it follows from the papers cited above and [12], ~3] that many results known for the unit disc can be generalized. One of such conditions is that ~ on and without cusps.

should be of bounded rotati-

RE FEREN CE S I,

2.

X B e ~ e a E ~ s e B.B. MeTo~ HHTerpa~OB w~na K o ~ B paspHB-HRX l~aH~l~Ix s ~ a ~ a x Teop~Gi rOJIOMOp~H2X ~ y H E L ~ O~J~io~ EOM!DIeEc-HO~ HepeMeHHO~i. "COBpeMeHS~e npo6xeM~ MaTeMaTHEH", T.7, MOOEBa, 1975, 5-162. ~ a H ~ a ~ E ~.H. Hepei~yx~H~e rpa~m~H~e s s ~ a ~ Ha IL~OCEOCT~, MOCEBa, HayEa , 1975.

316

3. D u r e n P.L., S h a p i r o H.S., S h i e 1 d s A.L. Singular measures and domains not of Smirnov type. - Duke Math. J., 1966, v. 33, N 2, 2%7-254. 4. T y M a p z ~ H r.~o P p a H H ~ e CBO~CTBa a ~ a x H T ~ e c ~ H x ~ , n p e ~ c T a B m ~ X ~HTerpa~a~m T~na E o ~ . -MaTeMoC6., I97I, 84 (126), 3 , 425--439.

5.H a a T a z B ~ x ~ B.A. 0 c m I t ~ y z ~ a p ~ HHTerpaxax Ko~m. - Coo6~. AH r p y s . c c P , I 9 6 9 , 53, ~ 3, 5 2 9 - 5 3 2 . 6.~ H H ~ K ~ H E.M. 0 paBHoMepHoM n p ~ 6 ~ e H ~ MHOrO~eHS~m B EOMr~eEcHo~ ~ O C E O C T ~ . - 8a~.Hay~H.CeMEH.~0M~, 1975, 56, 164--165, 7.~ H H ~ Z ~ H E.M. 0 paBHOMepHoM n p ~ 6 ~ e H ~ ~ B mop~aHOB~X 06~aCT~X. -- C~6.MaT.~. 1977, 18, ~ 4, 775--786. 8. A n d e r s s o n J a n - E r i k, G a n e I i u s T o r d. The degree of approximation by rational function with fixed p o l e s . - Math.Z., 1977, 153, N 2, 161-166. 9. T y M a p E ~ a r.h. I ~ a H ~ e CBO~CTBa E o H ~ p M m ~ OTO6ps~e-H ~ HeEoTopRx E~aCCOB o6~aoTe~.-c6."HeEoTopHe BonpocN CoBpeMeH-HO~ Teopm~ ~ y H E n ~ " , HOBOCH6HpcE, I976, I49-I60. I0. A x ~ ~ o p c ~. ~ e E n ~ no EBaS~EOH~OpMmm~ oTo6pa~em~M. MOCEBa, M~p , 1969. II. X a B ~ H B.H. l ~ ~ e CBO~OTBa ~HTe#ps~OB T E a K O ~ ~ IBp-Mo~eoE~ conp~l~eHs~X ~ y H E ~ B o6XaCT~X CO c n p ~ e M o ~ rpaH~~e~. -MaTeM.C6., 1965, 68 (II0), 499--517. 12. B e x N ~ B.H., M ~ E x ~ E o B B.M. HeEo~opwe CBO~CTBa E O H ~ O p ~ x ~ E B a S ~ E O H ~ O p ~ X OTo6pa~eHm~ ~ n p ~ e T e o p e ~ EOHCT-pyET~BH02 Teop~E ~ y ~ . -- HsB.AH CCCP, c e p ~ MaTeM.~I974, ~ 6, 1343--1361. 13. B e x ~ ~ B.H. E O H ~ O p M ~ e O T O 6 p a ~ e ~ ~ n p H 6 2 e H ~ e a H a J m T ~ e CKEX ~ B o6~a0T~X 0 EBaS~EOH~Op~Ho~ rpaH~e~. - MaT.c6., 1977, 102, ~ 3, 331-361. G. C. TUMARKIN

TYMAPEEH)

CCCP, 103912, MOCEBa, npocn.MapEca 18, M o C E O B O ~ reo~oropasBe~o~ ~CT~TyT

CO~NTARY A complete geometric description of the class ~p, ~/0

equals

0

IPI,~,0~ =~-4~ ~I~ =~-~/~. • be a simple closed oriented L~punov curve; ~4~..., ~

be points on

P , IPloss(IP '~K) ,~$5 )

in the space L ~ [ ~

it

be the essential norm of P

(L ~ (r, ~ k ~ on P

.asp~-o.,ed tha~"I P Io.~,.>..m.o..~~ (p,~,.~

with the weight ~ ~ T ) ~

(~ (p,.~)~-~.~i.,¢~,

~.~.~i~

deZi,,,~ ~,y (3)) • Then "i-Z[ 5] it ~pro'Ved If our Conjecture is

t'~'0 "P'IIo~. t.~= ~,~,,, I Pl ~'j'~ true thenl P l ~ = ~ (p, ~ , ) .

In conclusion we

T (~thout ,ei~t)

note that in the space

IPlo~ -IPl

~

on the circle

([3]). But i~ ~ene~l the ~o~

IP l depends on the weight and on the contour

r

(E3],[5] ).

327

REFERENCES

I. r o x d e p r H.ll., E p y ~ H ~ E H.H. 0 BopMe npeo6pasoBa~ r~depTa B npocTpaHcTBe LP . - SyH~.aaax. ~ ero n p ~ . , I968, 2, ~ 2, 91-92. 2. P i c h o r i d e s S.K. On the best values of the constants in

3.

4.

5.

6.

7.

in the theorems of M.Riesz, Zygmund and Kolmogorov. - Studia Kath., 1972, 44, N 2, 165-179. E p y n R x ~ H.~., H o x o ~ c K ~ ~ E.H. 0 sopMe onepaTopa cm~ryxspnoro ~Terp~poBan~s. - ~ y a ~ . aRax. • ero np~x., 1975, 9, 4, 73-74. B e p d ~ ~ E ~ i~ H.B. 0nem~a Hop~g~ ~ E I ~ HS IIpOOTI~OTBa X a p 2 ~epes HopMy ee BemecTBemmo~ ~ M ~ o ~ ~ac~z. - B cd."Ma~eM. ~cc~e~o~a~zs", l ~ m ~ e B , ~ ] T ~ a , 1980, J~ 54, 16-20. B e p d • ~ • ~ ~ H.B., E p y n ~ • ~ H.~. To~e EOHCTSHTI~ TeopeMax E . H . B a d e ~ o n B . B . X ~ e ~ s e od OI~paH~IeHHOCTI~ C ~ H r y ~ p moro onepaTopa. - Cood~.AH rpys.CCP, 1977, 85, ~ I, 21-24. r o x d e p ~ H.II., E p y ~ ~ n ~ H.H. B ~ e ~ e ~ e ~ T e o p ~ c~jxsp~x ~e~pax~x onepa~opoB. - E ~ m ~ e B , ~ u a , 1973. H ~ ~ o x ~ c ~ z ~ H.E. ~e~ o6 onepaTope c ~ m ~ a . M. : Hay~ , 1980.

8. H a r d y G.H., L i t t I e w o o d J.E., P ~ I y a G. Inequalities. 2nd ed. Cambridge Univ.Press, London and N e w York,

1952. I. E. VERBITSKY

(H. B. BEPBI~II~) N. Ya. ERUPNIK

(H.~. KPYnHHK)

CCCP, 277028, E m m m e ~ , H ~ C ~ T y T reoalmsmrz ~ r e o ~ o ~ AH MCCP CCCP, 277003, I ~ e B , ElmmKeBc~ rocy~apcTBe~ Ym~BepcxTeT

328 IS THIS OPERATOR I ~ R T I B L E ?

6.7.

Let ~ denote the group of increasing locally absolutely continuous homecmorphisms k of ~ onto itself such that ~t lies in the Muckenhoupt class A ~ of weights. Let Vk denote the operator defined by V ~ ( { ) - - t o k , so that V ~ is bounded on B M 0 ( ~ ) if

and only if

~ CG

(~ones [3]). Suppose that P

projection of BMO onto BMOA. Por which there exists a

for all ~

C~0

such that

k~G

is the usu~l

is it true that

II~V~(1)~M0

BMOA? Is this true for all W ~ G

~(~B~O

?

This questions asks about a quantitative version of the notion that a direction-preser~¢ing homeomorphism cannot take a function of analytic type to one of antianalytic type. For nice functions and homeomorphisms this can be proved using the argument principle, but there are examples where it fails; see Garnett-O'Farrell [2]. We should point out that, the natural predual ~ormulation of this U£#~#114 . I I14 l H'

J

"

This also has the advantage of working with analytic functions whose boundary values trace a rectifiable curve. An equivalent reformulation of the problem is to ask when

V- HV~

~4 ,,+ k

is inve~ible o~ ~o, if ,.

denotes the Hilbert

transform. This question is related to certain conformal mapping estimates; see the proof of Theorem 2 in [I]. In particular, it is shoIAr~X there that this operator is invertible if H ~ WU~M0 is small enough. REFERENCES I. D a v i d G. Courbes corde-arc et espaces de Hardy generallses. -Ann.Inst.Fouzier (Grenoble), 1982, 32, 227-239. 2. G a r n e t t J., 0 ' P a r r e 1 1 A. Sobolev approximation by a sum of subalgebras on the circle. - Pacific J.Math. 1976, 65, 55-63. 3. J o n e s P. Homeomorphisms of the line which preserve BMO, to appear in Arkiv for Natematik. STEPHEN SEMMES

Dept. of Mathematics Yale University New Haven, Connecticut 06520 USA

329

6.8.

AN ESTI~LiTE OF B~O NOR/{ IN TER~S OF AN OPERATOR NORM

Let ~ be a function in B~IO (~f~) with norm II~ I, and let K be a Calderon-Zygmund singular integrel operator acting on -----L~(~ ~) . Define

~

by

Kg(~)--g~ K(Cg~)

. The theory of weighted norm

inequalities insures that ~6 is bounded on ~ if II6 II, is small. In fact the map of 6 to ~6 is an analytic map of a neighborhood of the origin in BMO into the space of bounded operators (for instance, by the argument on p.611 of [3]). ~Iuch less is known in the opposite direction. QUESTION: Given 6~ ~

; if

I~-

K61

is small I must

The hypothesis i~ enough to insure that I16 ~ is finite but the naive estimates are in terms of ~ ] + ~ . If ~ = 4 and ~ is the Hilbert transform then the answer is yes. This follows from the careful analysis of the Helson-Szego theorem given by Cotlar, Sadosky, and Arocena (see, e.g. Corollary

( I I I . d ) of [1] ). A similar question can be asked in more general contexts, for instance with the weighted projections of [2]. In that context one would hope to estimate the operator norm of the commutator [ ~ , p ] (defined by [M~,P](~)=6P~-P(~-).,_,,~ ) in terms of the operator norm

of P -

P8 . RE~ERENCES

I. A r o c e n a R. A refinement of the Helson-Szego theorem and the determination of the extremal measures. - Studia ~¢eth, 1981, LXXI, 203-221. 2. C o i f m a n R., R o c h b e r g R. Projections in weighted spaces, skew projections, and inversion of Toeplitz operators. Integral Equations and Operator Theory, 1982, 5, 145-159. 3. ¢ o i f m a n R., R o c h b e r g R., W e i s s G. Factorization Theorems for Hardy Spaces in Several Variables. - Ann. -

Math. 1976, 103, 611-635. RICHARD ROCHBERG

Washington University Box 1146 St.Louis, MO 63130 USA

330

some OPE~ PROBnEMS CO~CER~n~G H ~

6*9. old

Am~ B M 0

I. A n int erp ola t ing BIs e chk e prod u c t is a Blsschke product having distinct zeros which lie on a n ~ ~ interpolating sequence. I s ~ ~ the un~fgrmly closed linear span of the interDolatin~ Blaschke pr0duc%s? See D ] , ~ ] -

It is

known that the interpolating Blasohke products separate the points of the maximal ideal space (Peter Jones, thesis, University of California, Los Angeles 1978), . Assume 2. Let ~ be a real locally integrable function on that for every interval

where ~I is the mean value of ~ over l , and where C is a con~tant. Does it follow that ~ = ~ ~ H~ . whoso ~ L~ snd~l -l--1, then I]~ [looand

llB-'llooshould be bounded independently of Y

Using the fact that

.

QI/~

has positive real part, i% is easy of ~ satisfy I~I ~ with ~ > 0 . We say that ~ is of P a r r e a u - W i d o m o

We first sketch some relevant results showing that such surfaces are nice. In the following, R denotes a surface of ParreauWidom type, unless Stated otherwise. (1) PARREAU [4]" (a) Every positive harmonic function on has a limit along almost every Green line issuing from any fixed point in ~ . (b) The Dirichlet problem on Green lines on ~ for any bounded measurable boundary function has a unique solution, which converges to the boundary data along almost all Green lines. (2) WIDOM ~ ] : For a hyperbolic Riemann surface ~ , it is of Parreau-Widom type if and only if the set ~@@(~,~) of all bounded holomorphic sections of any given complex flat unitary line bundle over ~ has nonzero elements. (3) HASUMI [6] : (a) Every surface of Parreau-Widom type is obtained by deleting a discrete subset from a surface of Parreau-Widom type, ~ , which is regular in the sense that ~ $ e ~ : @ ( @ , ~ ) ~ 8 is compact for any ~ 0 . (b) Brelot-Choquet's problem (cf. [7]) concerning the relation between Green lines and Martin's boundary has a completely affirmative solution for any surface of ParreauWidom type. (c) The inverse Cauchy theorem holds for ~ . In view of (3)-(a), w e a s s u m e i n what follows that ~

i s

a

r e g u 1 a r

s u r f a c e

o f

P a r-

r e a u - W i d o m t y p e. The Parreau-Widom condition stated in the definition above is then equivalent to the inequality

348

~ ~(~,i6):~ Z ( @ ) } < ~ , where Z(@) denotes the set of critical points, repeated according to multiplicity, of the function ~ ~. . we set

Moreover, let ~I be Martin's minimal boundary of ~ and ~ & the harmonic measure, carried by A I , at the point • . Look at the following STATEMENT (DCT): Let ~ such that

1'1,,,I

be a meromorphic function On

h~s a harmonic majorant on ~

. Then

A

=

, where

denotes the fine boun

W

A4

function for ~

.

(Note: DOT stands for Direct Cauchy Theorem).

(4) HAYASHI [8]: (a) (DCT) is valid for all points ~ in if it is valid for some @ . (b) (DCT) is valid if and only if each -closed ideal of ~o@(~) is generated by some (multiple-valued) inner function on ~ . (c) There exist surfaces of Parreau-Widom type for which (DCT) fails. We now mention SO~E PROBLEMS related to surfaces of ParreauWidom type. (i) Find simple sufficient conditions for a surface of Parreau-Widom t,Tpe t0 Satisfy (DCT).

Hayashi ~8~ has found a couple

of conditions equivalent to (DCT) including (4)-(b) above. But none of them are easy enough to be used as practical tests. (ii) Is there, any criterion for a surface of Parreau-Widom t.ype to satisf,7 the Corona Theorem?

Known results: there exist surfaces of Parreau-Wi-

dom type for which the Corona Theorem is false; there exist surfaces of Parreau-Widom type with infinite genus for which the Corona Theorem is valid. Hayashi asks the following: (iii) Does ~ , ~ ) for any

~

h@ve onl,y constant common inner factors?

(iv) Is a generali-

zed P, and M.Riesz theorem true f or measures on Wiener's harmonic boundar,7t which are o ~ h o ~ o ~ l rize those surfaces ~

to H ~ ? Another problem: (v) Characte-

for which ~ ( ~, ~)

ment without zero~ This

for ever7 ~

has an ele-

was once communicated from Widom and seems

to be still open. On the ether hand, plane domains of Parreau-Widom type are not very well known: (vi) Characterize closed subsets of the Riemann sphere (of. Voichick

M,

~

for which

~\ E

is of Parreau-Widom

Dol).

Finally we note that interesting observations may be found in

349 work o f Pommerenke ~1~, Stanton D2~, Pranger ~13~ and o t h e r s . REPERENCES I. H o f f m a n

K.

Banach Spaces of Analytic Functions. Prentice

-Hall, Englewood Cliffs, N.J., 1962. 2. H e 1 s o n H. Lectures on Invariant Subspaces. Academic Press, New York, 1964 . 3. G a m e 1 i n T. Uniform Algebras, Pretice-Hall. Englewood Cliffs, N.J., 1969. 4. P a r r e a u M. Th~or~me de Patou et probleme de Dirichlet pour les lignes de Green de certaines surfaces de R i e m a ~ . Ann.Acad.Sci.Penn.Ser.A. I, 1958, no.250/25, 8 pp. 5. W i d o m H. ~p sections of vector bundles over Riemann surfaces. - A n n . of N~th., 1971, 94, 304-324. 6. H a s u m i ~. Invariant r~bspaces on open Riemann surfaces. -Ann.Inst,Fourier, Grenoble,1974, 24, 4, 241-286; II, ibid. 1976, 26, 2, 273-299. 7- B r e 1 o t M. Topology of R.S. Martin and Green lines. Lectures on Functions of a Complex Variable, pp.I05-121. Univ. of Michigan Press, Ann Arbor, 1955. 8. H a y a s h i M. Invariant subspaces on Rieman~ surfaces of Parreau-Widom type. Preprint (1980). 9. V o i c h i c k ~. Extreme points of bDunded -~alytic functions on infinitely connected regions. - Proc.Amer.Math.Soc., 1966, 17, 1366-I 369. 10. N e v i 1 1 e C. Imvariant subspaces of Hardy classes on infinitely connected open surfaces. - Memoirs of the Amer.Nath.Soc.. 1975, N 160. 11. P o m m e r e n k e

Ch. On the Green's function of Fuchsian

groups. - ~ n . A c a d . S c i . F e n n . Ser. A. I, 1976, 2, 408-427. 12. S t a n t o n C. Bounded analytic functions on a class of open Riemann surfaces. - Pacific J.~ath., 1975, 59, 557-565. 13. P r a n g e r W. Riemann surfaces and bounded holomorphic functions. -Trans.Amer.Nath.Soc., 1980, 259, 393-400. MORISUKE

HASUMI

Ibaraki University, Department of Mathematics, Nito, Ibaraki, 310, Japan

35O

EDITORS' NOTE. A P a r r e a u - W i d o m surface w i t h a corona has been constructed in the paper N a k a i

M i t s u r u ,

Parreau-Widom t y p e . -

Corona problem for Riemann surfaces of

Pacif.J.Math.,

1982,

103, N I, 103-109.

351

6.19.

INTERPOLATING BLASCHKE PRODUCTS &~m

If

~-Z

is a Blaschke product, the interpo-

B:~ I~,r~l ~-~ z

lation constant of ~

, denoted

A well known result of L . Carlemon asserts that B is an interpolating Blaschke product if and only if ~(~) >0 . It is also well known that the following open problems are equivalent: PROBLEM I. Can every inner function be uniforml,y approximated b,y interpolating Blaschke products? i.e., Given an T Blaschke product B

and an

such that

~ >0

is there an interpolating Blaschke product ~I

IIB - B4 II < S

?

PROBLEM 2. Is there a function Blaschke product product

~

and a~y

~4 such that

6>0

~(6)

so that for ar4¥ finite

, there is a (finite~ B!aschke

IIB-~III


>0, (2) and

G

F=BG

where ~

is an interpolatin~ Blaschke product

is an oute r functio n satisfyin~

352

O< W IG(z)l.--0,

defines a l i n e a r operator b y the = l e

~...,~

o

~

on ~f

on A(~) and set ~ 4 = The differential polynomial

that maps ©~

into A(~I)

is a fundamental system

•

of solutions of the equation

~-k~--0

(~)

normalized by e q u a l i t i e s b~(P) (O,k) =~K,p+4 ' k = ~ , - . . , ~ p= 0 , . . . , ~ - ~ , the spectrum of ~ coincides with the set of solut i o n s of the characteristic equation

A(~)=o

, where

~(~)~()k~i={.

Since & is an entire function of ~ , the spectrum is, unless AmO,~ discrete set with possibly a unique limit point at infinity. In this case a root subspace of finite dimension corresponds to each point of the spcetrum. The PROBLEM mentioned in the title consists in obtaining a description of the A(ll)-closure of the linear span of root vectors of ~ . This problem is closely con2ectedwith completeness ques-

tio~fo=t~e

~yste~(Z,~)}

of solutions of eq~tion (i) in

A(ZI) , with the--O-constructionof general solutions of differential equations of infinite order with respect to ~ , with the theory of convolution equations and of mean periodic analytic functions. An analogous problem for differential operators on the real line is well-known. CONJECTURE. I~f ~

is convex and

A~O

of the linear span of ~oPt vec$ors of ~ of all its powers I i.e,. with th e subspace

then the closure Ok

coincides with the domain

375

0,,1,. . .]. The inclusion ~ C ~co follows immediately from the ~(~I) -continuity of $ and ~ . ~~ . The inverse inclusion is non-trivial and has been proved only in some particular cases: by A.F.Leont'ev [1] in the problem of completeness of the system { ~(2, ~i )} ; by Yu.N.~rolov [2] in the problem of constructing a general solution of equations of infinite order under some additional restrictions on & (~) , see also the papers of the same authors cited in [1~ and [2] ; by V.I.Matsaev [3] for a general k=~ but with ~ = C ; by the author [4] in some system { £~k} ~ weighted sp~ces of entire functions. In the case of an arbitrary convex domain ~ and ~ = ~ the question under consideration is equivalent to that of the possibility of spectral synthesis in the space of solutions of a homogeneous convolution equation; spectral synthesis is really possible in this situation, this has been proved by I.F.Krasickov-Ternovskii [ 5], [6], furSher generalizations can be found in [7]. The results of [5], [6] imply that the convexity condition imposed on ~ cannot be dropped. The question whether the above conjecture is true for an arbitrary convex domain and an arbitrary ~ remains open.

REFERENCES

I.

2; 3.

4.

5.

~ e o H T ~ e B A.$. E BOnpocy 0 noc~e~oBaTe~ocT~x ~ e ~ x al~eraTOB, oOpasoBaHHRX XS pemeH~ ~x~epeHnHaz~H~x ypaBHeH~. -MaTeM.c6., I959, 48, ~ 2, I29-I36. ~ p 0 X 0 B D.H. 06 O ~ O M MeTo~e p e m e ~ onepaTop~o~o ypaB~e6ecEoHeqRoro nop~Ea. -~mTeM.c6., I972, 89, ~ 3, 461-474. M a ~ a e B B.H. 0 pas~oxeH~ n e ~ x ~ y H E n ~ no COOCTBeH~M I ~ o o e ~ H e H H ~ M ~ J H E L ~ M O606~eHHO~ KpaeBo~ ss~a~o -- Teop.~ymc~., ~s.aHa~s H ~x np~.~ 1972, 16, 198-206. T E a q eHE O B.A. 0 pas~o~eH~ ne~o~ $ ~ Eo~e~o~o HO-p~a no EopHeB~M ~ ~ 0 ~ O ~ O ~ z ~ x ~ e p e ~ s ~ H O ~ O onepa~opa. ~awe~.c6., I972, 89, ~ 4, 558--568. Ep a c ~ ~ ~ o B- T e p H O B C ~ ~ ~ ~.~. 0 ~ H o p o ~ e ypaB-He~ T ~ a cBep~F~ Ha B ~ X o6~ac~x. - A o ~ . A H CCCP, IOVI, 197, • I, 2~-3I.

376

6.

7.

Kp a c z ~ E o B- T e p H O B C E ~ ~ H.*. HEBap~aHTH~e noAnpocTpaHCTBa a ~ T E ~ e c ~ z x S y B Z ~ . H. CneETpaa~B C~TeS Ha BU-o6aaCTSX. - MaTeM.c6., 1972, 88, ~ I, 8-30. T E a ~ e H E O B.A. 0 oHeETp6UIBHOM c~Tese B npocTpa~cTBax aHaxZT~xecE~x ~y~R~OHaXOB. - ~oEa.AH CCCP, 1975, 228, ~ 2, 307-309.

v.A. TKACHENKO (B.A. TEACh,K0)

CCCP, 810164, Xap~EoB npocneET ~e~Ha, 47, ~SHEO--TeXH~qecEm~ ~HCT~TyT H~SE~X TeMnepaTyp AH YCCP

COMMENTARY BY THE AUTHOR S.G.Merzlyakov has discovered that my CONJECTURE IS FALSE. Namely, pick up two entire even functions ~ and q of exponential type and of completely regular growth such that the zero-set of ~ is an ~ -set and all zeros are simple. For example

X4 fit. Let {~k} and Then the functions

{~k}

denote the zero-sets of q

"~4 (~(Ak)C~'(~'k)(A'Ak)

and ~

.

CA)=

'

are entire functions of exponential type and ~(~) + ~ ( ~ ) A ~(A)--~q(~) define continuous linear functiomals on (~i , being the interior of the indicator diagram of ~ The operator ~ defined by (~/~£)W and these functionals is an operator with the void spectrum because A(~)=A(A)q(~) +~(~)~(~)~4. However, the domain ~09 of ~ contains a non-zero element, namely, the holomorphic function defined in ~ by ,,

377

S.G.Merzlyakov has communicated that ANALOGOUS COUNTER-EXAmPLES EXIST FOR UNBOUNDED DONAINS AS WELL. Nevertheless, to my knowledge THE GENERAL PROBLEM of describing the closure of the family of roots vectors for an arbitrary operator RENAINS UNSOLVED.

378

TWO PROBL~S ON THE SPECTRAL SYNTHESIS

7,7~ old

1. Synthesis .is impossible. We are concerned with the synthesis of (closed) invariant subspaces of ~* , the a&Joint of the operator of multiplication by the independent variable ~ on some space of analytic functions. More precisely, let ~ be a Banach space of functions defined in the unit disc ~ and analytic there, a~d suppose that ~ X c X and the natural embedding X ~ ~ ( ~ ) is continuous,~ ~ b e i ~ the space of all functions holomorphic in ~. If # ~ X then k~(~; denotes the multiplicity of zero of ~ at a point ~ in ~ , and ~or any function k from ~ to nonnegative integers let

Xka.

%:>-k

A closed ~-invarlant subspace ~ of ~ is said to be DIVISORiAL (or to have THE ~-PROPERTY) if E = X k for some (necessarily k(~)~- ~EQ~) ~ ~ k~(~) , ~ ~)). CONJECTURE I. In every space sorial

~

as abov~ there exist non-divi-

Z-invaria~t subspaces.

The dualized ~-property means that the spectral synthesis is possible~ To be more precise, let ~ be the space dual (or predual) to X equipped with the weak topology 6"( ~, X) ( the duality of ~ and V is determined by the Cauchy pairing, i.e. ~ , $ ~ ~-A

=

E

space

~(~)~(~) E

of

~

for polynomials ~ ~

). A

~-invariant sub-

is said to be SYNTHESABLE (or simply

6 -SPACE) if

(I)

kE~..

with k ~ In other words ~ is an ~-space if it can be recovered by the root vectors of ~* it contains. All known results on Z-invariant subspaces (cf. KIS ) support Conjecture I. The main hypothesis on X here is that X shoula be a B a n a c h s p a c e . The problem becomes non-trivial if, e.g. the set of polynomials ~A is contained and dense in X and ~. ~G~ :~A I P(~)I IIP~:X f < oo} ...........~ . The existence of a single norm defining the topology should lead to some limit stable peculiarities of the boundary behaviour of elements of X , and it is

379

these peculiarities that should be responsible for the presence of non-divisorial ~-invariant subspaces. Spaces topologically contained in the Nevanlinna class provide leading examples. The aforementioned boundary effect consists here in the presence of a non-trivial inner factor (i.e. other than a Blaschke product) in the canonical factorization. Analogues of ~nNer functions are discovered in classes of functions defined by growth restrictions (~2], E3~, [4]); these classes are even not necessarily Banach spaces but their topology is still "sufficiently rigid" (i.e. the seminorms defining the topology are of "comparable strength"). On the contrary, in spaces X with a "soft" topology the invariant subspaces are usually divisorial. Sometimes the "softness" of the topology can be expressed in purely quantitative terms (for example, under some regularity restrictions on , all ideals in the algebra I ~ : ~ H O $ ( ~ ) , I(~(~)1 0(IC(~)), 0

O ~----C~

.....~-- ~

}

~ + co

are divisorial if and only if [5], [IO] ). This viewpoint can be given

,

a metric character; it can be connected with the multiplicative structure of analytic functions, with some problems of weighted polynomial approximation, with generalizations of the corona theorem, etc. (of.

D,3,6]). 2, A~proximative s~thesis is possible. Let us read formula (I) in the following manner: there is an increasing sequence { ~ } of ~W-invariant subspaces of finite dimension that approximates E :

E-~,~,,~ E~,~'~'-/-~{ ~: ~ X , -R~ ~,~(÷, E~,) : i,~

o} .

w.,

Removing one word from this sentence seems to lead to a universal description of ~-invariant subspaces. CONJECTURE 2. Let ~ ~*-invariant subspace of with

~*E.cE., ~

be a space from section I and ~

E

• Then there exist subspaces

E.C(~-l~l) ~

for some

, is ~ X

c~clic?)

is the Bergman space, that is

analytic ~unctions-ll~]l¢=~ IS Jz ~ ° ° " is in the Ber~man s p a c e and if I~(~)I>

O, @ > 0

, then

S

is cyclic.

If correct this would imply an affirmative answer to QUESTION I when X is the Bergman space. The conjecture is known to be correct under mild additional assumptions (see [2], [3], [4~ ). In particular it is correct when ~ is a singular ~ n e r function. In this c a s e the condition in the hypothesis of the conjecture is equivalent to the condition that the singular measure associated with ~ has modulus of continuity 0 ( ~ @ ~ ~/i) (see [1]). CONJECTURE 2. A singular ~ e r function is c2clic in ~he Bergman

,space

if and onl.T if its @ssociated sinRular m e a d e

on an~ ~ r l e s o n set.

puts no mass

(For the definition of Carleson set see [5~,

pp. 326-327. )~) For more discussion of the cyclicity of inner functions see §6 of ~6], pages 54-58, where the possibility of an '~nner-oute~' factorization for inner functions is considered. ,

u)

i

or else

9.3

- Ed.

383 QUESTION 2. Does there exist a Banach space of anal,v~ic functions. satisfyin~ (i) and (ii). in which a function

and onl~ i f

~t ~ s no

zeroj in ~

i

is c~clic if

?

N . F ~ N i k o l s k i i h a s shown E7] t h a t no w e i g h t e d sup-norm s p a c e o f a c e r t a i n t y p e has t h i s p r o p e r t y . I f such a space X e x i s t e d t h e n t h e o p e r a t o r o f m u l t i p l i c a t i o n by ~ on X would have t h e p r o p e r t y t h a t i t s s e t o f c y c l i c v e c t o r s i s n o n - e m p t y , and i s a c l o s e d s u b s e t o f t h e s p a c e X \ {0} ( t h i s f o l l o w s s i n c e t h e l i m i t o f n o n - v a n i s h i n g a n a l y t i c f u n c t i o n s i s e i t h e r n o n - v a n i s h i n g o r i d e n t i c a l l y z e r o ) . No example o f a n o p e r a t o r w i t h t h i s p r o p e r t y i s known. ( T h i s ~ y no l o n g e r be c o r r e c t ; P e r E n f l o h a s announced a n example o f a n o p e r a t o r on a Ban~ch s p a c e w i t h no i n v a r i a n t s u b s p a c e s ; t h a t i s , e v e r y n o n - z e r o v e c t o r i s c y c l i c . The c o n s t r u c t i o n i s a p p a r e n t l y e x c e e d i n g l y d i f f i c u l t . ) H ° S . S h a p i r o h a s shown t h a t f o r a n y o p e r a t o r t h e s e t o f c y c l i c v e c t o r s i s always a ~ s e t ( s e e [ 8 ] , §11, P r o p o s i t i o n 40, p ° 1 1 0 ) . F o r a d i s c u s s i o n o f some o f t h e s e q u e s t i o n s from t h e p o i n t o f view o f w e i g h t e d s h i f t o p e r a t o r s , s e e [ 8 ~ , ~ 1 1 , 12. QUESTION 3. Let_ X

c

clic.

be a s before, and l e t #

,

~, ~ e X

wish

clic?

This question has a trivial affirmative answer in spaces like the Bergman space, since bounded analytic functions multiply the space into itself. It is unknown for the Dirichlet space (that is, the space of functions with ~l~'I~< eo ); the special case ~ = constant is established in [9] •

REFERENCES

1. S h a p i r o

H a r o I d S. Weakly invertible elements in certain function spaces, and generators in ~ . - Mich.Math.Je,

1964, 11, 161-165. 2. S h a p i r o H a r o i d S. Weighted polynomial approxlwation and boundary behaviour of holomorphic functions. - B ~ . :

CoBpeMeH~e npo6xe~ Teop~ aRa~zT~ec~x ~ m ~ ,

M., Hay~a,

1966, 326-335. 3. ~ a n ~ p o r.

He~oTopHe saMe~zam~t,,~ o Becoso~ no,~z~o~,ma~z,Ho~ annpo~czMam~ rO~OMOp~HX ~ y m ~ . -MaTeM.cS., 1967, 73, 320-330.

4o A h a r o n o v

D., S h a p i r o

HoS.,

S h i e I d s

A,L~

Weakly invertible elements in the space ef squ~re-summ~ble holo-

384 morphlc functions. - J.London Nath.Soc.~ 1974, 9, 183-192. 5. C a r I e s o n L. Sets of uniqueness for functions regular in the unit circle. - Acta Nath.~ 1952, 87, 325-345. 6. D u r e n P.L., R 0 m b e r g B.W., S h i e I d s A.L. Linear functionals on ~ ~ spaces with 0< ~~ I . - J.fur reine und angew.Math.~1969,

238, 32-60.

7. H ~ E o x ~ c ~ z ~ H.E. C n e E T p a x ~ m ~ C~mTes ~ sa~a~a mecoBo~ a n n p o ~ c ~ m m m B upocTpaRcTBaX asa~mTE~ec~x ~ y ~ r ~ . - Hs~.AH Ap~. CCP. Cep.MaTe~., I97I, 6, ~ 5, 345-867. 8. S h i e 1 d s A I I e n L. Weighted shift operators and analytic function theory. - In: Topics in operator theory, ~ath.Surveys N 13, 49-128; Providence, Amer.Math.Soc., 1974. 9, S h i e 1 d s A 1 I e n L. Cyclic vectors in some spaces of analytic functions. - Proc.Royml Irish Acad.~1974, 74, Section A, 293-296.

AT,TRN L. SHIELDS

Department of Nathematics University of Michigan Ann Arbor, Michigan 48109 U.S.A.

O0~ENTARY QUESTION 1 has been answered in the negative by Shamoyan

• and l e t

X~

ITZ~l

d,eno¢e the spa,oe of ~11 f u n c t i o n s ~

= 0 ( 0 ~(~)) f~r i~[o

, such t ~ t ~ - l ~ l ~ f ( ~ / I ~ l ,

E) k

for all ~ . (Here ] denotes the Euclidean metric. ) The following unpublished theorem of B.A.Taylor and the author provides solutions to problems (I) and (2) in a special case.

mmo~,,

(a) As,s~eZ

±)=Z (I). z~

o r d e r tm~t

necessar 2 and sufficient that the z ergs of B contact

at

Bi

it is

have finite degree of

Z~(D. z__~Bl=C~ ,then multiplication

,b,y B

,iS oontinu-

ous,,oni, 5i is closed,and the , inv~.rse operation is...continuous. (b) Ass~e ~l)n 8~-~ ~ I ) .

~et E

Blaschke factors of the non-zero '~ c t i o n s

~e t.ae.,g,c.d,, of the in I • In order

that(~/E)l

b.e closed i,t,,,,i,s,~ecessary and suffioient that t.he. zeros of ~ ~inite degr¢,~,,of contact at

~ql)

have

•

THE PROOF of sufficiency in (a) is primarily a computation of the growth of the derivatives of ~ near ~ ( ~ ) . The computation has also been done by James Wells [61 . The proof of necessity in (a) requires the construction of outer functions. (One can assume without loss of generality that the g.c.d, of the singular inner factors of the non-zero functions in I is ~ . ) In section 3 of [5] it is demonstrated that there is an outer function ~ , ~ e O A , vanishing to infinite order precisely on ~oo(l) , and such that ~ I ~ ( ~ @ ) [ ~ ~--00(-~?(~)), where ~ is continuous, const.~(~%)~ ~(~) O : I ----~(~)iff ~ourier transforms of ~ have no common zero in the strip I I ~ l ~ ~ and

E-J •,~

~ : e ~ ~-.--~o ~'P ~

(3)

S c r u t i n i z i n g c o n d i t i o n s (S) we can see that in the algebra 4L~(]~) there are closed proper ideals contained in no maximal ideal. Such ideals will be called p r i m e i d e a I s c o r r e s p o n ding t o infinity p o int s of the strip Independently Nyman's result was rediscovered by B.Korenblum, who described completely the prime ideals corresponding %o infinity points of the strip II~4 ~ I ~ ~ . The question concerning Wienertype theorems in algebras ~i (~) , where the weight ~ satisfies

(I)

and

R

oo

,

(4)

411

~+-~-~_ ~ 0 , is still open. The methods used earlier don't work in this case. The author does not know a necessary and sufficient condition for the validity of approximation theorem even under very restrictive conditions of regularity of the weight ~

(for example ~(~)~ C~p

(4+I$Ii) )"

For a weight ~ , satisfying (I), (4). one can find chains of prime ideals corresponding to infinity points (see, for example, [6]). The reason for the existence of prime ideals of this sort is that by (4) %here are the functions in L~(~) with the greatest possible rate of decrease of Pourier transform (see [7], [8] ). It remains ~own whether all prime ideals are of this sort. There is a similar question for the algebra I.~(~+) . We have the following theorem [9] : Let ~ satisfy (I), (2), and let I~ be the_ _cl°sure , of the linear span of all right translates of~. Then I ~ % ~ ' ~ ( ~ ) if and only if the following two conditions are fulfilled: I ) there is no interval I adjacent to the origin and such that all functions in ~ vanish a.e. on I , 2) Fourier transforms of functions in ~ have no common zero i n l ~ 0 . It is worthwhile to note, that this case is simpler than the case of h'~(~) because there are no prime idemls of the above type. (The case when ~ = ~ $ doesn't differ from the case when ~ = ~ ). We conjecture that the previous assertion remains true also for the case when ~ satisfies condition (4). In particular the following conjecture can be stated: cO

If the Fourier transform ~ and if ~_~_~ ~

~0~I~({~)I ~

j ~(~)@(~+~)~ =

0

= VT> 0

doesn't 0 ,

va sh in

0

then the ~equali%,7 implies ~ ( ~ ) = 0

..a,e, ..

once.

There are some reasons %o believe the conjecture is plausible. We have no possibility %o describe them in detail here, but we can note that condition (2) is much stronger than the condition

412 I

which is a well-known condition of "non-quasianalyticly" of 4~(~÷). A more detailed motivation of the above problem and a list of related problems of harmonic analysis can be found in DO]. In conclusion we should like to call attention to a question on the density of right translates in ~ ( ~ ) . Let ~ ~ ( R ) , let be the closure in ~I(~) of linear span of all right translates of ~ . It is easy to prove that I I ---implies that

and that ~ i s nowhere zero. However these conditions are not sufficient. There are some sufficient conditions but unfortunately they are far from being necessary. I think that it deserves attention to find necessary and sufficient conditions.

REFERENCES

l. r e ~ B ~ a H ~ 14.M., P a ~ E o B ~.A., ~ I~ Jl o B r.E. KO~yTaTHBHHe HOpMHpOBaHHHe EOXB~a, M., $.-M., 1960. 2. B E H e p H. HHTeI~a~ $ypBe H HeEoTopNe ero H p ~ o ~ e ~ , M., ~.-,M., 1963. 3. B e u r i i n g A. Sur les integrales de Fourier absolument convergentes et leur application a une transformation fonctionelle. Congres des N~th. Scand., Helsingfors, 1938. 4. N y m a n B. On the one-dimentional translations group and semigroup in certain function spaces. Thesis, Uppsala, 1950. 5. K o p e H 6 a D M B.H. 06odmeHEe Tay6epoBo~ TeOpeM~ B~Hepa rapMoHH~ecE~ aHaxHs OHcTpopacTym~x ~ y H E ~ . -- Tpy~H MOCE.MaTeM. o6-~a, I958, 7, I2I-I48. 6.

V r e t b 1 a d A. Spectral analysis in weighted ~-spaces on • - Ark.Math., 1973, 11, 109-138. 7. ~ • p 6 a m a H M.M. TeopeM~ e~EHCTBem~ocT~ ~ npeoSpasoBaH ~ ~ypBe H ~ 6eCEOHe~HO ~E~epeHsEpyeM~x ~ y H I g / ~ . -- H S B . ~ ApM.CCP, cep.~.-M., I957, I0, ~ 6, 7-24. 8. r a 6 e H E 0 K.M. 0 HeEoTop~x F~accax npocTpSHCTB 6ecEoHe~Ho ~H~epeHnEpye~x #yH~. - ~ o E x . A H CCCP, I960, I32, ~ 6, I23I-I234. 9. F y p a p ~ ~ B.H., C ~ B ~ e E B npocTpaHcTBe

Z e B E H B.H. 0 HOXHOTe C ~ C T m ~ ~(0,~) C BecoM. -- 8an.MeX.-MaT.~-Ta

XY7 E XM0, I964, 30, cep.4, I78-I85.

413

I0. r y p a p H ~ B.H. r a p M o H ~ e c E ~ a ~ s B npoc~ps2cTBaX C BecoM. -- T p y ~ MOOE.MaTeM.O6--Ba, 1976, 35, 21--76.

¥. P. GURAR!I

(B.H. IVPAPM~)

CCCP, 142432, EepHoroxoBEa, MOCEOBOEa~ 06~aCTB, ~ E S ~ E-Ta X~ecE0~3~LEE AH CCCP

414

7-18.

TWO PROBLEMS OF HARMONIC ANALYSIS IN ~rEIGHTED SPACES

We consider the space ~ (~] of measurable functions on with the norm II~II= ~ ~ I~(~)I/q(~) , The weight ~ is supposed to be measurable and to satisfy the conditions

A ssign to each function ~ g ~ e (~) the smallest ~*-closed subspace of L ~ (~) (denoted b; ~ ) invariant under all translations and containing ~ . The set

is called the spectrum of

~

.

Denote for { 0 ). Then the dual s p a c e X t is a space of distributions. We say that a closed set ~ in ~ admits X-s p e c t r a I s y n t h • s i s if every T in X [ that has support in ~ can be approximated arbitrarily closely in X / by linear combinations of measures and derivatives of order < ~ of measures with support in ~ PROBLEM. Do all closed sets admit

X -spectral synthesis for

~]%e above s~ces? The PROBLEM c a n a l s o b e g i v e n a d u a l f o r m u l a t i o n . If measure with support in ~ such that a partial derivative

•

is a ~ k

belongs to X [ , then one can define S ~ ~ for all ~ in X Then ~ admits X -spectral synthesis if every ~ such that ~k~ ~.~ ~

0

for all such ~

and all such multiindices ~

. can

be approximated arbitrarily closely in X by test functions that vanish on some neighborhood of ~ . The PROBLEM is of c o u r s e analogous t o the famous spectral synthis terminology thesis problem of Beurling, but in the case of ~ was introduced by Fug!ede. He also observed that the so called fine Dirlchle% problem in a domain ~ for an elliptic partial differential equation of order ~5 always ~ s a unique solution if and only if the complement of ~ admits W~-spectral synthesis° See [I ; IX, 5. I~. In the case of ~ the PROBLEM appeared and was solved in the work o f V°P.Havin [2] and T.Bagby [3] in c o = e c t i o n w i t h t h e p r o b l e m o f a p p r o x i m a t i o n i n l,P b y a n a l y t i c functions. F o r W ~ the solution a p p e a r s a l r e a d y i n t h e w o r k o f B e u r l i n g a n d Deny ~4]. I n f a c t , i n these spaces all closed sets have the s~ectral synthesis property° This result, which can be extended to ~ , $~ ~< ~ , depends mainly on the fact that these spaces are closed under truncations° When 5 > ~ this is no longer true, and the PROBT.RM is more complicated. Using potential theoretic methods the author [5] has given sufficient conditions for sets %o admit spectral synthesis in ~ ( ~ ) ~ ~6~, . These conditions are so weak that all closed sets if ~>W~IX(~/~, ~-4/~)

they are satisfied for , thus in particular if

436

e - ~ and ~ - ~ or 3. There are also some still unpublished results for ~ and B ~ P showing for example that sets tha~ satisfy a cone condition have the spectral synthesis property. Otherwise, for general spaces the author is only aware of the work of H.Triebel ~ , w h e r e he proved, extending earlier results of t, i o n s and Nagenes, that the b o u n d a r y of a ~ domain admits spect-

ral s y n t h e s i s f o r

and

.

REI~ERENCES I. S c h u I z e B,-W., W i I d e n h a i n G. Methoden der Potamtialtheorle fur e~liptische Differautialgleichungen beliebiger 0rdnm~g. Berlin, Akad~m~e-Verlag, 1977.

2. X a B ~ H B.H. A ~ O ~ C m ~ S n ~ B cpe~HeM aHax~T~ecm~M~ ~ y m ~ M~. - ~ o ~ . A H CCCP, I968, Iva, I025-I028. 3. B a g b y T. Quasi topologies and ratioual approximation. - J. l~amct.Anal.,1972, 10, 259-268. 4. B e u r 1 i n g A., D e n y J. Dirichlet spaces. - Proc.Nat. Acad°Soi., 1959, 45, 208-215. 5. H e d b e r g L.I. Two approximation problems in function spaces. - Ark.ma%.~1978, 16, 51-81. 6. T r i e b e 1 H. Boundary values for Sobolev-spaces with weights. Density of ~ (~i) . - A~.Sc.Norm.Sup.Pisa,1973, 3, 27,

73-96. LARS INGE HEDBERG

Department of Mathematics University of Stockholm Box 6701 S-11385 Stockholm, Sweden

CO~S~ENTARY BY THE AUTHOR P

For the Sobolev spaces W S , ~ < P 0 aoe° with for ~ = ~ l l ~nd all ~ [3]. ~hus (4) can fail even if (5) holds.

~,I~ >-~

QUESTION 2. Assm~e that %he inte~r~l in that &

is ~iven b x (6). o,r that

E ,EcT

se~

~---~

(~) is finite! or even

. Is ~here a measurable

,:',.~.h

where %h e first summand consists of " ~ l y % i c " an E

C,o~%ain an,y arc on which ~ ( ~, ~)

functions? ~i~h% such

(or a suitable an%lo~ue)

tends %o zero sufficien%l y slowly as I--~ 0 ? If ,~here is no such F

with M E

> 0 , ~x~ctl~ how can the variou s conclusions of Theo-

rem 2 fa,,ilt,if indeed ~he,y can? QUESTION 3. Let W(~) Assumin~ the integral in ~ )

be smooth with a single zero at ~-----0 . is f tni%¢, describe the ,invarian% sub-

spaces of the operator "multiplication b~ • " on H~(~) of the rates of decrease of W(~)

near 0 and

~(~)

in t e ~ near I.

Perhaps more complete results can be obtained than in the similar situation discussed in [8]. Finally we mention that the study of other special classes may be fruitful. Recently A.L.Volberg has communicated interesting related results for measures 9 t W ~ ~ where ~ is supported on a radial line segment. (See [I0], [13] in the reference list after Commentary. - Edo )

442 REPERENCES

I. C I a r y

S. Quasi-similarity and subnormal operators. - Doct. Thesis, Univ.Michigan, 1973. 2. H a s t i n g s W. A construction of Hilbert spaces of analytic functions. - Proc.Amer.r~ath.Soc., 1979, 74, N 2,2295-298. 3. K r i e t e T. On the structure of certain H (~) spaces. Indiana Univ.Math.J., 1979, 28, N 5, 757-773. 4. B r e n n a n J.E. Approximation in the mean by polynomials on non-Caratheodory domains. - Ark.Nat. 1977, 15, 117-168. 5. M e p r e x ~ H C.H. 0 n o ~ o T e CzCTeM a ~ a ~ z m ~ e c E H x S y H ~ . Ycnex~ MaTeM.RayK, I958, 8, ~ 4, 3--68. 6. K r i e t e T., T r e n t T. Growth near the boundary in M~) spaces. - Proc.Amer.Math.Soc. 1977, 62, 83-88. 7. T r e n t T. ~(~) spaces and bounded evaluations. Doer. Thesis, Univ.Virginia, 1977. 8. K r i e t e T., T r u t t D. On the Cesaro operator. Indiana Univ.Nath.J. 1974, 24, 197-214. THOMAS KRIETE

Department of Math. University of Virginia Charlottesville, Virginia 22903,

USA

COMMENTARY THEOREM (A.L.VoI'Berg) such that

m2(~)

There

exists W

splits even for ~ ~ ~

, W>0

T

a,e, on

.

The theorem gives an affirmative answer to QUESTION I. It may be seen from the proof that ~ (~,S) tends to zero rather rapidly for every ~ . The proof follows an idea of N.K.Nikolskii [9], p.243. PROOF. It is sufficient to construct a function W , ~ > 0 a.e. on T and a sequence of polynomials {P~ } ~ 4 such that 2

I$

(P.IT, Let

{~}~4

ID) =C0,

in

the Hilbert space IZ(WcI,I'II,)~I~L(~

be any sequence of positive numbers satisfying

443 Z S. ¢ we shall denote by 05 the capacity naturally associated with the Sobolev space W I'$ and A~ will stand for ~-dimensional Hausdorff me&sure. A comparison of these two set functions together with their f@rmal definitions can be found in the survey article of ~az' ja and Havin [5] • Let ~ be any function in ~$(O W~X~) , ~ p/(~-~), with the property that ~ Q ~ , ~ # , ~ = 0 for all pol~onials Q and form the Cauchy integral

~L

~-~

Evidently, ~ v~nishes identically in ~ and so, by "continuity", ~ 0 a.e. -C$ on ~lao . To establish the completeness of the polynomials we have only to prove that 5-----0 a.e. -C$ on the rest of the boundary as well, this approach having first been suggested by Havin E6] (cf.also

E3] and E4] ). The argument is then carried out

463

in two stages; one verifies that: STEP 1. ~ 0 a.e. with respect to harmonic m e a s ~ e on ~ l STEP 2, ~--0 a.e. with respect to the capacity C~ on ~ Horeover, in the process it will be convenient to transfer problem from ~ to ~ by means of confoz~nal mapping. With ~ = let ~ = ~(~ . ~or each ~ > 0 let A ~ = { ~ :I ~o¢~1 < 4 - ~. } and put

{~(~)= Thus,

~

I

~(~ ~# ~-r~

a n d ~a

are

; . the ~-~

a n d ~6 ~-- "~6 (~) •

both defined

on ~

a n d ~8

is analytic

near

3D. STEP 1. By c h o o s i n g ~ ~ 0 a n d s u f f i c i e n t l y s m a l l we c a n f i n d a corresponding ~> 0 such that the following series of implications are valid for any Borel set E ~- ~ll :

0

(1.1)

Here $ ~ p/(p-J) and p < ~ + ~ ° The first implication (i) is essentially due to Frostman K7~- Although he considered only N e ~ o n i a n capacity, his ar6ument readily extends to the nonlinear capacities which enter into the completeness problem (cf.~az'ja-Havin [5~ ). The second assertion (ii) is a consequence of a very deep theorem of Carleson [8]. Because I = 0 a.e. -05 on ~ o o and ~(~-oo) > 0 it follows that ~ 0 on some boundary set of positive harmonic meas~LTe. Consequently, taking radial limits, ~-- 0 on a set of positive are length on S~ . We may now suppose that W ( ~ ) ~ 6-~($) , where ~(~)~ + co

a s t , i o . Then, n s i ~

the f a c t t ~ t

~l~'l~&~

< ~

forp 5 . If ~£~ , let

]~, (~)~ I~ ~ ~ : I I~- ~I< ~ }. If als On

Zf

k ~ {4,~}

L~ t ~ ? ~ )

is fixed, define the (positive) function

k'

aS %he inverse Pourier transform of

= (i i" i~I ~ )- I~

~

, and for each

A C~

,f)1,(~) = Jwl~ [ II~ II Lp,(~) Q

ta~t C >0 for

All functions w i l l be real-valued.

t°O, 4} ,Ulet ~ "denote the vector space Of all polynomi~ which are homogeneous of degree ~ , with inner product

define the Bessel capacity

if

kp' < M,

~0

On

, t h e r e e= s s a co

-

C -~ .< ~Kp (B~t°))/~IP'-k " 0 , stud each function ~ ~ LP(X) which is harmcnlo on the interior ~ A , there exists a harmonic function ~ on an open

neighborhood of X THEOREM 1. I f

X

such that any

l[ ~- { II J(X) < ~

"

one of the f c l l o w i n ~ conditions holds, the~

~ s ~he LP h..,,p. a) ( [ 8 ] , [ z ] ) b) ([8])

p'>~v,

e'< ~

and there exists a constant

~4,p,(B~tm)\X)>~!2~ ~tP'-4 c) ( h i , [51 ) ~or each

a~t~o~ is .et: (±) kp' > v~ se~ E k with

~K,p' (Ek) : 0

me~X

if

k ~ {i,~}

~'0

such that

and 0~ or (ii) kp1% • and there exists a set E k with ~K,p'CEk) =O such that

,{01 0 and

:aSX\E k.

If ~ has the ~ h.a.p., then ~ has property ~*) ; this follows from Theorem 2, (I) and the Kellogg property ~5, Theorem 2]. Our question is whether the converse holds: if ~ has Rro~erty (~) , does it have the L e h.a,p.? We remark that if this question were answered affirmatively, then part c) of Theorem I would follow by use of (I).

469

2. If h.a~.

p=~

follows

and

~5

, a different criterion for the

~P

from work of Saak [9]. What is the relation between

saak,s work and Theorem 2?

3. If H ~ 0 U~ ~{0} and ~ is an open set. then o~j~(', ,~') is an increasing set function defined on the subsets . Can one characterize the compact sets ~ v i n ~

the ~

h.a.p.

b,~ means of inoreasim~ set functions which are countabl,~ subadditive a_nd have the propert,y that all Bore! sets are oapacitable? that a set

~

is capacitable with respec~ to a set function

eans t h a t

open, ~ D E

(To say

K

~. ;

See

[3],[~].

compact,

°

Per the case

p=~

, see [9 , Le.~ma

2]. REFERENCES I. B a g b y T. Approximation in the mean by solutions of elliptic equations. - Trans.Amer.~ath.Soc.

2. B y p e H ~ O B B.~. 0 n p ~ 6 m m ~ H ~ ~yH~m~ ~3 npocTpa~CTBa W~ (i~I ~z~ma~ ~ys=~ ~ npo~sBox~oro o ~ H ~ o ~ ~O~ . -- Tpy~N MaTeM.~--Ta NM.B.A.CTeF~oBa AH CCCP,1974, ISI, 51-63. 3. c h o q u e t G. Porme abstraite du theoreme de capacitabiliZeoTBa

re. -Ann.Inst.Pourier (Grenoble), 1959, 9, 83-89. 4. H e d b e r g L.I. Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem. - Aota Math., 1981, 147, 237-264. 5. H e d b e r g L.I., W o 1 f f T.H. Thin sets in nonlinear potential theory.Stockholm, 1982. (Rep.Dept.of Math.Univ. of Stockholm, Sweden, ISSN 0348v7652, N 2~. 6. L i n d b e r g P. A constructive method for ~? -approximation by analytic functions. - Ark.for Mat., 1982, 20, 61-68. 7. M e y • r s N.G. A theory of capacities for functions in Lebesgue c l a s s e s . - Math.Scand., 1970, 26, 255-292. 8. P o 1 k i n g J.C. Approximation in ~? by solutions of elliptic partial differential equations. - Amer. J.Math., 1972, 94, 1231-1244.

9. C a a E

%.M. ~ o c ~

Ee~ ~ p ~ x x e ~

Ep~Tep~ ~

odxacT~ c yc~o~mBO~ sa~a-

SJ~nT~qecENx y p a B H e H ~ BNC~HX ~ o p ~ E o B . -- Ma-

470

~em.c6., 1876, 100(142), ~ 2 (6), 201--208o I0. B H T y m E ~ H A.Y. AHaJn~T~ecEa~ eMEOCT~ MHO~eOTB B s~ a~ ax Teop~ ~p~0~eH~. - Ycnex~MaTe~.~ayE, I967, 22, B~n.6, 14I--

-199. THOMAS BAGBY

Indiana University Department of Mmthmmatics Bloomington, Indiana 47405, USA

EDITORS' NOTE. Many years before the appearance of C6S the constructive techniques of Vitushkin was applied to the LP -approximation by analytic functions by S.O.Sinanyan (CMHaH~H C.0. ANHpOECHMa~HS aHa~HTMqecEH~ ~tHE~H~MH H HO~MHOMaMH B cpe~HeM no n ~ o ~ H . Ma~eM.c0., I966, 69, ~ 4, 546-578. )See also the survey MexbHHEOB M.C., C w a H ~ C.0. BonpocH TeopHH rrpH6~ixeH~i~ S y H ~ O~HOI'O Komn~exCHOre nepeMeHHoPo. - B KH: "I~TOrH HayKH ~ TeXHHF~". CoBpemeHHme npo_ 6~eM~ MaTemaTMKH, T.4, MOcKBa, 1975, HS~-BO B~T~, 143-250.

471 RATIONAL A P P R O ~ T I O N

8.11.

0F ANALYTIC P~CTIONS

old

1.

Local approximations. Let

l¢)O

and let ment ~

~ be a complete analytic function corresponding to the ele, For any ~ ~ define ~ ( ~ ) = s~p{~(~-~): ~ ~

where ~ ($) is the multiplicity of the zero of ~ at ~ is the set of all rational functions of degree at most ~. Por any ~ there exists a unique function 9 ~ , ~ ~ such that ~ n ( ~ ) = ~ (I-~) " It is called the ~ - t h diagonal Pad~ approximant to the series (1). Let ~ @ > ~ be an arbitrary fixed number and let I . I = @ -%(') ; then ~ is the function of the best approximation to ~ in ~ w i t h respect to the metric: ~ ( ~ ) = detailed discussion on the Pad~ approximants (the definition in slightl~ differs from the one given above). For any power series (I) we have

A

being an infinite subset of ~

depending on

[2]

~.

A functional analogue of the well-known Thue-Siegel-Roth theorem (see [3], Theorem 2, (i)) can be formulated in our case as follows: is ~ is an element of an algebraic nonrational function ~;hen for any ~ 8 e ~ ~ , the inequality ~ (~) > ~8~ holds on2y for a finite number of indices ~ , ~rom this it follows easily that in 01.L~ e a s e -4

C3) Apparently, this theorem is true for more general classes of analytic functions. CONJECTURE I. If

f un~ction ~

~

is an element of a multi-valued analytic

with a finite set of singular points then (3) is valid.

472 In connection with CONJECTURE 1 we note that if

~-47~(I) = + ~

(4)

A,A>0

then ~ is a single-valued analytic function; but for any the inequality ~ - 4 9~(~) > A is compatible with the fact that ~ is multi-valued (the first assertion is contained essentially in [4],[5], the second follows from the results of Polya [6]). Everything stated above can be reformulated in terms of sequences of normal indices of the diagonal Pad6 approximations (see[7], [I] ). In essence the question is about possible lacunae in the sequence of the Hankel determinants

5~.

Thus (3) means that the sequence { ~ } has no "Hadamard lacunae" and (4) means that {F~} has "Ostrowski lacunae" (in the terminology of [8]), Apparently many results on lactmary power series (see [8]) W e their analogues for diagonal Pade approximations. 2. Uniformapproximation, We restrict ourselves by the corresponding approximation problems on discs centered at infinity for the functions satisfying (1). Let ~ > ~ l , ( is ho-

} ".

lomorphic on E ) and ~R = {E: the best approximation of ~ ~(~)=H no:l~ on

2, ~) on E by the elements of ~ n : "

{II~-%~E: % ~ ~

~,

Denote by U'~

is the sup-

E e

Let ~ be t h e s e t of a l l compact8, F , F C ~)R (with t h e conn e c t e d complement) such that ~ admits a holomorphic (single-valued) continuation on C \ F , Denote by C D (F) the Green capacity of F with respect to ~ (the capacity of the condenser (E, F) ) and define •

473 For every

This inequality follows from the results of Walsoh ([9], oh.VIII). CONJECTURE 2. For ar4y

ct7/ Inequalities (5), (6) are similar to inequalities (2). To clarify (here and further) the analogy with the local case one should pass in section I from 9~ to the best approximations k ~ . In particular, equality (3) will be written as

i

CONJECTURE 3. I_~f $ which

is an element of an analytic f ~ q t i o B

D

has a finite set of sin~u,lar points then

(7) |

If tmder the hypothesis of this conjecture $ is a single-valued analytic function, both parts of (7) are obviously equal to zero. CONJECTURE 3 can be proved for the case when all singular points of 4 lie on ~ ( f o r the case of two singular points see [10]). In contradistinction to the local case the question of validity of (7) remains open for the algebraic functions also.

REFERENCES 1. P e r r o n

O.

Die Lehre von den Kettenbx4/chen, II, Stuttgart,

1957. 2. B a k e r 1975.

G.A. Essentials of Pad~ Approximant, New-York, "AP",

474

3. U c h i y a m a S. Rational approximations to algebraic functions. - Jornal of the Faculty of Sciences Hokkaido University, Serol, 1961, vol.XV, N 3,4, 173-192.

4. r o H ~ a p A.A° ~OEa~BHOe yC~OB~e O~HO3Ha~HOCTE aHaJn~T~eCEEX ~y~. -MaTeM.C6., 1972, 89, 148-164. 5. r o H ~ a p A.A. 0 C X O ~ M O C T H 2n~ROECEMsrU~ ~ e . - ~2TeM.C6., 1973, 92, 152-164. 6. P e I y a G. Untersuchungen uber Lucken und Singularitaten yon Petenzreihen.- Math.Z., 1929, 29, 549-640. 7.

r o H ~ a p A.A. 0 C ~ O ~ M O C T H a n n p o E c H M a ~ Ha~e ~ HeEoTopHx ExaCCOB MepoMop~m~X $ ~ . -- MaTeM.Cd., I975, 97, 605 - 627. 8. B i e b e r b a c h L. Analytische Fortsetzung. Berlin - Heidelberg, Springer-¥erlag, 1955. 9. W a I s h J.L. Interpolation and approximation by rational functions in the cemplex domain. AMS Coll.F~Bl., 20, Sec.e~.1960. I0. r o H ~ a p A.A. 0 cEopocTH pan~oHax~HO~ annpoEc~Mau~ EeEoTop~x aHaJL~TEeCE~X ~ y 2 E ~ . -MaTeM.C6., I978, I05, I47-I88. A,A. GONC~AR

CCCP, 117966, Mocxma

(A.A.r0RNAP)

yx.Bamzxoma 42, CCCP.

475 8.12.

A CONVERGENCE PROBLEM ON RATIONAL APPROXIMATION

old

IN SEVERAL VARIABLES

1. The one-variable case, ~ e 6 . Let me first give the background in the one-variable case. Let ~(~)~ ~ C ~ ~ , R e 6 , be a formal power series and P/Q , Q ~ 0 , a rational function in one variable ~ of type (~,~) , i.e. P is a polynomial of degree ~ ~ and Q of degree ~ 9 . It is in general not possible to determine P / Q so that it interpolates to ~ of order at least ~* 9 @ I at the origin (i.e. having the same Taylor polynomial of degree ~ v 9 as ~ ). However, given ~ and ~ , we can always find a unique rational function P / ~ of type (~,~) such that ? interpolates to ~Q of order at least ~ + ~ + ~ at the origin, i.e. ( ~ - P ) ( ~ ) ~ 0 ( ~ + $ + 1 ) . This function P/~ , the[~P a d e a p p r o x i m a n t to ~ , was first studied systematically by Pade in 1892; see [I]. In 1902 Montessus de Ballore [2] proved the following theorem which generalizes the well-known result on the circle of convergence for Taylor series. 1

THEOREM. Suppose ~ Rhic in ~ I < ~ Then the [I,,#3 formly to ~

with ~

is holomorphlc at the origin and meromorpoles (counted with their multiplicities),

-Pade approximant to

~ , ~ / ~

, conver~es uni-

, with ~eometric de~ree of convergence I in those com-

pact subsets of I~I < ~

which do not contain ar47 poles of ~ .

With the assumption in the theorem it can also be proved that P /Q diverges outsiae I I= if is chosen as large as possible [3, p.2693 and that the poles of Pw / ~ w converge to the poles of { in l~l < ~ • Furthermore, when ~ is sufficientS,y large, ~ / Q ~ is the -n~que rational function of type (~, ~) which interpolates to ~ at the origin of order at least I,+9 + I. Montessus de Ballore's original proof used Hadamard's theory of polar singularities (see [4]). Today, several other, easier proofs are known; see for instance [51,[6] ,[7] and [8]. Pad6 approximants have been used in a variety of problems in numerical analysis and theoretical physics, for instance in the numerical evaluation of functions and in order to locate singularities of functions (see [I] ). One reason for this is, of course, the fact that the Pad6 approximants of ~ are easy to calculate from the power series expansion of ~ . In recent years there has been an increasing interest in using analogous interpolation procedures to apprexi-

476

mate functions of several variables (see E9~). I propose the problem to investigate in which sense it is possible to generalize Montessus de Ballore's theorem to several variables. 2. The two-variable case, ~ ( ~ i , ~ ) ; ~,~e~. We first generalize the definition of Pad6 approximants to the twovariable case. Let ~ ( ~ ) ~ ~ G~K~~ ~ be a formal power series and let ~/~ , ~0 , be a rational function in two variables ~I and ~ of type (~,$) , i.e. P is a polynomial in ~I and ~ of de~ree ~< ~ and ~ of degree ~ ~ . By counting the number of coefficients in P and ~ we see that it is always possible to determine P ~K~-O

and 0

so that, i f ( ~ f - P ) ( ~ ) ~

for (~,K)6 ~

, where ~

chosen subset of ~ x ~

K

~

, then

, the interpolation set, is a

with ~(We~)(~+$)+~(9+~)(9+~)-

elements. There is no natural unique way to choose ~ but it seems reasonable to assume thatI(~,K):~+K~W} c ~ and t h a t ( ~ , K ) ~ =>(~,W%) 6 ~ if ~ ~ } and ~ K . In this way we get a r a t i o n a i a p p r o x i m a n t P/@ of type (w, 9) to

corresponding

to

. With a s

table choice of

, P/Q

is unique [7 , Theorem 1.I~. The definition, elementary properties, and some convergence results have been considered for these and similar approximants in [9], ~0] and [7]. The possibility to generalize Montessus de Ballore's theorem has been discussed in [6],[73 and E11] but the results are far from being complete. PROBLEM I. In what sense can Montessus de Ballore's theorem b@ ~eneraliz~d to several variables? I% is not clear what class of functions ~ one should use. We consider the following concrete situstion. Let ~ = ~ / 6 , where is holomorphlc in the p o l y d i s c ~ = ( ~ , ~ ) : l ~ I < ~ , ~=~,~ ~ and & is a polynomial of degree 9 , ~(0) ~ 0 • By the method described above we obtain for every ~ a rational approxin~nt P W / Q w of type (~,~) to # corresponding to some chosen interpolation set ~ = ~w" In what region of ~ does P w / ~ converge to ~ ? Partial answers %o this problem are given in [7] and [11] (in the latter with a somewhat different definition of the approximants). If @ = ~ , explicit calculations are possible and sharp results are easy to obtain [7 , Section 4~. These show that in general we do not have convergence in{~: I ~ l < ~ , $=~,~\{~:e(~)~---O} . This proves that the general Analogue of the Montessus de Ballore's theorem is not true. It may be

477 added, that it is easy to prove - by just using Cauchy's estimates that there exist ration~l functions ~ $ of type (~,~) interpolating to ~ at the origin of order at least ~ + I and converging ifo=ay,

as

, to

in compact

subsets

A disadvantage, however, of ~ $ compared to the rational approximants defined above is that % ~ is not possible to compute from the Taylor series expansion of ~ (see ~ , Theorem 3.3~). In the one-variable case the proof of Montessus de Ballore's theorem is essentially finished when you have proved that the poles of the Pad~ approximants converge to the poles of ~ . In the several-variable case, on the other hand, there are examples ~ , Section 4, Counterexample 2];when the rational approximants P~/~ ~ do not converge in the whole region ~ ~: ~ ( , ) = 0 } in spite of fact that the singularities of P l ~ / ~ converge to the singularities of ~ / ~ . This motivates:

\|

PROB~

2. Und.er what c ondi,t.!ons does Q~

conver~e to ~

?

The choice @f the interpolation set ~t~ is important for the convergence. For instance, if $-----J and ~i_~_~_~_oo , we get convergence in ~ \ ~ ~: ~ ( ~ ) = 0 } with a suitable choice of ~ ET, Section 4~. On the other hand if we change just one point in ~ without violating the reasonable choices of ~ indicated in the definition of the rational approximants - we get examples [7, Section 4, Cotuuterexample 1],where we do not have convergence in any polydisc around ~ 0 . PROBLE~ 3. How is the convergence the choice of the interpolation s~t ~

P~/Q. ......~.. ~

influenced b~

.~

Since we do not get a complete generalization of Montessus de Ballore's theorem it is also natural to ask: PROBLEM 4. If, the sequence .o,frat!ona,l,,,approximants aces not conver~e t is there a subs equence t ~ t

c,onv~rges _t0 ~ ?

(Compare ~7, Theorem 3.4] ) l~inally, I want to propose the following conjecture.

and the interpolation set Corollary above. )

~

2~ and the case 9 =

is suitably chosen. (Compare Ell, ~

referred to just after PROBLE~ 2

478

REFERENCES I. B a k e r

C.A.

Academic Press, 2. d e

Essentials of Pad6 Approximants.

N e w York,

1975.

M o n t e s s u s

d e

B a I I o r e

fractions continues alg~briques.

R.

Sur les

- Bull. Soc•Math. France ~ 1902~

30, 28-36. 3- P e r r o n

O.

Die Lehre yon den Kettenbr~chen. Band II.

Stuttgart, Teubner, 4. G r a g g

1957.

W.B. On Hadamard's theory of polar singularities.

In: Pad6 approximants and their applications

-

(Graves-Morris,

P.R., e.d.), London, Academic Press, 1973, 117-123. 5. S a f f

E.B.

A n extension of Montessus de Ballore's theorem

on the convergence of interpolation rational functions. - J. Approx.T., 1972, 6, 63-68. 6. C h i s h o 1 m J.S.R., G r a v e s - M o r r i s

P.R. Gene-

ralization of the theorem of de Montessus to two-variable approximants. - P r o c . R o y ~ l Soc.Ser.A., 1975, 342, 341-372, 7- K a r 1 s s o n J., W a 1 1 i n H. Rational approximation by an interpolation procedure in several variables.rational approximation York, Academic Press,

8.

In: Pad~ and

(Saff, E.B. and Varga, R.S., eds.), New 1977, 83-100.

r o H q a p A.A. 0 CXO~HMOCT~ O606~eHR~X annpo~c~Many~ ~a~e Mepo~opSHax ~ y H E ~ . -~4aTez.cS., 1975, 98, 4, 563-577.

9. C h i s h o 1 m

J.S.R.

N

-variable rational approximants.

-

In: P a d 6 and rational approximation (Saff, E.B. and Varga, R.S., eds.), N e w York, Academic Press,

1977, 23-42.

IO. F o H ~ a p A.A. JIOEaKBHOe yC~OB~e O~O3HaqHOCTE aHa~I~T~lqeoK~X ~ y ~ Hec~o~X nepeMeHHax. - ~Te~.C6., 1974, 93, ~ 2,

296-313. 11. G r a v e s - M o r r i s

P.R.

Generalizations of the theorem

of de Montessus using Canterbury approximant. tional approximation Academic Press, HANS W A L L S

- In: Pad6 and ra-

(Saff, E.B. and Varga, R.S., eds.), N e w York,

1977, 73-82. Ume~

U n i v e r s i t y

S-90187 Ume~, Sweden

479

COMMENTARY BY THE AUTHOR In a recent paper A.Cuyt (A Montessus de Ballore theorem for multivariate Pad~ approximants, Dept. of Math., Univ. of Antwerp,,_ Belgium, 1983) considers a multivariate rational approximant ~/ to ~ where ~ and Q are polynomials of degree ~ ÷ ~ and ~ $ ~ , respectively, such that all the terms of P and Q of degree less t h a n ~ $ vanish. It is then possible to determine P and Q so that ~Qhas a power series expansion where the terms of degree ~+~+~ are all zero. Por this approximant P / Q she proves the following theorem where P / Q - P~ / Q ~ and ~ and ~ have no common non-constant factor: Let ]=F/6 where F i , holomorphic in the polydisc {Z:IZ~I(R~} and ~ is a polynomial of degree ~ , ~(0) ~ 0 , and asstm~e that ~ ( 0 ) ~ 0 for infinitely many ~ . Then there exists a polynomial Q(Z} of degree $

uch that converges ,~niformly to ~

u oo uo oo of{P,/Q4

on compact subsets of

[;~.lzi, I
0 , one possible approach to prove CR(E)>o

E -1-T-z

is

E,0(e(2

tO consider the set A point ~ is not in ~ if and only if the line passing through ~ and whose distance to the origin is I~I , misses the set . It is not hard to see C R(E)> o i f i f E has positive area. Uy [ 24] *) has recently shown that a set F has positive area if Bad only if there is a Lipschitz continuous function which is analytic on ~ \ F . so one might try to construct such a function for the set ~ , A related question was asbed by A.Beurling. He asked, if ~ and i f E has no removable points, then must the part of the boundary of the normal fundamental domain (for the universal covering map) on the unit circle have positive length? This was shown to fail in [26]. Finally, I would like to mention that I see no reason why C ~ ( E ) is not comparable to analytic capacity. In other words, does there exist a constant K With 4/K "C~(E)~ ~ ( E ) ~ ~ C ~ ( E ) ? If this were true, it would have application to other problems. ~or example-, it would prove that analytic capacity is semi-subadditive.

E

(E)>o

REFERENCES I. A h 1 f o r s L.V., B e u r 1 i n g A. Comformal invariants and f~uction-theoretic null sets. - Acta Math., 1950, 83, 101-129 2. P a i n 1 e v ~ P. Sur les lignes singuli~res des fonctions analytiques. -Ann,Fac.Sci. Toulouse, 1888, 2. 3, A h 1 f o r s L.V. Bounded analytic functions. - Duke Math.J., 1947, 14, 1-11. 4.

B m T y m ~ ~ H

A . F . A ~ J ~ Z T m ~ e c ~ e eM~CTI, MHoxec~'B B s s ~ a x Teolm~ ~ p m 6 ~ z e ~ . - Ycuex~ ~mTeu.HsyK,I967,22,~, I4I-I99. 5. z a 1 c m a n L. Analytic capacity and Rational Approximation - Lect.Notes Math., N 50, Berlin, Springer, 1968. 6. C o 1 1 i n g w o o d E.P., L o h w a t e r A.J. The Theory

of Cluster Sets. Cambridge, Cambridge U.P., 1966~ 7. B e s i c o v i t c h A. On sufficient conditions for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated singular points. - Proc.London Math. Soc°, 1931, 32, N 2, I-9. See [27] for a short proof. - Ed.

489 8. B ]~ T y m E B H

A.r.

lIpilMep MEtoxecT~t ZZO~ZZZTe~Z~Ot ~ m ~ ,

~ro

Hy~eBoM SN~Jt~TB~eoEo~ eNEOOTg.-~0~.AH CCCP, 1959, 127, 246-249. 9. G a r n e t t

J, Positive length but zero analytic capacity

-

Proc.Amer.Math. Soc., 1970, 24, 696-699.

l O . H B a ]K o B

~.~.

11. O r o f t o n

BS1SI~aI~MB MHozecTB lit ~ y H l ~ ] ~ .

M.,

"Hayza",I975.

M.W, On the theory of Local Probability.

-Philos.

Trans.Roy.Soc., 1968, 177, 181-199. 12. S y 1 v e s t e r J,J. On a funicular solution of Buffon's "Problem of the needle" in its most general form~ - Acts Math , 1891, 14, 185-205. 13. M a r s t r a n d

J°M. Fundamental geometrical properties of

plane sets of fractional dimensions.

- Proc.London Math.Soc.,

1954, 4, 257-302. 14. D e n j o y A° Sur lea fonctions analytiques uniformes ~ singularit~s discontinues. 15. X a B M H C O H

-CoR,Acad.Sci,Pmris,

C.H. 06 ~ a a ~ T M ~ e c E o ~

1909, 149, 258-260,

e~EOCT~ MHoxecTB, CoBue-

O~Ol) H e T p M B a a ~ O O T ~ ImSJta~KUX ~laccoB aHaJt~T~qecz~x ~ y m ~ c ~ ~ e ~ e ~Bap~a B npoMsBo~m~x 06~aOTSX. - MaTes.c6., 1961, 54,

J

~ I , 3-50. 16. I~ B a H o B

~.~. 06 aHa~Tsqecxo~ eKI~OOT~ ~J~He~h~x IE~O~eOTB.

--Ycnexa MaTea.Hay~, I962, I7, I43-I44. A.M. Analytic capacity and approximation problems. 17. D a v i e Trans.Amer.Math.Soc., 1972, 171, 409-444. 18. C a 1 d e r ~ n A.P. Cauchy integrals on Lipschitz curves and related operators. - Proc.Nat,Acad.Sci, USA, 1977, 74, 1324-1327 P.R, Schwarz's lemma and the Szeg'o kernel 19. G a r a b e d i a n function. -Trans.Amer.~lath,Soc=, 1949, 67, 1-35. 20. X a B ~ H B.H. l~aHa~Hue CB0~OTB8 asTerpa~oB Tana Koma a rapao-

~ecEa

Conp~eHHuX ~ m U ~ a ~ B 06~a0TSX C0 c~pe~seao~ rpaHa~e~. -~aTe~.c6., 1965, 68, 499-517. 21. X a B z H B.H., X a B ~ H c o H C.~. HeEOTOI~e o~eH~a aHa~ a ~ e c ~ o ~ eaEoc~z. - ~ o ~ . A H CCCP, 1961, 138, 789-792. 22. B e s i c o v i t c h A. On the fundamental geometrical properties of linearly measurable plane sets of points I. - Math.Ann.,

23.

1927, 98, 422-464. II: Math.Ann., 1938, 115, 296-329. B e s i c o v i t c h A. On the fundamental geometrical properties of linearly measurable plane sets of points I I I . -

Math.Ann.,

1939, 116, 349-357. 24. U y N. Removable sets of analytic functions satisfying a Lipschitz condition.

25.

e d e r e r

-Ark.Mat.,

1979, 17, 19-27.

H. Geometric measure theory. Springer-Verlag,

Bar-

490 lin, 1969. 26° N a r s h a 1 1 D°E. Painlev~ null sets, Colloq, d'Analyse Harmonique et Complexe. Ed,: G.Detraz, L,Gruman, J.-P.Rosay. Univ. Aix-Marseill I, Marseill, 1977. C.B. HpOcTOe ~oz,asa~em~cTBo ~eopeuu o0 y C T I m ~ M ~ X ooo6e~ooT~ a E a ~ e c z ~ x ~ m o ~ , ~o~.~e~Bop.mo~x yO~OB~ ~Lmn=az~a. - 8 a ~ . H a ~ . o e a . ~ 0 M N , I 9 8 I , I I 3 , I 9 9 - 2 0 3 .

27. X p ~ ~ e B

DONALD E. MARSHALL

Department of MathematicspUniversity of WashingtonjSeattle, Washington 98195 USA

Research supported in part by National Foundation Gramt No MCS 77-01873

491

8.16. old

ON PAINLEV~ NULL SETS Suppose that

E

neighbourhood of s e t

is a compact plane set and that

~

. i set is called a

~

is an open

P a i n 1 e v e

n u 1 1

(or 2.N. set) if every function regular and bounded in ~\ E

can be analytically continued onto that

E

has

z e r o

E

. In this case we also say

a n a 1 y t i c

c a p a c i t y.

The problem of the structure of P.N. sets has a long history. Painlev~ proved that if dorff) measure

E

has linear (i.e.

zero, then E

~ - dimensional Haus-

is a P.N. set,

this result was first published by Zoretti

though it seems that Painleve' " s theorem

[I].

has been rediscovered by various people including Besicovitch who proved that if outside

E

, and if

~

is continuous on E

E

[2]

, as well as regular

has finite linear measure,

analytically continued onto

then

~

can be

. Denjoy [3] conjectured that if

lies on a rectifiable curve, then E

E

E

E

is a P.N. set if and only if

has linear measure zero. He proved this result for linear sets.

Ahlfors and Beurling

[~

proved Denjoy's conjecture for sets on ana-

lytic curves and Ivanov [ 4

for

Sets on sufficiently smooth curves.

Davie [6] has shown that it is sufficient to prove Denjoy's conjecture for

C 4 curves. On the other hand Havin and Havinson [ ~

and Ha-

vin [8] showed that D e n j o y ' s c o n j e c t u r e follows if the Cauchy integral operator is bounded on ~ for ~ curves. This latter result has now been

proved by Calder6n

[gB so that Denjoy's conjecture is

true. I am grateful to D.E.Narshall [10] for informing me about the above results. Besicovitch

[11] proved

that every

compact set

linear measure is the union of two subsets

~I

'

~

~

of finite

. The subset E~

lies on the union of a finite or countable number of rectifiable Jordan arcs. It follows from the above result that set unless

~4 has linear measure zero. The set

~4

~

is not a P.N.

on the other

hand meets every rectifiable curve in a set of measure zero, has projection zero in almost all directions and has a linear density at almost none of its points. The sets tively

r e g u 1 a r

and

E4 and

~

i r r e g u 1 a r

were called respecby

Besicovitch

[11]. Since irregular sets behave in some respects like sets of measure zero I have tentatively conjectured

[12, p.231]

be P.N. sets. Vitushkin

[14] have given examples

[13] and Garnett

that they might

of irregular sets which are indeed P.N. sets, but the complete conjecture is still open. A more comprehensive conjecture is due to Vitushkin

[15 , p . 1 4 ~

492 He conjectures that

E

is a P.N.

set if and only if

~

has zero

projection in almost all directions. It is not difficult to see that a compact set

~

is a P.N. set

if and only if for every bounded complex measure distributed on E

,

the function

(I)

E is unbounded outside

E ~). Thus

E

is certainly not a P.N. set on E

if there exists a positive unit measure

such that

f t~-gl E

is bounded outside

E

i.e. if

E

has positive linear capacity [16,

p.73]. This is certainly the case if respect to some Hausdorff function

E

has positive measure with , such that

9,-----i-~ < oo 0

([17]). Thus in particular

E

is not a P.N. set if

E

has Haus-

dcrff dimension greater than one. While a full geometrical characterization of P.N. sets is likely to be difficult there still seems plenty of scope for further work on this intriguing class of sets.

RE FEREN CE S I. Z o r e t t i

L.

Sur les fonctions analytiques uniformes qui

possedent un ensemble parfait discontinu de points singuliers. J.~th.Pures

Appl.,

-

1905, 6, N I, 1-51.

2. B e s i c o v i t c h A. On sufficient conditions for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated singular points. - Proc.London ~ath. Scc°, 1931, 32, N 2, I-9. 3. D e n j o y ° ~

gularltes

A.

Sur les fcnctions analytiques uniformes a sin-

discontinues.

- C.R. Acad. Sci.Paris,

1909, 149, 258-

-260. ~)

See E d . n o t e a t

t h e end o f t h e s e c t i o n .

- Ed.

493 0d aHS~HT~eoEo~ eMEOCTM MHOXeOTB, COBpasm~x IuIaOOOB a ~ T H ~ e c E ~ X ~yHEI~ aeMMe IUBap~a B ~ p o H s B O ~ X 06zaoT~x. - NmTeM.c6., 1961, 54, I, 3-50. 5. H B a H 0 B ~.~. 0 lauoTese ~aHxya. - Ycnex~ MaTeM.HayK, 1964, 18, 147--149.

4. X a B H H C o H

C.~.

MeoTHO~ H e T p H B H ~ H O O T I

6. D a v i e

A.M.

Analytic

capacity and approximation

problems.

1972, 171, 409-444. X a B ~ H C O H C.H. He~oTop~e o~e~Fa a~a- ~ o F ~ . A H CCCP, 1961, 138, 789-792.

-Trans.Amer.Math.Soc.,

7. X a B ~ H ~T~ecEo~

e~OCTH.

8. X a B H H

B.H.

HEec~

B.H.,

F p ~ e

conp~eB~x

M~TeM.C6., I965, 9. C a I d e r ~ n

B o6~aCT2X CO c n p ~ e M o ~

A.P.

related operators.

Cauchy integrals

on Lipschitz

-

curves and

USA, 1977, 74, 1324-1327. Preprint, 1977.

- Proc.Natl.Acad.Sci. D.E.

The Denjoy Conjecture.

11. B e s i c o v i t c h perties

rp~e~.

68, 499-517.

10. M a r s h a 1 1

Ann.,

CBO~CTBa HHTeI~S2IOB T~na Ko,~ ~ rapMo-

~

A.

On the fundamental

of linearly measurable

geometrical

pro-

plane sets of points I. - Math.

1927, 98, 422-464. II: Math .Ann., 1938, 115, 296-329.

12. H a y m a n

W.K.,

K e n n e d y

Vol. 1. London - N.Y., Academic

13. B H T y m E ~ H

A.~.

~ep

HyxeBO~ a ~ a ~ T ~ e c E o ~

-249. 14. G a r n e t t

Positive

TeOp~ np~6~eH~.

16. C a r 1 e s o n

--~oEx.AH

~ , Ho 1969, 127, 246capacity.

-

AHSJH~TEeCEB~ eMNOCTB MHo~eCTB B s8~aqax - Ycnex~ MaTeM.HayE, I967, 22, ~ 6, 14I-I99.

L.

Selected

problems •

0.

A.F.

on exceptional

N 13, Toronto,

Potentiel

Medded.Lunds~Univ.~t.S~.,

E ~ H

CCCP,

length but zero analytic

.

•

1935,

sets.

-

1967.

Van Nostrand, °

d'equ~l~bre

sembles avec quelques applications

18. B ~ T y m

1976.

Press,

A.r.

Van Nostrand Math.stud.,

17. E r o s t m a n

Functions

1970, 24, 696-699.

Proc.Amer.~'2th.Soc.,

15. B ~ T y m E Z H

Subharmonic

~Ho~ecTBa H o ~ O ~ T e ~ H o ~

eMEOCT~.

J.

P.B.

•

et capac~te

des en-

a la theor~e des fonctions.

-

3, 1-118.

06 O~HO~ Bs~a~e ~ a ~ y a .

- HsB.AH CCCP,

cep.MaTeM., 1964, 28, ~ 4, 745--756. 19. B a ~ ~ c E ~ ~ P.8. HecEox~EO s a M e ~ a ~ o6 o ~ p a H ~ x e ~ x aHa~ecE~x ~y~n~x, npe~cTaB~M~x ~HTe~pa~oM T~na Kom~-CT~T~eca. -C~6.MaTeM.~., I966, 7, ~ 2, 252--260. W.K.HAY~,~N

Imperial

College, Department

Mathematics,

of

South Kensington,

L o n d o n SW7 England

494

EDITORS' NOTE. As far as we know the representability of a 1 1 functions bounded and analytic off E and vanishing at infinity by "Cauchy potentials" (I) is guaranteed when E has finite Painlev~'s length whereas examples show that this is no longer true for an arbitrary E (~18],[19]). We think THE QUESTION £f existence of potentials (I) bounded in

~ \~

(provided

E

is no t a P.N. set) i_~s

one more interestin ~ problem (see also § 5 of [ 4]).

495 8.17.

ANALYTIC CAPACITY AND RATIONAL APPROXIMATION

old

Let

all

E

be a bounded subset of C ^and functions ~ in ~ analytic on ~ \ m

I~1 < 4 on~O 3

.

Put

A(E,O

B(E,~)

= { ~ ~ BQE,~) "

be the set of

and with ~(co)= 0 , ~ is continuous

. The number

is called the analytic capacity of

E

. The number

~A(E,O ~--~ is called the analytic C-capacity of E • The analytic capacity has been introduced by Ahlfors [I] in connection with the Painlev~ problem to describe sets of removable sin~alarities of bounded analytic functions. Ahlfors [I] has proved that these sets are characterized by ~(E) = 0 . However, it would be desirable to describe removable sets in metric terms. CONJECTURE I. A compact set E capacity iff the projection of E

,

E cC

, has zero anal~tic

onto almost evelV direction has

zero length ("almost every" means "a.e. with respect to the linear measure on the unit circle). Such an E is called irregular provided its linear Hausdcrff measure is positive. If the linear Hausdor~f measure of E is finite and ~(~) = 0 then the average of the measures of the pr@jections of E is zero. This follows from the Calderon's result [2] and the well-known theorems about irregular sets (see [ 3], p.341-348). The connection between the capacity and measures is described in detail in [4]. The capacitary characteristics are most efficient in the approximation theory [ ~ , [6] ,[7], [8S. A number of approximation problems leads %o an unsolved question of the semiadditivity of the analytic capacity:

%(EUF) ~ c [ ~ ( E ) + ~ ( F ) ]

,

496 where C is an absolute constant and E, F are arbitrary disjoint compact sets. Let A (K) denote the algebra of all functions continuous on a compact set K , K:-C , and analytic in its interior. Let ~ ( K ) denote the uniform closure of rational functions with poles off and, finally, let ~o~ be the inner boundary of ~ , i.e. the set of boundary points of ~ not belonging to the boundary of a component of C \ K . Sets K satisfying A ( ~ ) = ~ ( ~ ) were characterized in terms of the analytic capacity [6]. To obtain geometrical conditions of the approximability a further study of capacities is needed. CONJECTURE 2. If ~ ( ~ ° ~ ) = 0

then

A(~) =~(K)

.

The affirmative answer to the question of semiaddivity would yield a proof of this conjecture. Since ~ ( [ ) = 0 provided ~ is of finite linear Hausdorff measure this would also lead to the proof of the following statement. CONJECTURE 3. If the linear Hausdorff measure of ~o~

( K

b e i n g a compact ..sub.s.et o f

~.,

) then

is zero

A(K] --C(K). f

The last equality is not proved even for K s with ~°K of zero linear Hausdorff measure. It is possible however that the semiadditivity problem can be avoided in the proof of CONJECTURE 3. The semiadditivity of the capacity has been proved only in some special cases ([9], [10-13] ), e.g. for sets ~ and ~ separated by a straight line. For a detailed discussion of this and some other relevaal% problems see [14] o

REFERENCES I. A h 1 f o r s L.V. Bounded analytic functions. - Duke Math.J., 1947, 14, 1-11. 2. C a 1 d e r ~ n A.P. Cauchy integrals on Lipschitz curves and related o p e r a t o r s . - Proc.Nat.Acad.Sci., USA, 1977, 74, 1324-1327. 3. H B a H 0 B ~.~. B a p ~ a m ~ MHo~eCTB ~ ~ y H E m ~ , M., "HayEa", 1975. 4. G a r n e t t J. Analytic capacity and measure. - Lect.Notes Math., 297, Berlin, Springer, 1972. 5. B ~ T y m E ~ ~ A.r. A H a X ~ T ~ e c E a ~ eMEOOT~ ~ o ~ e c T B B sa~avax

497

Teop~npEdx~meH~. 6. M e ~ ~ H ~ E 0 B

- Ycn~xHMaTem.HayE, 1967, 22, ~ 6, 141-199. M.C., C ~ H aH ~ ~ C.0. Bonpoc~ T e o p ~

np~6~m~eH~a ~ y ~ z ~ O~HOrO K O M ~ e E C H O P O HepeMeHHOrO. - B EH.: CoBpeMem~ge n p o 6 x e M H M a T e M a T ~ T.4, MOCKBa, BHHETH, I975, I43-250, 7. z a I c m a n L. Analytic capacity and Ration~l Approximation. - Lect.Notes Math., 50, Berlin, Springer, 1968. 8. G a m e I i n T.W.. Uniform algebras, N.J., Prentice-Hall, Inc. 1969. 9. D a v i e A.M. Analytic capacity and approximation problems. - Trans.Amer.Math.Soc., 1972, 171, 409-444. IO. M e x ~ H ~ E 0 B M.C. 0~eHEa ~HTerpaxa K o ~ no a~a~TH~ecEo~ EIDEBO~. -- MaTeM.Cd., 1966, 71, ~ 4, 503--514. II. B ~ T y m E ~ H A.F. 0 ~ e ~ a ~ T e ~ a ~ a E o m ~ . - MaTeM.c6., 1966, 71, ~ 4, 515--534. 12. ~ ~ p 0 E 0 B H.A. 06 O~HOM CB0~CTBe a H a ~ T E e c E o ~ eMEOCTE. -BeCTH~E ~IY,cep.MaTeM., ~ex., aCTpOH., 1971, 19, 75-82. 13. ~ ~ p 0 E O B H.A. HeEoTop~e 0BO~CTBa aHa~IETEeCEo~ eMEOCT~. -BeCTH~E~I~J, cep.MaTeM., MeX., aCTpOH., 1972, I, 77--86. 14@ B e s i c o v i t c h A. On sufficient conditions for a function to be analytic and on behaviour of analytic functions in the neighbourhood of non-isolated singular points. - Proc.London Math.Sec., 1931, 32, N 2, I-9. A. G. V I T U S H K I N

(A.r.BHTY~Gm)

M. S.MEL 'NIKOV

(M.C.MF/6m~0B)

CCCP, 117966, MOCKBa, y x . B a B ~ o B a , 42, MEAH CCCP CCCP, 117234, MOCEBa, Mexa~aEo-MaTeMaT~ecE~ (~m~ym_~TeT MOCEOBCEOrO yH~BepC~TeTa

498 8.18. old

ON SETS OF

ANALYTIC CAPACITY ZERO

Let K be a compact plane set and Aoo (K) the space of all functions analytic and bounded outside K endowed with the sup-norm. Define a linear functional ~ on ~ o o ( K ) by the formula

I~-$ with

~

>

ttM3/J6 {1~ I : ~ ~ K ]

The norm of

L

is called

t h

e

a n a 1 y t i c c a p a c i t y o f k . We denote it by ~(K). The function ~ is invariant under isometries of C . Therefore it would be desirable to have a method to compute it in terms of Euclidean distance. E.P.Dolzenko has found a simple solution of a similar question related to the so-called %-capacity, C1]. But for ~ the answer is far from being clear. I would like to draw attention to three conjectures. CONJECTURE 1. There exists a positive number a%7 compact set

where

~(K,~)

the line through

such that for

K

T denotes the l e n g t h o f the p r o ~ e c t i o n o f K

0

and ~ ~ T

CONJECTURE 2. There an,y compact set

C

onto

•

exists a positive number C

such that for

K

Y(K)~c I ~ (K,~)#~(~). T These CONJECTURES are in agreement with known facts about analytic capacity. For example, it follows immediately from CONJECTURE I that ~ ( K ) > 0 if ~ lies on a continuum of finite length and has positive Hausdorff length. In t~rn, CONJECTURE 2 implies that ~(K)=0 provided the ~avard length of K equals to zero. At last, let be a set of positive Hausdorff M-measure (a surveE of literature < co on the Hausdorff measures can be found in ~3] ). If ~ ~(~2~"

~t

O

then the Favard length of

K

is positive. This ensures the existence

499

K~cK

of a compact K4 , function

~,

~(K0>O

, such that

~ ( ~ ) = I. - ~

, is continuous on

the Hausdorff ~-mea~ure. Hence ~(K) >~ ~ (KO > 0 easily follows from CONJECTURE 2. CONJECTURE 3. Pot an E increasing function with

I ~(~)/~ ~----oo

and

0

there Qxists a set

K

and the ,~ being which also

~ :(0,+°°)--~(0,+°°) satisfyin~ ~ ( K ) > 0

.

To corroborate this CONJECTURE I shall construct a function

~nd a set E

such that

~(~=0,

~(E)>O

but ~ ( E ) - O •

Assign to any sequence 6 ~ { 8 . } , ~ev~O (~), a compact set E(8). • Namely, let ~a(8) [0,4J . If ~. is the union of disjoint segments , of length^j ~.(8) then ~.+~C~) is the union of ~n sets ~L\ ~n ^j , A n being the interval of length I

~(6)~-8~)

and let ~

concentric with the segment

be a constant ~e~uenoe, ( ~ ) ,

=C

~

. Put

. ~inally let

E=

= E(~°). Tt is known (see [~]) that ~(E) = 0 existence of a function ~ such that

~ (~(13)=0 and t-,-o

. This ~mplies the

~'(E(6))0 . ~ will be used as a generic notation for compact subsets of ~ . For any locally integrable (complex-v~lued) function ~ on ~ we denote by

its mean value over the disc ~ ( ~ , , ) = ~ : ~ will stand for the class of all functions integrable to the power ~ and satisfy

~ ~

on

I~-~I~0 , and ~ , ~ > 0 ,

503

where the infimum is taken over all sequences of sets M ~ , M ~ c with ~ M ~ ~ - ~/2 . sie ist des weiteren stetig und gen'ugt sogar einer H~ider-Bedingung, falls R e ~ > 0 ist. Wenn im Band 0 > B e ~ > ~ - ~/2 Eigenwerte des genannten Operators existieren, dann besitzt die Ausgangsrandwertaufgabe verallgemeinerte L~sungen, die in einer beliebigen Umgebung des Eckpunktes des Konus unbeschr~akt sind, und yon einer Regularitat nach Wiener kann man selbst . bei einem konisohen Pumkt mioht reden. Es zeigt sioh ([15], [ 1 6 ] ) , ~ wir auf solche unerwarteten Erscheinungen schon bei stark elliptischen Gleichungen zweiter Ordnung mlt konstanten Koeffizienten =--a ~ -

512 stolen, falls nicht alle Koeffizienten reell sind; In [16] (eine ausfS/L~liche Darstellun~ erscheint evtl- in ,,MaTe~aTN~eoE~ O6Op~NE") wird das homogene Dirichletproblem au~erhalb eines d'6nnen Konus l'~'~=(t~ Z-'~E~ ~: ~,,.)' O~ U.,~I~ q~untersuoht, wobei ~ ein kleiner _ F,, (. )~' iiJ. rTM ~-i ~-I "J ~ ~rv-, farame~er, CLT~---- ~.~ e IK : ~ ~ e u I i ~nd w ein Gebiet im IK ist.

~s wir~ ~ie ~,~su~ ~(~)---I~IX(~)~(£,~/I~I)

aes sta~k

elliptischen Systems Qz~ ( ~ ) ~ (6~ ~C) = o betrachtet, wobei ~ ( ~ ) eine ~atrix mit homogenen Polynomen der Ordnung ~ als Elementen und ~ (6) ----0 (~) f'/r ~-~+ 0 ist. Hauptergebnis ist eine asymptotische ~ormel fur den Eigenwert k (£) , welche f't~r den eimfachsten ~all der Gleichung (6) die Gestalt •

X (~)= £n-s{ _ _ .,- z I ~

ca,pp.(D~t,O)

.n

.(~-a)/~

]

÷O(1)

(,~) (d,et; I~iKII~,K-,)

]

( (~ei: II ~, ~eX > (2-~)/2

so ~ m e n , ~a9 ~ e U~leio~u~

erfullt ist. Im Fall

h(2)= t21 ~o~21)-'(~ o0)) +

f~

~=3

gilt

~ - - +0.

Folglich erf~It j ede verallgemeinerte LBsung die H~Ider-Bedimgung, falls der 5ffnungswinkel des Konus K~ genugend klein ist. Es ist nicht ausgeschlossen, da~ die Forderum~ nach einem kleinen ~ffnungswinkel unwesentlich ist. Dies ist gleichbedeutend mit folgendem Satz HYPOTKESE 5. F ~

~=3

b_i~en elliptisehen Operator gular math Wiener.,

ist ein konischer Punkt fur einen belie-

P~(D,) nit,komplexen

Koeffiziente ~ re-

513

F~tr den biharmonischen Operator im ~ und fur die Systeme yon Lam~ und Stokes im --]R~wurden derartige Ergebnisse in "-[17J, "-[18] erhalten,

REFERENCES

I. W i e n e r N. The Dirichlet p r o b l e m - J,Math. and Phys~ 1924, 3, 127-146~ 2. W i e n e r N~ Certain notions in potential theory - J~Math and Phys. 1924, 3, 2 4 - 5 1 3. ~I a H ~ ~ c E.M. YpaBHe~S BTO10OrO I I O 1 0 S ~ ~ J U ~ n T ~ e c ~ O ~ O ~ na-

pa6o~m~eci¢oro T~a, M., Hayz,a, 197I. 4. M a 3 ~ ~ B.r. 0 n O B e ~ e H ~ B 6 ~ S Z r p S H ~ H pemeH~R saXaqR ~ p s x Jle ~ 6 R P s p M o ~ , e o E o r o oIIepSTOpS. - ~oEJI.AH CCCP, 1977, I8, ~ 4,

15-I9. 5. M a 2 ~ s B.r. 0 !DSI~JI~pHOOTm H8 I~H~I~8 IDSmSHR~ @JI~IITBRS01~X ylmBHe~mi~ ~ KoH~olxmoro OTO6paxe~ms. - ~or~I.AH CCCP, I963, 152, 6, I29V-I300.

6. M a s ~ s B . r . 0 ~oBe~eH~ ~ 6 ~ s rlmHs~u peme~s sa~a~s ~ p s x ae ; ~ s ~ z ~ n ~ a ~ e o ~ o ~ o ypa~se~as B~OpO~O nops;m~a B ~ B e p r e H T S O ~ ~OpMe. - ~ a ~ e a . s a a e ~ F m , 1967, ~ 2, 209-220. 7. M a s z s B . r . 0 ~ e n p e p u z ~ o c ~ z B r l m S a ~ o ~ ~ o ~ e pemeHz~ r m a s a J I ~ H e ~ X @ ~ a i i T ~ e c F ~ x ypaBHeHZ~. - BeOTH.~[rY, 1970, 25, 42-55 (nonlmBF~: BeO~H.~rY 1972, I, 158). B. G a r i e p y R., Z i e m e r W,P. A regularity condition at the boundary for solutions of quasilinear elliptic equations. Arch,Rat.Mech.Anal., 1977, 67, N I, 25-39. 9. E p o ~ ~ ~.H., M a ~ ~ s B.r. 06 O T O y T O T ~ ~enpepB~ooT~ Henl~elmBHOCTZ uo re~epy pemeHz~ K B S S ~ H e ~ x ~m~Tz~ec~mx ypB~H e H ~ B6JL~3~ HepSryJlSpHOH TOV~ER. - T I ~ MOCE.MaTBM.O--BS, 1972,26,

73-93. I0. H e d b e r g L. Non-linear potentials and approximation in the mean by analytic functions. - Math,Z,, 1972, 129, 299-319, II.A d a m s D.R., M e y e r s N. Thinness and Wiener criteria for non-linear potentials. - indiana Univ,Math.J., 1972, 22, 169197. I2. M a s ~ s B.T., ~ o H ~ e B T. 0 p e r w ~ q p H o o ~ no B ~ e p y rlmKZ~HO~ TOPAZ ~ S u o ~ r a l X ~ O H ~ e o x o r o onepaTopa. - ~o~.Bo;az,.AH,

514

1983, 36, 2 2. I3.

m a z ' y a V~G. Behaviour of solutions to the Dirichlet problem for the biharmonic operator at the boundary point, Equadiff IV, Lect.Notes Math , 1979, 703, p 250-262

14. E o H ~ p a T ~ e B B.A. ElmeB~e s a ~ s ~ ~ smmn~ecF~x YlmB-HOHI~ S 06JIaCTSX C EOH~0CF~M~ ~Jl~ yraOBHMH TO~I~SMH. T I ~ MotE. aa~eM.o-Ba, 1967, I 6 , 209-292. 15. M a s ~ s B.T., H s 3 8 p 0 B C.A., H a a M e H e B C ~ m ~ B.A. 0TCyTCTB~e T e o p e ~ TmnS de ~ o p ~ s ~as C~a~HO s a a ~ T ~ e c z ~ x y p S B H e H ~ O KOMII~IeEoHHM~ l ~ o s ~ u a e H T a ~ . - 3aTr.Hayq.OeM~H.ZOMM, I982, IIS, 156-I68. 16. M a s ~ s B.F., H a s a p o B C.A., I I a a ~ e H e B C ] ~ ~ B.A. 06 o/o~opo/~ux p e m e H ~ x sa~a~z Jl~p~x~e Bo B~e=HOOT~ TOH~O~O ~¢o]¢~0a. -/~oI~a.A~ CCCP, 1982, 266, • 2, 281-284. 17. M s s ~ B.F., H a a ue H e Be ~ ~ ~.A. 0 n l ~ a ~ n e ~ m E csw~Ma ~ 6aral~OHa~eCl~OrO ypaBHeHas B 06Y~aGTB C EOHJ~NOCI~M~ TO-~ . - 14SB.BY3oB, 1981, ~ 2, 52-59. 18. M a s ~ s B.L, H a a m e H e B C ~ a ~ B.A. 0 CBO2Cr~ax l~m e H ~ TpeXMepH~X sa~sq T e O p ~ ynlmjrocT~ ~ r ~ p O J ! ~ H ~ K ~ B O6aSCTSX C ~SOaSpOBaHHHM~ OC06eHHOOTm~S. -- B 06. : ~ H S S ~ E 8 O]UtOmHO~ c p e ~ , HoBoc~6spc~, 1981, BUn.50, 99-121.

V. G.MAZ'YA

(B.r.MAS~)

CCCP, I98904, JIelmHrlm~, He TpO~mOl~ea, ~ e H g H r l ~ o ~ m ~ ~ooy~81oc TBSHHR~ ~H~BepC~TeT,

MaTeMaTaXo-aexa~eo~caJl

~a~vev

515

THE EXCEPTIONAL SETS ASSOCIATED WITH THE BESOV SPACES

8.21.

For ~ real and 0 < p , ~ < o o , we will use Stein's notation P~ for the familiar Besov spaces of distributions on ; see IF] and [S~ for details. The purpose of this note is to generally survey and point out open questions concerning the general problem of determining all the inclusion relations between the classes ~ ~, $ ~>0~ of exceptional sets naturally associated with the spaces A~ for various choices of the parameters &,p )~ ; c.f.[A~S], These exceptional sets can be described as sets of Besov capacity ze-

AA

on

~)~

l~l~p¢

some fixed smooth dense class in the spaces

the n o = (q~si-no=) of ~

is extended to all subsets of

E~B~p~

iff A ~ p ~ < E )

bitrar~ compact ~rite A ~ s

set

K

such

, hold. Now when 4 ~ p,~ < OO , there is quite a bit that can be said about this problem. First of all, one can restrict attention to ~ ~ 4 ~/& . Functions in A ~ for p > ~ / A are all equivalent to continuous functions and hence AA, p, $ (E) > 0 iff E ~ . Continuity also OCCURS, f o r example, when ~ - I'l,/& and ~= 4 . Secondly, i n t h e range 4 < p .< ~/~., ~< ~ co , there appear to be presently four methods for obtaining inclusion relations. They are: I. If

~

C ~j~

( continuous embedding), then clearly

. Such e~Bbeddings but not ve~ypoften. However, since & . ~ >0 , with ~ denoting the usual class of Bessel potentials of ~P functions on ~ (see [ $] ), and since the inclusion relations for the exceptional sets associated with the Bessel potentials are all known [AM], it is easy to see that A>,%, s co)

~(~) >0 ; 2) the integrals

Ak=kl g

are finite. Putting

, consider entire functions of

A o=

- 2

Z

:

(.-kI 0

00

~k

and

wC ®) e

~et, finny, {~k}4

(o < l~kl . 0 is a ~ K , L =) -set (though we know it is when C= 0 or when E satisfies (c)). All this is closely connected with our PROBLE~ (or better to say with its slight modification). DEFINITION. I) A Lebesgue measurable function q on the line is said to be ~-s t a t i o n a r y on the set E~ E c~, if there are functions ~4 ~ . . - , ~ ~ W ; (~) (i.e ~ ( ~ ) , absolutely continuous and with the L¢(~) -derivative) such that

~ I E =~41E, ~, 2) A set t y

S~

~IE, .... ~-~1

E, (or E

is said

E(S~)

~ H~(IR),

~IE have

IE

O.

t

r o p e r -

) if ~ ~-stationary on

E ~

~-=-0.

It is not hard to see that

E ~ C~s E, E e t C ) ,

l~es E > O ~

Ee(S~),

~=~,2,...

co

and that if

E ~ ~-i S t

then

E

is a (k, ~ k )

-set for eve-

ry semirational k [3]. moreover if there exists a ~ H 2 ( ~ ) ~ ~ @ 0, stationary on the set E , then E is not a (K,~k) -set for a semirational k (which may be even chosen so that k agrees with a linear fmnction

on (-co,O)

).

Another circle of problems where ~ -stationary analytic functions emerge in a compulsory way is connected with

539

3. Jordan operators. We are 6oing to discuss Jordan operators (J.o.) ~of

the form

~+~ where ~ is unitary, ~ =O,~G-Q~ (in this case we s a y T is of order ~ ). It is well known that the spectrum of any s u c h ~ lies on ~ so that ~ is invertible Denote by ~ ( ~ ) the weakly closed operator algebra spanned by ~ and the identity I We are interested in conditions ensuring the inclusion

T-' ~ P,, (T).

(**)

EXAMPLE. Let E be a Lebesgue measurable subset of T and be the direct sum of (~+I) copies of L ~ ( T \ E ) The operator

I = Y( E,~)defined by the

I

( ~ 4 ) × ( ~ * {)

H

-matrix

~. I

.

( ~ being the operator of multiplication by the complex variable is a J.o. of order ~ . It is proved in [3] that

J-'e g(l)

< :- E ~(S;).

Here ( ~ ) denotes the class of subsets of T defined exactly as C8~) in section 2 but with ~ replaced by the class of all functions absolutely continuous o n ~ . The special operator I - I c E ~ ) is of importance for the investigation of J.o. in general, Namely [3],if ~ is our J.o. (~) of order ~ and ~ stands for the spectral measure of ~ then

•herefore i~ E e ~=,ACS'~)

( ~ particular ~ ~eS E >0 ~d

E ~ CC) ) then (**) holds whenever ~ U ( E ) 0 Recall that for a unitary operator T (i.e. w h e n T = ~ Q =

0

540 in (*)) the inclusion (~*) is equivalent to the van lshing of ~ on a set of positive length A deep approximation theorem by Sarason [61 yields spectral criteria of (**) for a normal T . Our questions concer~ing sets with the property ~ and analogous questions on classes ( ~ ) , ( ~ ) are related to the following difficult PROBLEM: which spectral condltions ensure (**~ for T ~ ~ ~ ~ is normal and .....~ is a nilpotent com~ntin~ with ~ ?

where

REPERENCES 1. X p y m e B C.B. Hpo62eMa O~HoBpeMeHHO~ annpoEc~Ma~EE z cT~paH~e Oco6eHHOCTe~ ~Hwerpa~oB T~na Kom~. - T p y ~ MaTeM.EH--Ta AH CCCP, I978, IS0, I24-195. 2. E p ~ K E e B., X a B ~ ~ B.H. H p ~ R ~ Heonpe~e~H~ocT~ onepaTopoB, nepecTaHOBO~HRX CO C~BErOM I. -- 8anEcE~ Hay~H.Ce~H. ~0M~, I979, 92, I34-170; H. - ibid., I98I, II3, 97-I34. S. M a E a p o B H.F. 0 CTa~zoHapH~x ~yR~IE~X. - BeCTH~K ZIY (to be published). 4. H a v i n V.P., J o r i c k e B. O n a class of uniqueness theorems for convolutions. Lect,Notes in Math , 1981, 864, 143170. 5. X a B E H B.H. H p z H ~ H Heonpe~e2~HHoc~z ~x~ O~HoMepHax H o T e ~ a 2OB M.P~cca. - ~ o E ~ . A H CCCP, 1982, 264, ~ 8, 559-568. 6. S a r a s o n D. Weak-star density of polynomials. - J.reine umd a~ew.Math., 1972, 252, ~-15.

V. P. H A V I N

(B.H.XABHII)

CCCP, 198904, ZeH~Hrlm~, IleTpo~mope~, JleR~HrpajIcE~ r o c y ~ a p C T B e H ~ yH~Bep-CE TeT M a T e M a T ~ K o - M e x a H ~ e c E ~ ~BEyJIBTe T

B. JSRICKE

Akademie der Wissenschaften der DDR Zentralinstitut f~ur Mathematik und Mec~ DDR, 108, Berlin Mohrenstra~e 39

N. G. M A K A R O V

CCCP, 198904, JleHEHrpa~, HeTpo~Bopen, JIeHEHrpaJIc~zi~ rocy~apc TBeHHNI~ yH~BepOZTeT MaTeMaTI~Eo-~exaH~ eCKEI~ ~Ey2BTe T

(H.LMAKAPOB)

541

PROBLEM IN THE T ~ O R Y

9.5.

OF FUNCTIONS

In 1966 I published the following theorem: There exists a constant @ • 0 nomials Q

such that a%7 collection of poly-

of the form

k

)

with

is a norma ! famil,y in the complex plane. See Acta Math., 116 (1966), pp.224-277; the theorem is on page 273. This result can easily be made to apply to collections of polynomials of more general form provided that the sum from I to oo in its statement is replaced by one over all the non-zero integers. One peculiarity is that the constant @ > 0 r e a I I y m u s t b e t a k e n quite small for the asserted normality to hold. If ~ is I a r g e e n o u g h, the theorem is f a I s e. The results's proof is close to 40 pages long, and I t h ~ k very few people have been through i%. Canoone find a shorte r and clearer proof?

This is my question.

Let me explain what I am thinking of. Take amy fixed ~ , 0 < 2 < < ~

and let ~

be the sllt domain

CbO

C\ U If

~

i s any p o l y n o m i a l , w ~ t e

By d i r e c t harmonic e s t ~ t i o n much t r o u b l e t t ~ t

in

~3

one can f i n d w i t h o u t too

542

_~

' 7 + ~ ~'

where K~ (~) depends only on ~ and ] . (This is proved in the first part of the paper cited above. ) A natural idea is to try to obtain the theorem by making ~ ~ 0 in the above formula. This, however, cannot work because Kf (~) tends to oo as ~ - ~ 0 whenever $ is not an integer. The latter must happen since the set of integers has logarithmic capacity zero. For polynomials, the estimate provided by the formula is too crude, The formula is valid if, in it, we replace ~ I ~ ( ~ ) [ by an~ function subharmonic in ~ having sufficiently slow growth at ~ and some mild regularity near the slits K ~-~, ~ ÷ ~ ] • P o 1 y n o m i a 1 s, however, are s i n g 1 e - v a 1 u • d in ~ . This single-valuedness imposes c o n s t r a i n t s on the subharmonio function ~ [ ~(~)I which somehow work %o dimlniah ~8(~) to something bounded° (for each fixed • ) as ~--~ 0, provided that the sum figuring in the formula is sufficiently small. The PROBLE~ here is to See quantitativel,7 how the constraimts cause this d~m~n~shin~ t o take place. The phenomenon just described can be easily observed in one simple situation. Suppose that U(~) is subharmonic in ~ ,%hst ~(~)~< ~[~l there, and that E~(~)] + is (say) continuous up to the slits ~ - ~ , ~ + ~ . If U(x)~< ~ on each of the intervals El,!,- ~ , 'H, * jo]

, then

This estimate is best possible, and the quantity on the right blows up as ~ ~ 0 . However, if U ( ~ ) = ~ i ~(~)l where ~(~) is a s i n g i e v a I u e d entire function o f exponential type A < ~ , we have the better estimate

M with

a constant

CA

cA • A i

n d e p e n d e n t

of

~

. The improved

543

result follows from the theorem of Duffin and Schaeffer. It is no longer true when with L ~ oo •

A ~ ~

--

consider the functions

~(~)~~

The whole idea here is to see how harmonic estimation for functions analytic in multiply connected domains can be improved by taking Into account those functions' simgle-valueduess. PAUL KOOSIS

Institu% Mittag-Leffler, Sweden McGill University, Montreal, Canada UCLA, Los Angeles, USA

544 9.6. old

PEAK SETS FOR LIPSCHITZ CLASSES The Lipschitz class

analytic In D

(~ s~bo~

, consists of all functions

, continuous on ~ D

I{(~,) { ( ~ ) I . < ~ I ~ A closed set

A A , 0< % ~ ~

ql ~

q

and such that

(~)

~T

E , E c T , is called p e a k s e % f o r A~ E ~ % ). i~ the~ e~s~s a ~unction ~, I ~ A~

(the so called

p e a k

f u n c % i o n) such that

I{1/J , and if $ 2 ~ is uniformly continuous, then IT IS CONJECTURED that no nonzero function exists in the domain of ~ ~ishes

which

in a set of positive measure. K~FEI~NCE

I. d e B r a n g e s L° Espaces Hilbertiens de Fonotions Entieres., Paris, Masson, 1972o L.DE •RANGES

Purdue University Department of Nath. Lafayette, Indiana 47907

USA

553 FROM THE AUTHOR'S SUPPLEMENT,

1983

The problem originates in a theorem on quasi-analyticity due to Levinson [2~ . This theorem states that ~ cannot vanish in an interval without vanishing identically if it is in the domain of ~ where K is sufficiently large and smooth. L a r g e means that the integral in (*) is infinite, s m a I 1 that it is finite. The smoothI is non-decreaness condition assumed by Levinson was that ~ K sing, but it is more natural E3~ to assume that ~ I K I is uniformly continuous (or satisfies the Lipschitz condition). A stronger conclusion was obtained by Beurling E4~ under the A Levinson hypothesis. A function ~ in the domain of ~ cannot vanish in a set of positive measure unless it vanishes identically. The Beurling argument pursues a construction of Levinson and Carleman which is distinct from the methods based on the operational calculus concerned with the concept of a local operator~ The Beurling theorem can be read as the assertion that certain operators are local with a trivial domain. It would be interesting to obtain the Beurling theorem as a corollary of properties of local operators with nont~ivial domain~ . . . ~ . The author thAn~S Professor Sergei Khrushchev for informing him that a counter-example to our locality conjecture has been obtained by Kargaev. REPERENCES 2. L • v i n s o n

N. Gap and Density theorems

Providence, 1940. 3. d e B r a n g e s

- Amer.Math.Soc.,

L. Local operators on Pourier transforms

Duke Math.J., 1958, 25, 143-153. 4. B e u r 1 i n g A. Quasianalyticity and generalized distributions, unpublished manuscript, 1961.

CO~AENTARY

P.P.Kargaev has DISPROVED the PIRST and the LAST CONJECTURES. As to the first, he has constructed an entire function k not only of minimal exponential type, but of z e r o o r d e r such that ~ is not local. Moreover, the following is true.

-

554

THEOREM (Kargaev). Let

be a positive function decreasi~

to zero on [0 ,+@@) . Then there exist a function ("a divisor"),

a set

e c

, and a function

~ t h the

I~(~)

followin~ properties:

i~e~e >O, ~

* ~

is bounded away from zero on

eooo~ n ~(,)=th(~+~)] but

-4

e

, where

.o see k~ ~o o~ ~oro order,

K~ "s ot l o c a l . The LAST CONJECTURE i s disproved by the f a c t (also found by

Kargaev ) t h a t there e x i s t r e a l f i n i t e

v e ~ large lacunae in {(~

~

~-~

))~=4

~p~

Borel meam~es ~

Then h k

with

(i.e. there is a sequence

of intervals free of

I~I

, ~

< ~

,~=~,~,"-

tending to infinity as rapidly as we please) and with

vanishing on a set of positive length. Take

where

onn ~

~

b ~- ~ q

,

is a suitable mollifier and k : ~ C ~ ( ~ , ~ p ~ ) ) is a Lipschitz f~ction and

z~ ~=+~,

if • grows rapidly enough. Then the inverse Fourier transform vanishes on a set of positive length and belongs to the domain of ~ . Kargaev's results A l l soon be published. The THIRD CONJECTURE is true and follows from the Beurling-~alliavin multiplier theorem (this fact was overlooked both by the author and by the editors). Here is THE PROOF: there exists an entire function ~ of exponenti~ t~e ~ ~ ~0 satis~yin~ t lk I ~ on ~ . Then ~ ' ' ~ ~[X) is in the domain o f K •

555

NON-SPANNING SEQUENCES 0F EXPONENTIALS ON RECTIFIABLE PLANE ARCS

9.10.

Let A : (~) be an increasing sequence of positive numbers with a finite upper density and let ~ be a rectifiable arc in ~ . Let C (~) denote the Banach space of continuous functions on with the usual sup-morm. If the relation of order on ~ s d e n o t e d b y < and if Eo and Z 4 are two points on ~ such that ~ 0 < Z! we set

The following theorem due to P.Malliavin and J.A,~Siddiqi [ 7]gives be a necessary condition in order that the sequence (eXZ)~A.~

~on-spanning in C(y; THEOREM. If the class

c°° (Mr.,,

i,,s non-empty .for some

Zo,ZIC

M~= ~p_ ~ ~hen

(eX~) ~ eA

?

, where

~cA

is ~on-spa~in~ in C (y).

It had been proved earlier by P.~alliavin and J.A.Sidaiqi [6] that if ~ is a piecewise analytic arc then the hypothesis of the above theorem is equivalent to the :5/Itz condition ~ ~-~ < c o In connection Wlth the above theorem the following problem remains open. PROBL~

I~ Given any non-quasi-analytic class o.f functions.....on

in the sense .of Den,~o~-Carleman , to f..ind a non-zero

function. ~e.l.on-

~im~ to that class and hay in ~ zeros of infinite order at two peint.s

ix }

With certain restrictions on the growth of the sequence partial solutions of the above problem were obtained by T.Erkamma [3] and subsequently by R.Couture [2],J.Korevaar and M.Dixon [4] &nd

556

M. Ltundin ~ 5 ]. Under the hypothesis of the above theorem, A.Baillette and J,A. Siddiqi [1] proved that ( e A~ ) A e A is not only non-spanning but also topologically linearly independent by effectively constructim~ the associated biorthogonal sequence.

I n this connection the following

problem similar to one solved by L.Schwartz

[8] in the case of linear

segments remains open. PROBLEM 2, To characterize the closed linear span of

(eA~)AeA

i_.nn C (V) when it is non-spanning. @

REFERENCES I. B a i 1 1 e t t e

A.,

S i d d i q i

ctions par des sommes d'exponentielles

J,A. Approximation de fonsur u n arc rectifiable

-

J.d'Analyse Math., 1981, 40, 263-26B. 2~ C o u t u r e R. U n th~or~me de Denjoy-Carleman sur une courbe du plan complexe. 3. E r k a m m a

- Proc,Amer,Math,Soc.,

d'approximation de Muntz.

- C,R,Acad.Sc. Paris,

4, K o r e v a a r J,, D i x o n

et le th~or~me

1976, 283, 595-597.

M. Non-spanning sets of exponenti-

als on curves. Acta Math.Acad.Sci.Hungar, 5~ L u n d i n

1982, 85, 401-406.

T. Classes non-quasi-analytiques

1979, 33, 89-100.

M. A new proof of a ~t[utz-type Theorem of Korevaar

and Dixon. Preprint NO 1979-7, Chalmers University of Technology and The University of Goteborg, 6. M a 1 1 i a v i n P°, S i d d i q i J.A. Approximation polynSmiale sur un arc analytique dans le plan complexe. C.R.Acad. Sc. Paris, 1971, 273, 105-108. 7. M a 1 1 i a v i n P., S i d d i q i

J.A. Classes de fonctions

monog&nes et approximation par des sommes d'exponentielles sur u n arc rectifiable de ~ , ibid., 1976, 282, 1091-1094. 8~ S c h w a r t z L. Etudes des sommes d'exponentielles. Paris,

Hermann,

1958.

J.A°SIDDIQI

I

Department de Mathematlques Universit~ Laval Quebec, Canada, GIK 7P4

557

T It is a well-known fact of Nevanlinna theory that the inequality in the title holds for boundary values of non-zero holomorphlc functions which belong to the Nevanlinna class in the unit disc. But what can be said about s!,m~ble functions S with non-zero Riesz projection~_~=~= 0 ? Here~_~ %e~ E ~ ( - ~ ) ~ , l ~ J ~ 1 • ~>Oc Given a positive sequence ~ M ~ } ~>~ 0 dsfine

It is assumed that

a)

M~q, j=~,...,,},

~D(o,c) - uz(.) ~ Co,c).

j~ , ~ ~HC~(_Q,O))

is continuous on the set

~cm)= ,T. ~.J~Cm), is well-defined in ~ proved in [I]. If there exist

C

Define

~ CO ,c),

~D( ~ , c ) =e~Uz(~' ~)~ CO,c) , ~ppose t ~

.

n :{~n:

~

the restriction ( n , O)

. Then

m~}, ~ ~=~~'...'d~

. The following uniqueness theorem has been C = ( C 4,C 2 ~ , , . , C ~ ) ~

~+

and functions

a~t(=, an)> nclt=~c~.c~+ ...+c,~ }.

Note that the theorem is important for studying homogeneous convolution equations in domains of real ( R ~) or complex (C") spaces (see [I] , [2] , [3] ).0ne might think that,=--0 on ~ , as it occurs in the one-dimensional case. However there exists an example (see [I] )~ vhere all conditions of the uniqueness theorem are satisfied, but

562

~0 in ~ (for sufficiently large IICII ).Hence the appearance of the set ~C is therefore inevitable although ~ C does not seem to be the largest set where ~ = 0 • PROBLEM. Pind the maximal open subset of the domain ~

where

REFERENCES

I.

2.

3.

H a ~ a ~ E o B B.B. 06 o ~ o ~ TeopeMe e~HCTBeHHOCT~ B Teop~ ~m~ ~mi~x EO~,~eEcHHX uepeMeHHHX X o~opo~mHe ypsmHeH~ TZna c~epT~Z ~ Tpy6~aT~X 06AaCT~X ~ -- HSB.AH CCCP, cep.~aTeM. 1976, 40, ~ !, 115-132. HauaaEoB B.B. 0 ~ o p o ~ e CHCTeMH ypasRem~ T ~ a cBepT-EH Ka B~ny~HX O6maCT~X ~. -~o~x.AH CCCP, 1974, 219, ~ 4, 804-80V. Ha~ax~oB B.B. 0 pemem~x ypa~Rem~ 6ec~oHe~oro ~op~•a B ~e~CTB~TeX~O~ C4AaCTH. -- MaTeM.c6., 1977, 102, ~ 4, 499--510. V. V. NAPALKOV

(B.B.HAIIA~OB)

CCCP, 45O057, Y#a yx. Ty~aeBa, 50 F m m . l p c ~ @ ~ a x AH CCCP C e ~ o p MaTeMaTNEM

CHAPTER

10

INTERPOLATION, BASES, MULTIPLIERS

We discuss in this introduction only one of various aspects of interpolation, namely the

f r • •

(or Carleson) interpolation by

analytic functions. Let X

be a class of functions analytic in the open u ~ t

. We say that the interpolation by elements of X is free if the set XIE

(of all restrictions ~I E

disc

on a set E c

,Se X ) can be desc-

ribed in terms not involving the complex structure inherited from ~ . So, for example, if ~ (see formula (C) ments of H ~

satisfies the well-known

in Problem 10.3 below), the interpolation by ele-

on E

rich, bounded on ~

is free in the following sense: , belongs to H ~ I ~

lation for many other classes that the space ~ I ~ =>~I

Carleson condition

X

a n y

lunc-

. The freedom of interpo-

means (as in the above example)

is ideal ( i . e . ~ X I ~ , ] ~ l ~ I ~ l

on ~ >

~ )" Sometimes the freedom means something else, as is the ca-

se with classes

~

of analytic functions enjoying certain smooth-

ness at the boundary (see Problem 10.4), or wlth the Her~ite interpolation with unbounded multiplicities of knots (this theme is treated in the book H.E.H~KO~BCE~, ~eE~N~ 06 oHepaTope C~BEIB, MOCKBa, HayEa , 1980, English translation, Springer-Verlag, 1984; see also the article B~HoPpa~oB C.A., P y E ~ H C.E., 8an~cEE Hay~H~X CeMzHapoB ~ 0 ~ , I982, I07, 36--45).

564

Problems I0.1-I0.5 below deal wit~ free interpolation which is also the theme (main or peripheral) of Problems 4.10, 6.9, 6.19, 9°2, 11.6. But the imfoxlmation, contained in the volume, does not exhaust the subject, and we recommend the survey BNHOPpa~OB C.A., XS3~H B.~., 8an~cF~ Hay~H.ce~HapoB ~0MM, 1974, 47, 15-54;1976, 56, 12-58, the book Garnett J., "Bounded analytic functions" and the recent doctoral thesis of S.A.Vinogradov "Free interpolation in spaces of analytic functions", Leningrad, 1982. There exists a simple but important connection of interpolation (or, in other words, of the moment problem) with the study and classification of biorthogonal expansions (bases). This fact was (at last) widely realized during the past I~-20 years, though it was explicitly used already by S.Banaoh and T. Carleman~ Namely, every pair

of biorthogo~l

, ~'/" { ~ A' } l ~

families ~-----{~I}AG ~

tors in the space V , ~

( &A are vec-

belong to the dual space, < ~ ' ~ > = ~ A ~

)

generates the following interpolation problem: %o describe the coefficient space ~ V (~S ~

~,~>}Ae~

)

of formal Fourier

expansions ~

~ ~, ~> ~ . There are also continual analoA gues of this connection which are of importance for the spectral theory. "~reedom" of this kind of interpolation (or, to be more precise, the ideal character of the space ~ V

) means that ~

is an

unconditional basis in its closed linear hull. This observation plays now a significant role in the interaction of interpolation methods with the spectral theory, the latter being the principal supplier of concrete biorthogonal families. These families usually consist of eigen- or root-vectors of an operator ~ is oftem

differentiation or the backward shift, the two being iso-

morphlc) : T ~ l = ~-~ unt of

(in Function Theory

( ~

. Thus the properties of the equation

is the given function defined on 6" ) depend on the amo-

m u 1%

sending ~A

~I A , ~ E g

i p 1 i e r s

to ~(~) ~A

, where ~

of ~

, i~e. of operators ~ - ~

denotes a function C--~ ~ or the

565 multiplier itself). These multipliers ~ of ~ ~ - ~ C T ) ) (given ~

may turn out to be functions

and then we come to another interpolation problem

, find ~

). The solution of this "multiplier" interpola-

tion problem often leads to the solution of the initial problem ~

~ . Interpolation and multipliers are related approximately in this

way in Problem 10.3,whereas Problem 10.8 deals with Fourier multipliers in their own right. These occur, as is well-known, in numerous problems of Analysis, but in the present context the amount of multipliers determines the convergence (summability) properties of standard Pourier expansions in the given function space. (By the way, the word "interpolation" in the title of Problem 10.8 has almost nothing to do with the same term in the Chapter title, and means the interpolation

o f

o p e r a t o r s . We say "almost" because

the latter is often and successfully used in free interpolation). We cannot enter here into more details or enlist the literature and refer the reader to the mentioned book by Nikol'skii and to the article H1~a~6$v S.V. ,Nikol'skii N.K., Parlor B.S. in Lecture Notes in Math. 864, 1981. Problem 10.6 concerns biorthogonal expansions of analytic functions. The theme of bases is discussed also in 10.2 and in 1.7, 1 10, 1.12. Problem

10.7

represents

an

interesting

and

vast

aspect

of interpolation, namely, its "real" aspect. We mean here extensio~ theorems b. la Whitney tending t o the constructive description of traces of function classes determined by global conditions. Free interpolation by analytic functions in ~ functions in ~

(and by harmonic

) is a fascinating area (see, e.g., Preface t o Gar-

nett's book). It is almost unexplored, not counting classical results on extensions from complex submanifolds and their refinements. Free interpolation in $ ~ is discussed in Problem 10.5.

566

10.1. old

NECESSARY CONDITIONS POR INTERPOLATION BY ENTIRE ~UNCTIONS

Let ~ be a subharmonic function on C such that~(¢+Izl)= ~(~(~)) and let A# denote the algebra of entire functions such that I ~(~)I~ < A for some A 2 > 0 Let V denote a discrete sequence of points { a%l of C together with a sequence of positive integers ~p~] (the multiplicities of {~n] )" If~A#, ~#~ 0 , then V(~) denotes the sequence ~a~} of zeros of and ~% is the order of zero of ~ at @ ~ . In this situation, there are THREE NATURAL PROBLemS to study. i. Zero set problem. Given ~ , describe the sets V(~) , ~ A 3 . Ii. Interpolation oroblem. If { ~ , ~ - V c V(~) for some ~, ~ A# , describe all sequences { ~I$,K } which are of the form Kr ~cK)(~)

'

O~< J 0 and ~ (E) < + ~ are important for the problems of interpolation theory in ~P as well as in other spaCeSo [3]. Everything said above makes plausible the following conjecture. *) The{ "(C) does not i m p l y ~ E ( ~ ) = proved with help of [3].

~ ( E ) , ~B E ~

N

NPA

where

stands for the B l a s c ~ e product ~enerated

byE. CONJECTURE 1 follows from CONJECTURE 2. To see this it is sufficient to apply the Earl theorem [5] about the interpolation by Blaschke products. It is not hard to show that the zero set F of the corresponding Blaschke product can be chosen in this case satisfying 6"(E)P0, ~ ( E ) < ~ (see [6], §4 for details), ~p~°° ) that every follows fr= = + ~=i~ inner function I *~ ~ satisfies

M (v

for ~ es, ts [8], [91 ead to the lowing question. QUESTION. Is there a sin6~lar inner L function ~n U ~ I~ ? ,, REFERENCES 9. B ~ H O r p a ~ o B C.A. My~TZna~xaT~Bm~e CBO~CTBa CTeHeHEaX p~OB C noc~e~oBaTe~HocT~ XO~m~eHTOB ~8 ~P -. ~O~a.AH CCCP, 1980, 254, ~ 6, 1301-1306. (Sov.Math.Dokl., 1980, 22, N 2~ 560-565) I0. B e p d z ~ E z i~ I~.3. 0 ~yx~TZn~EEaTopax IIpOCTpaHCTB ~ ~rH~.aHa~ms z e r o npEJI., 1980, 14, BI~II.3, 67-68.

575 10.4.

FREE INTERPOLATION IN REGULAR CLASSES

Let ~ denote the open unit disc in C and let X be a closed subset of ~ . For 0 < % < I , let ~ & denote the algebra of holomorphic functions in ~ satisfying a Lipschitz condition of order . The set X is called an interpolation set for ~ & if the restriction map

A~

~,,,L~p(~,X)

is onto. The interpolation sets for ~ & , 0< & 0.

In the limit case ~ =~ there are (at least) three different of posing the problem: I. We can simply ask when the restriction map A4-~ Lip ('[, X) is onto, A~ being the class of holomorphic functions in ~) satisfying a Lipschitz condition of order I . 2. We can also consider the class

ways

A~ = H(~) a c ~ (D) and call ~ an interpolation set for (the space of Whitney jets) such that A with , on X .

A~ if for all ~ C ~(X) ~ 0 there exists { in

576

3. Finally one can consider the Zygmund class version of the problem. Let A , denote the class of holomorphic functions in h having continuous boundary values belonging to the Zygmund class of . We say that ~ is an interpolation set for ~, if for any ~= in the Zygmund class of ~ there exists ~ in ~ such that

on X In [I] and [2], it has been shown that Dyn'kin's theorem also holds for ~ -interpolation sets. For A 4 interpolation sets the Carleson condition must be replaced by (2C) ~ N ~

is a union of two Carleson sequences.

Our PROBLE~ is the following: which are the interpolatiom sets for the Zygmund class? Considering the special nature of the Zygmund class, I am not sure whether the condition describing the interpolatinn sets for A~ (one can simply think about the boundary interpolation, i.e. ~ C ~ ) should be different or not from condition (K). Recently I became aware of the paper [4], where a description of the trace of Zygmund class (of ~ ) on any compact set and a theorem of Whitney type are given. These are two important technical steps in the proofs of the results quoted above and so it seems possible to apply the same techniques. REFERENCES I. B r u n a J. Boundary interpolation sets for holomorphic functions smooth to the boundary and B~i0. - Trans.Amer.Eath.Soc.~1981, 264, N 2, 393-409~ 2. B r u n a J., T u g o r e s F. Free interpolation for holomorphic functions regular up te the boundary.-to appear in Pacific J. Math°

3. ~ U H ~ ~ ~ ~ E.M. ~[~o~ecTBa C B O 6 0 ~ O ~ ~ e p n ~ ~z~ F~ac-COB r ~ e p a . - ~aTeM.c6opH., 1979, 109 (151), ~ I, 107-128 (Math.USSR Sbornik, 1980, 37, 97-117). 4. J o n s s o n A., W a 1 1 i n H. The trace to closed sets of functions in ~ with second difference of order 0(~). - J. Approx.theory~ 1979, 26, 159-184. J.BRUNA

Universitat autBnoma de Barcelona Secci~ matematiques. Bellaterra (Barcelona) Espa~a

577

old Let ~N b e t h e u n i t b a l l o f ~N ( ~ > ~ ) a n d d e n o t e b y I~ H~(]~ N) the algebra of all bounded holomoz~hic functions i n ]~ , An a i ~ a l y t i c s u b s e t ~ o f ]~1t is said to be a z e r o - s e t

f o r H ~ ( R N) (in symbois, E~EH'(R ~) ) if there exists a non-zero function # in H ' ( B N) with E=~-~(O) ; E i s said to be a n i n t e r p o I a t i o n s e t f o r ~o(~N) (in symbols: E ~ I ~ ( ~ N) ) if for any bounded holomorphic function ~ on E there exists a function ~ in H °°(~N) with ~IE = ~ . The problem to describe the sets of classes ~ H ~ ( B N) and I ~ao(~N) proves now to be very difficult. I would like to propose some partial questions concerning this problem; the answers could probably suggest conjectures in the general case. Let A be a countable subset of ~ . Set

1,.What are the sets

~

such that

~& ~ H ~ ( ~

N)

?

PROBLEM 2. What are the sets

A

such that

T& E l Ho~(~N)

?

~ 0 ~

It follows easily from results of G.N.Henkin [I] and classical results concerning the unit disc that the following two conditions

are ~ecessary

~e~

for

7~ ~ ZH®r.~BN)-

'"' ) 0} be the biorthogonal family to [ ~ : k(~)> O} , ooe

L;( )

~-~-e,_ O.,t, , IfflO. 0

Condition (2) implies that qX are analytic outside ~ , continuous up to the boundary and bounded (by the constants, which may depend on ~ ). Let B(~) be the class of all functions ~ analytic in int ~ , continuous in ~ and such that

0 Putting

C =~,

~K,X----~A ,

4 k(~)>0

in (I), associate with

581

every function ~ ~~o

(~e

E)

582

It was shown in [2] and [3] how the general case (i.e. the case of an arbitrary ~ analytic in int ~ ) can be be reduced to the case ~ ~(~0) • PROBLEM 3. Show that for any domain int tion

~

with the properties

~

there exists a func-

(1), a), b), c).

REFERENCE S

I.

~ e o H T ~ e B

2.

.[ e o H T ~ e B A.~. K Bonpocy 0 npe~cTa2~e~E aHaJn~THqeoENX ~yHELG~ B 5ecEoHe~HO~ B~LUyF~O~ odaaCTH p ~ a M ~ ~ p H x ~ e . - ~OE~.

A.$.

P~j~ SECnOHeHT. M., Hs~Ea, 1976.

AH CCCP, 1975, 225, ~ 5, I013-i015. 3.

~ e o H T B e B A.~. 06 O~HOM r@e~CTaB~eHH~ a H ~ T ~ e c K o ~ # y ~ E u ~ B 6ecEoHeqHo2 B ~ m y ~ o ~ o6aacT~. -- Anal.Math.s 1976, 2, 125-148. A. F. LEONT iEV

(A.$.~EOHTBEB)

CCCP, 450057, Y~a y~. TyEaeBa, 50 ~Hpcz~ ~s~ AH CCCP

583

10.7.

RESTRICTIONS OF THE LIPSCHITZ SPACES TO CLOSED SETS K

The Lipschitz space of the semi-norm

A~(~) ~

K

is defined by the finiteness

~Cl~l) k

Here as usual A~ = ( ~ - ~) and %~ #(G)) = ~(SC+~). The majorant : ~÷ --~ ~ + is non-decreasing and~,GJ~+0) = 0 . Without loss of generality one can suppose that ~0($~$K is non-increasing. Let X ~ ) be the closure of the set Co°° in A w ( ~ ) . This n o t e d e a l s w i t h some problems c o n n e c t e d w i t h t h e space o f t~oes A t (E) -- At, (RT I~ and with its separable subspace ~,(F) -~, where~F~ %" is an arbitrary closed set. Among ~ spaees~u~der consideration there are well-known classes C ,C and A ~ + 4 whose importanCen ~is~indubitable" Recall that C & @ ~-th derivatives satisconsists of all functions ~ U with fying HSlder comdition of order @ . Replacing here H'~Ider condition by Zygmund condition K) we obtain the definition of the class

A g+t CONJECTURE 1. There exists a linear continuous extension M

tor

~'

op~ra-

k

A~(F)-~A~(~ ).

A nonlinear operator of this type exists by Michael's theorem of continuous selection [I]. The lineartiy requirement complicates the matter considerably. Let us review results confirming our Conjecture I. Existence of a linear extension operator for the space of jets ~ , o L ( F ) connected with ~ -CL,~(-'I~ ~) is proved in the classical_ ___W~hitney th\eorem --[2J . But the method of Whitney does not work for A~+~(~) ° Recently the author and P.A.Shwartzman have found a new extension process proving Conjecture I for ~ = ~ (the case ~-----~ is well-known, see for example E3 ] ). The method is closely connected with the ideology of the local approximation theory [4] ). The following version of Conjecture I is intresting in connection with the problem of interpolation of operators in Lipschitz spaces.

~ction

$

satisfiesZ y ~ d

conditionif I A ~ I = 0 0 ~ I )

584

CONJECTURE 2. Let c0~,

~ =t,~

, be majerants. There exists a K

linear exstension operator K

~ : C C F ~ C C ~ ~) mapping

~C

F)

A~C~), ~=~,~.

into

The above mentioned extension operators do not possess the required property. If Conjecture 2 turns out to be right we would be able to reduce the problem of calculation of interpolation spaces K for the pair A ~ ( F ) , $ : ~, ~ , to a similar problem for ~ . PROBLEM. Pind condition necessary and sufficient for a ~iven function

~

# ~CCK)

to be extendable to a

}. ~n

C~ ~)

restriction

~E

K

n

other words we ask for a description of the K

~C~)IF

(or X~Cg~)IF ).

The problem was solved by Whitney for the space C K C ~ ) in 1934 (see [~)° In 1980 P.A.Shwartzman solved the problem for the space A % C ~ ) and in the same year A.Jonsson got (independently) a solution for the space A~+~C~) (see [6,~ ). The situation is much more complicated in higher dimensions; there is nothing but a non-effective description of functions from the space A ~ C ~ ) involving a continual family of polynomials, connected by an infinite chain of inequalities (see [8] for power majorant; general case is considered in [9] in another way). Analysis of the articles [5-7] makes possible the following CONJECTURE 3. Let

N=N(K,~,F)

be the least integer with the

following propert~ ~): if the restriction of a on any subset

H~F

with card

H~ N

function

~EC(F)

is extendable to a

K

function

~H ~ A t ( ~ )

and

HS ~

I~H I~ < co

, then ~

belongs

A~CF). K

to

Define

N(k,n)

by the formula

N (K,~) =,s~p NCK,~o,F). Then the number N(K,t~) is finite. It is obvious that N(4 ~)=~ ; the calculation of N (K,~) for k > I is a very complicated problem. P.A.Shwartzman has proved recently that NC~,~) = S' ~ - 4 ~) One can prove that N c k , ~ I ~ F ) < O0,

585

and using this result has obtained a characteristic of functions from ~ F ) , Fc~ ~ by means of interpolation polynomials (see [6]). When ~ > ~ the number N(~,~) is too large and the possibility ef such a description is dubioms. In conclusion we note the connection of the considered problems with a number of other interesting problems in analysis (spectral synthesis of ideals in algebras of differentiable function, HP space theory etc.)

REFERENCES I. M i c h a e I

E.

Continuous selections. - Ann.Math.,

1956, 63,

361-382. 2. W h i t n e y H. Analytic extensions of differentiable functions defined in closed sets. - Trans.Amer.Math.Soc., 1934, 36, 63-89. 3. D a n z e r L., G r u n b a u m B., K 1 e e V. Helly's theorem and its relatives. - Proc.Symp.pure math., VIII, 1963. 4. B p y ~ H H 2 D.A. Epoc~paHcTBa, onpe~ea~eM~e c noMom~m aoEax~H~x npE6am~eH~. - T p y ~ ~ 0 , I97I, 24, 69-I32. 5. w h i t n e y H. Differentiable functions defined in closed sets, I. -Trans.Amer.Math.Soc., 1934, 36, 369-387. 6. ~ B a p ~ M a H

H.A.

0 cae~ax ~yHE~HR ~Byx n e p e M e m m x ,

B o p a m ~ x ycaoBm0 8~r~ys~a. - B c6."Hccae~oBaHm~ no T e o p ~ ~moz~x B e ~ e c T B e H m ~ nepeMeHH~X". - HpocaaBa~, I982,

y~oBaeT~mu~ I45-

168. 7. J o n s s o n A. The trace of the Zygmund class to closed sets and interpolating polynomials. - Dept.Math.Ume~,1980, -

AK(~)

7. 8. J o n s s o n A., W a i I i n H. Local polynomial approximation and Lipschitz type condition on general closed sets. - Dept. Math.Ume~, 1980, I. 9. B p y ~ H H ~ D.A., m B ap n M a H H.A. 0n~caH~e c~e~a ~yHEIU~ ES 0606~eHHO~O rfpOOTpaHCTBa ~ r m m ~ a Ha n p o E s B O ~ H ~ i EOM-naET. - B c6."Hccae~oBaH~ no ~eopHH ~ y H E n ~ MHO~HX Be~ecTBeH~mX nepeMemmx".

Hpocaa2a~ I982, I6-24.

Yu.A.BRUDNYI

(~.A.BPY~)

CCCP, 150000, HpooJza~l.~, HpooJIaBCEH~ I D c y ~ p C T B e H H ~ yH~Bepc~eT

586 MULTIPLIERS, INTERPOLATION, AND

10.8.

A(p)

SETS

Let G be a locally compact Abelian group, with dual group F An operator T ' I p (G)--~p(~) will be called a multipliAer provided there exists a function T ~ I Qo~P) so that T(~)A--T~ , for all integrable simple functions ~ . The space of multipliers on }?(G) is denoted by Mp(G) . ~et CMp(@=IT~Mp(G):T~C(V) }.

.

In response to a question of J.Peetre, the author has recently shown that for the classical groups, C M p ( G ) is not an interpolation space between M ~ ( G ) = M ( G ) and C M ~ ( G ) = L ~ ( P ) ~ C ( P ) . More specifioall~, we obtained the following theorem (see [2] )° THEOREM I. Let G

denote one of the groups ~ ~,

Then there exists an operator T

(b) TIM(G) is a (c) T ICMp(G)

L~(F)NC(F).

bounded operator on

i_~

n o t

NCG).

a bo~de~ operator on CMp(@,p~t@.

Observe that T is i n d ep e n d e n Our method of construction makes essential use concerning ~(~) sets. Recall that a set E ~ type A(~) (~ < ~ < oo) i f whenever ~L~(T) all ~ 9 ~ E , we have ~ E I % ~ T ) . We used the sult of W.RUdd.u [1].

S >~

THEOREM 2. Let N >5

{0, I,

, and let

2,...,M} (a) F

, o r Z n.

so that

is a bounded ope,z~tor on

(a) T

~

% of p , ~ < p < g . of certain results Z is said to be of and ~(~)=0 for following elegant re-

be an integer, let N

M = 5 S-IN s-~

be a prime with

. Then there exists a set

F~_

so that contains exactl,y

(b) II ~ ~25 ~< cIl

~ II2

N

points,, an d

, fo r evex7 t r i ~ o n o m e t r i c vo!yaomial ~ ,

A

wlth

~(~)=0

(Here C

is

for

~I,E¢:F

(Suoh

are called

~

i n d e p e n d e n t

of

~-pol~momials,).

N ).

AS a consequence of Theorem 2, Rudin showed that there exist sets of type A(~s) which are not of ~rpe A(25+6) , for all 6>0 (see [I] )o An obvious conjecture arisimg from Theorem I is the following: CONJECTURE I. Let

4~ 0 . The difficulty is in proving t3at F~,N is of type ~ (p) (with all constants uniform in N ). One may seek to accomplish this by w r i t i n g an ~F~,M - p o l y n o m i a l _~ i n a J u d i c i o u s way a s a sum o f p r o d u c t s o f F - p o l y n o m i a l s , and c a r e f u l l y examining t h e r e s u l t a n t r e p r e s e n t a t i o n o f ~ . However, new e s t i m a t i o n t e c h n i q u e s f o r ~p norms would s t i l l be v e r y much a n e c e s s i t y i n o r d e r t o c a r r y o u t t h i s program. REFERENCES I. R u d i n W. Trigonometric series with gaps. - J.~ath.Mech., 1960, 9, 203-227.

588

2. Z a f r a n M. Interpolation of Multiplier Spaces, Nath., to appear. MISHA ZAFRAN

Department of ~thematics University of Washington Seattle, WA 98195 USA

Amer.J.

CHAPTER

11

ENTIRE, MEROMORPHIC AND SUBHA~ONIC I~UNCTIONS

This mid and ramified theory, the oldest one among those presented in this collection, hardly needs any preface. By the same reason ten papers constituting the Chapter cannot reflect all tendencies existim~ in the field. But even a brief acquaintance with the contents of the problems shows that the main tendency remains invariable as though more than a quarter of the century, which passed since the appearance of the book by B°Ya.Levin "Zeros of Entire Functions", has shrunk up to an instaut~We reproduce here the first paragraph of the preface to this book: "One of the most important problems in the theory of entire fumctions is the problem of connection between the growth of an entire function and the distribution of its zeros

Nmny other problems in

fields close to complex function theory lead to this problem"~ The only discrepancy between then and now, apparently, consists in more deep and indirect study of this problem A good illustration to the above observation is provided by Problem 11.6. It deals with description of zero-sets of sine-type functions and is important for the purposes of Operator Theory. Problem 11.2 is, probably, "the most classical" one in the Chapter. The questions posed there look very attractively because their formulations are so simple.

590

The theory of subharmonic functions is presented by Problems 11.7, 1 1 . 8 . Problems 11.3 and 11.4 deal with

exceptional values in the spi-

rit of R.Nevanlinna Theory. Problem 1 1 . 1 0 concerns the limit behavi~trof entire functions. An important class of entire functions of completely regular growth is the subject of Problem 11.5. "Old" Problem 11.9 by B. Ya.Levin includes three questions on functions in the Laguerre-P61ya class. Problem 11.1 is rather a problem of approximation theory The problems 11.1, 11.5, 11.6, 11.8, 11.9 are "old" and the rest are

new.

591 11.1. old

THE INVERSE PROBLEM 0F BEST APPROXIMATION 0F BOUNDED UNIFOR/KLY CONTINUOUS FUNCTIONS BY ENTIRE FUNCTIONS 0P EXPONENTIAL TYPE, AND RELATED QUESTIONS

Let E be a separable infinite-dimensional Banach space, let ~ c E , C ... be a chain of its finite dimensional subspaces such t ~ t ~l"4, Em= kt, and U ~ is dense in ~ . For ~ £ ~ we define the sequence of "deviations" from

e4 { II - iII: S

~

by

,

....

S,N,Bernstein [I] (see also [2~ has proved, that for every sequence { ~ 0 of non-negative numbers such that ~ n ~0 there exists ~ £ E , with

This is a (positive) solution of the inverse problem of best approximation in a separable space in the case of finite dimensional subspaces. Strictly speaking S.N.Bernstein has treated only the case of E =C [ ~] , E ~ being the subspace of all polynomials of degree 46-4 , but his solution may be reproduced in general case without any change. Now let ~ (~) be the Banach space of all bounded uniformly continuous functions on ~ with the sup-norm; let B ~ be its closed subspace consisting of entire functions of exponential type %~ (or, to be more precise, of their restrictions to ~ ). S.N.Bernstein has shown [3] that many results concerning the best approximation of continuous functions by polynomials have natural analogues in the theory of best approximation in B(~) by elements of We define the

deviation of

~ "

from ~

A function ~ being fixed, the function ing properties: 1. A(~, ~) ~ A ( I : ~ )

by

A(~,~)

has the followfor ? < ~ .

592

PROBLEM 1. Let a bounded function

6~-

s a t i e f ~ c o n d i t i o n s 1- 3 . Is there a f u n c t i o n

PROBLEM 2. Let 3 ~-0 B~_ 0

A(I,~)

~

~

(0 4 ~ < o o )

, ~eS(~)

be the c l o s u r e of ~ 2

iS a propgr subspace of B ~

PROBLEM 3. L et

~ F(~)

be a Boh z

such

. S~

. What is itsf~+codimension in

almostrp~@~

function. Is

necessarily a ,jump function?

PROBLEM 4. Let

A (I,~')

be a ~ump funotion. Is ~

almost-pe-

rlodic? REFERENCES I.

2. S.

B e p H m T e ~ H C.H. 0d odpaTHO~ ss~a~e T e o p ~ Ham~y~mero np~d~e~ Henpep~_BH~X ~JHmq~. - B EH. : Codp.co~., T.2, M., MS~-BO AH CCCP, 1954, 292-294. H a T a H c o ~ H.H. EOHOTpyET~BHa~ T e o p ~ ~ y s ~ , M.-~., I~TT~, 1949. B e p ~ m • e ~ ~ C.H. 0 H a r e m n p ~ d ~ m ~ e ~ Henpep~Bm~x ~ ~a Bce~ Be~ecTBem~o~ oc~ np~ n O M O ~ nexm( ~ y ~ E ~ ~am~o~ cTeneH~. - B EH.: Codp.co~., T.2, M., MS~-BO AH CCCP, I954, 371-395. CCCP, 3IO0(E, Xap~EOB Xap~EOBCE~ ~CTHTyT ~xeHepoB x o ~ s ~ I o r o

M. I.KADEC (M.~.~M)

CTpO~Te~OTBa

OC#n"I~AItY Problems 1, 2 and 4 have been s o l v e d by A.Gordon (A.H.rop~o~) who k i n k y s u p p l i e d us w i t h t h e f o l l o w i n g i n f o r m ~ t i o n .

~WW

1. (A, Gordon). Le__~t ~

A ~ , e)

~(~) , ~ 0 .

s a t i s f ~ 1N~ above. Than theze

593 PROOF. P i c k a d e n s e s e q u e n c e {~K}~>~o i n (O, e o o ) end consid e r a monotone s e q u e n c e o f p o s i t i v e numbers { ~ } ~>~o such that

K Let { tR} t e n d t o ~r ~o so f a s t t h a t the intervals T K d,,e~ K~>0 . . = ~ e ~,: I~-~K < ~ - ~ } do not overlap, St~c~srd a r g e n t s show

(~) ~ o f o r each } e [1%

~ (t-~z) and --~--- 0 .

Here ~E

(2)

stands as usual for the ohal~oteristic

function of a set

E. ~iveu )~~

~ > 0

Clearly ~ ~ ~

let

. The d e s i r e d

~

is defined as follows :

K~O The s e r i e s c o n v e r g e s a b s o l u t e l y a n d , m ~ f o r m l y on compact s u b s e t s o f (see (1)},which implies ~eB(~) . Fix 5> 0 end consider

~)

d~ ~

p(~).~(~, ~,~_~)

.

AK~ Since ~ ~ i s c l o e e d u n d e r b o u n d e d a n d p o i n t ~ s e c o n v e r g e n c e on ~ At i s c l e a r t h a t ~¢~ B~. On t h e o t h e r h a n d ( I ) i ~ l i e s the inequality

Suppose mew A(;~,o~) ~ ( f f )

of i n t e g e r s

such that

~K~

. Then t h e r e e x i s t ~:'6", ~ ( ~ , f f ) < } ~

. t~operty 2 of P

e n d (2) imply

that

We may assume w i t h o u t l o s s o f ~ e n e r a l i t y

that the sequence

594 It follows dist (~t ~, ~ g) ~ ~ refore A(.-~6") ~ ~I(.6") . @

THEO~

2. (A.Gordon).

mOOF, L e t ~ K ] above for the constant ce ~=liSK]K~

~k~

in contradiction

to

[3] - T h e -

(B~/B~_o)=+ =

and t'~K} be the sequences c o n s t r u c t e d as s e q u e n c e ~K------- 6~ . F o r e v e r y b o u n d e d s e q u e n -

O define

~)

d~ ~

~,$_~).

~K~,

k~0

Clearly ~ t ~ B ~

and it is easy to check that

-

Indeed, ~ r ~

K

~(~-4/~, 8,~) =

q ( ~ 6,~)

i n B ( ~ ) . Therefore

Usimg t h e same argumen%s a s i n t h e p r o o f o f Theorem I , we o b t a i n

~(~K,B~)~ ~ I ~I f o r ~ ~ ~ . i~ f o n o . , the f a c t o r - s p a c e ' ~ / ~ ~_ o contains a subspace and t h e r e f o r e i s n o t separable. • Note that the function ~ with ~ properly t h e n e g a t i v e a n s w e r t o P r o b l e m 4. I n d e e d , i f ~ I ~ K ~) O,

~'~ ~

•

f r o . (~) t ~ t isometrlo to ~ %

I

ohoosen ~ives ---~ ~ then

595

SOME PROBLEMS ABOUT UNBOUNDED ANALYTIC FUNCTIONS

11.2.

A famous theorem of Iversen [I, p.284a] says that if ~ is a transcendental entire function then there is a path ~ along which tends to ~e . Thinking about this theorem has led us to formulate the following four problems, which we would like to solve, but cannot yet solve. I. I_~f ~ is a transcendental entire function, must there exist a ~th

~

alon~ which e v e r y

derivative Of ~

Short of that, how about just havin~ both

tends to oe ?

and

~,ity alon~ ~?(It renews, say, by w ~ - w l i r o n

tend to infi-

theory, that i f

is a transcendental entire function, then there exists a s e q u e n c e (En) such that I(~) (Z~) - - - ~ as B - - ~ for each k= ~ , ~ , . . . , but to obtain a p a t h on which this happens seems much more difficult. ) 2. If I is an unbounded anal,Ttic function in the open disc ~ ,

~ust there exist ~ sequence (Z~)

of points of D

ev e ~

~--=

k= 0 , ~ , £ , . . . , ~

just h a v i ~

(L)

(z~)-~

~(Z~)--~

that one can always find If ~

~

? Short of that, how about

and ~t"(Z~)-,~ : (~)

such that for

(~he authors have sho~

So that both

(Z~)'~O@ and

I' (Z~)-~.~

grows fast enough, i.e., if either

or

=~,

then we can show that there exists a

(~)

(2)

with

as

~-~ for ~ = 0 , ~ , ~ , . . . , but we do not know hew necessary these growth assumptions are. Note that conditions (I) and (2) are not strictly comparable. The proof involving condition (I) quotes a theorem of Valiron, while that involving (2) uses some Nevanlinna theory.) 3.

Is there an eas 2 elementar~ proof I u s i n ~ n e i t h e r Wiman-Vali-

ron t h e o r ~ n o r advanced N e v a n l i n n a t h e o r ~ t h a t i f

~

is a transcen-

596 dental entire function. . then there exists a .sequence ~(k~(~) ---~ ~

a_ss~ - - ~

f0r

~

0,4, ~ . . .

(~)

such that

?

4, The following possibility is suggested by many examples: the differential equation

where the ~ no solutions

are polynomials in one variable, not all constants, has ~(~) that are analytic and unbounded in the unit disc . For example, the equation ~ f z _ m I can be solved explicitly by means of the standard substitution~ V ~ - ~ V ~ V ~ and it is -'4 easily seen that it has no unbounded analytic solution in any disc, lending support to the above hypothetical statement~ The QUESTION re-

mains whether that statement is true or not°

REFERENCES 1. T i t c hma Oxford S 932.

JAMES LANGLEY LEE AoRUBEL

r s h

E.C. The Theory of Functions

2nd Edition,

Dept.Math, University of Illinois Urbana, IL 61801 USA

597 11.3.

COMPARISON OF SETS OF EXCEPTIONAL VALUES IN THE SENSE O!~ R.NEVANLINNA AND IN THE SENSE OF V.P.PETRENKO Let

~

where [ @ ~

be a meromorphic function in C

and put

is the spherical distance between ~

and ~

. Demote

in the sense of V.P.Petx~nko an~ by E~C~)=IOv£~:~(~,~)>0} . the set of aeZicie~t values o~ { . It'~s ~cl~ear that The set E~(~) is at ~ost countable if ~ is of finite order [I]. There are examples of ~ 's of finite order with E~({)~ E~(~)

E'~(,,()CE#(_~).

b

41.

-

,,

PROBLEM

I. Let E~CE~cC

be arbitrary at

sets, Does there exist a meromorp..hic function I

with

most countable of f.inite order

Eech=E,?

implies

EN({)= E~({)

I~O:B:I~ 2. Let {

[6]. be an,, ,entire f ~ c i ; i o n of f.i.mii;,e,,,,order,

REFERENCES

I. 2.

3.

4.

H e T p e H K o B.H. POCT MepoMop$s~X ~ , Xap~xOB, "Bm~a m~oxa", 1978. ly r p i~ m ~ H A,~. 0 cpaBHe,~ ~e~eETOB 05(@) . -- Teop~ ~ys~, ~yHE~.a~a~. z ~x np~., X ~ K O B , 1976, • 25, 56-66. r o ~ ~ ~ 6 e p ~ A.A. K Bonpocy O CB~S~ M e ~ y ~eSexTOM o T ~ o H e ~ e M MepoMop~HO~ ~ym~mz. - T e o p ~ ~ , ~ymu~.aHs~. ~x np~., Xap~oB, I978, ~ 29, 31-35. C o ~ ~ H M.~o 0 c o o ~ o s e ~ M ~ MHOZeCT~ ~e~eETm~X 3~a~e~ s OTF~OHe~ ~ MepoMoI~HO~ ~ EOHe~O~O n o p ~ a . -C~6.MaTeM,~/pHa~, I98I, 22, ~ 2. I98-206.

598

. E p e M e H E 0 A.3. 0 ~e~eETax ~ O ~ O H e K ~ q X $ym~m~ EoHe~oro nops~a (B nepal). A.A. GOL 'DBERG

(A.A.IY~hEBEPr)

A. E. EREMENK0

(A. ~.EP~EHE0)

MepoMop~x

CCCP, 290602, JI~BOB JL~BOBC~ r o c y ~ a p c T B e ~ y~epc~TeT CCCP, 310164, Xap~oB np.JleH~Ha, 47, ~sm~o-Te~ec~ ~HCTH~yT ~S~HX ~eMnepaTyp AH YCCP

599

11.4.

VALIRON EXCEPTIONAL VALUES OF ENTIRE FUNCTIONS OP COMPLETELY REGULAR GROWTH

Let E# be the class of all entire functions of order p , ° 0 and ~(0)~--~ , where ~ is the open unit ball in 6 ~ , Thus ~(~) is convex (and compact in the compact open topology). We think that the structure of N(~) is of interest and importance. Thus we ask: What

are

the extreme points of ~(~)

Very little is known, Of course if ~ - ~

=

? , and if

c4+ 8-)Ic4-~),

(1)

then ~ is extreme if and only if #(~) ----- C~ , where O ~ T . It is proved in Eli that if S ( ~ ) ~ $ ~ and if ~ is the Cayley transform (I) of £ , then $ ~ E £ ~ ) , where E(~) is the class of all extreme points of N (~) . Let K~- (~,..., K N) be a multi-index and consider monomials K

~(~) ~

C$ K in 6~

where by

such that I~(~)I ~ ~

if ~

~

. ThuslCl~([~[)"~

we mean K,+...

let

~(~)=(~+OK~K)/(~--0K~K). It is proved in [2] that ~ ( ~ ) if and only if the components of k are relatively prime and positive. 2. We have ~ ~ ~(~) , however it is a corollary of the just mentioned theorem of [2] that ~ 6 6 5 ~ ( ~ ) , where the closure is in the compact open topology. Thus E(~)=~= c~o~ (~) . (If ~ - ~ ,

then E ( ~ ) =

c ~ E c ~) ).

It is also known that if ~ is an extreme point of ~(~) and if (I) holds(that is to say if ~=(~-~)/(~÷I) ), then ~ is irreducible. This is a special case of Theorem 1.2 of [3]. The term "irreducible" is defined in ~]. If ~ ~ , then ~ is extreme if and only if~ is irreducible. However for ~>/ ~ , the fact that ~ is irreducible does not imply that ~ is extreme. 3. The extreme points ~ in section I and the extreme points that can be obtained from them by letting ~ $ (~) act on ~(~) have the property that the Cayley transform I=(~-I)/(~+~) is

holomo~hic on ~ U 8 ~ Is this the cas,e for eve,ry ~

in E(~)

?

If the answer is yes, then it would follow (since ~

) that

824 the

F. and M.Riesz theorem holds for those Radon measures on a

whose Poisson integrals are pluriharmonic. In particular there would be no singular Radon measures =~= 0 with this property, which in turn would imply that there are no nsnoonstant inner functi6ns on 8

.

REDOES I. P o r • I i i

F.

Measures whose Poisson integrals are pluri-

harmonic I!. - l l l i n o i s J.Math. ~ 1975, 19, 584-592. 2. F o r e I i i F. Some extreme rays of the positive pluriharmonic functions. - Canad.Math.J., 1979, 31, 9-16. 3. F o r e i i i

F.

A necessary condition on the extreme points of

a class of holomorphic functions. - P a c i f i c 81-86. 4. A h e r n

P., R u d i n

W.

J. Nath.~1977, 73,

Factorizations of bounded holomor-

phic functions. - Duke Math. J., 1972, 39, 767-777. PRANK ~ORELLI

University of Wisconsin, Dept. of Math., M a ~ s o n , Wisconsin 53106, USA

COMMENTARY The second question has been answered in the n e g a t i v e mentary in S. 10.

See Com-

625

12 •3 .

PROPER MAPPINGS OF CLASSICAL DOMAINS

A holomorphic mapping

is called

p r o p e r

if

q;~

~

of a bounae d domaln ~ C ~

~

~l~"~O

~(C~C~),

~

for e v e ~ se-

A biholomorphism (automorphism) of ~

is called a t r i v i a 1 p r o p e r m a p p i n g of ~ . If .0. is the l-dimensional disc ~ the non-trivial proper holomorphic mappings ~:~-'~ do exist. They are called finite Blaschke products. The existence of nontrivial proper holomorphic mappings seems te be the characteristic property of the l-dimensional disc in the class of all irreducible symmetric domains. CONJECTURE I. For an irreducible bounded s,~etric domain in C ~ , ~

# ~

, every proper holomor~hic mappin~

~--~

is an

aut omorphiem. According to the E.Cartan's classification there are six types of irreducible bounded symmetric domains. The domain ~ p ~ of the f i r s t type i s the set of co lex

matrices

Z

,

,

such that the matrix I--Z ~ Z is positive. The following beautiful result of H.Alexander was the starting point for our conjecture.

T~o~M i.e. ~ = ~ ? , ~

I (H.Ale~nder [I]). ~et ~ and let

be ths ~ i t ball in C P ,

p>/~ . Then ' e v e ~ proper holomorphic map-

,~i~ (~:_qp,~ ~I'9.p,~

is an. autom.o.~ph.i~mof the ban.

Denote by S the distinguished boundary (Bergman's boundary) of the domain ~ . A proper holomorphic mapping @: ~ •~ is called s t r i c t i y i r o p e r if ~ ( ~ ( ~ u ) } ~)--~0 for every sequence ~ E with the property ~ ( ~ S)-~0. The next result generalizing Alexander's theorem follows from [2] and gives a convincing evidence in favour of CONJECTURE 1. THEOREM 2 (G.M.Henkin, A.E.Tumanov

le bounded symmetric domain in

,proper holomorphic mappin~

C ~ and

~ :/i--'/~

[2] ). ~

If' -0.

~ D

is an irreducib-

, then,,any strictly

i,~,,,~,~,,~,utomo,rp~!sm,

Only recently we managed to prove CONJECTURE I for some symmetric domains different from the ball, i.e. when ~ 2 ~ S .

THEOEEM 3 (G.M.Re.'~in,

R.~.~ovikov). Let, ~ c C

b e the classical domain of the 4-th type, i,e.

~

, ~~,

626

$I ={ z:z.~', ((,~'~')~- Izz'l~)V~< ~}, where

E=(~,...,E~)

and

Z l stands forthe transposed matrix

The n every prgperholomorphic mappin~ ~ - ~

is an automorphism.

Note that the domain I~,~ of the first type is equivalent to a domain of the 4-th type. Hence Theorem 3 holds for ~L~,~ e We present now the scheme of the proof of Theorem 3 which gives rise to more general conjectures on the mappings of classical domains. The classical domain of the 4-th type is known to have a realization as a tubular domain in C ~ ~ ~ , over the round convex Gone

boundary of this domain coincides with the space T h • d i s t i Then boundary g u i9 2 s contains h e together d R =t~C • ~= 0 } with each point Z ~ \ ~ the l-dimensional analytic component 0 ~ = ~ :~EC~ I~ >0}. The boundary ~ 0 ~ • ~f this component is the nil-line in the pseudoeuclidean metric ~5

=~o-~®,-...~ e n

the disti~uished bo~d~ry S

.

If ~ is a map satisfying the hypotheses of the theorem, an appropriate generalization of H.Alexander's [1~ arguments yields that outside of a set of zero measure on ~ a boundary mapping (in the sense of nontangential limits) ~ :9 ~ --~ ~ of finite multiplicity is well-defined. This mapping POsseses the following ~ro~erty: for almost every analytic component 0 Z the restriction q l ~ is a holomorphic mapping of finite multiplicity of ~ into some component ~ W " ~urthermore, almost all points of ~ are mapped (in the sense of nontangential limits) into points of ~ . It follows then from the classical Frostman's theorem that ~I~Z is a proper mapping the half-plane ~ into thehalf-plane Y v W o it follows that the boundary map ~ defined a.e. on the distinguished boundary ~ C ~ has the following properties: a) ~ maps ~ into ~ outside a set of zero measure; b) ~ restricted on almost any nil-line ~ E coincides (almost everywhere on ~ Z ) with a piecewise continuous map of finite

627 multiplicity of the nil-line S~Z into some nil-line With the help of A.D. Alexandrov' s paper

[3]

S~Wone can prove

that the mapping ~: ~ - - - ~ satisfying a), b) is a conformal mapping with respect to the pseudoeucli:dean metric on S . It follows that is an automorphism of the domain II To follow this sort of arguments, say, for the domains /Ip,~ where p ~ , one should prove a natural generalization of H.Alexander's and O.Prostman's theorems. Let us call a holomorphic mapping ~ of the ball l~p, 1 a 1 m o s t p r o p e r if ~ is of £inite multiplicity and for almost all ~ S i p , 4 we have ~(~)~ ~'~-p,4 , where ~(~) is the nontangential limit of the mapping ~ defined almost everywhere on ~Ap, I CONJECTURE 2. Let ~ be an almost Droner maDpin~ of ~ip,1 and p~ ~.

Then

~

is an automorphism~

If we remove the words " 9

is of finite multiplicity" from the

above definition, the conclusion of Conjecture 2 may fail, in virt~e of a result of A.B.Aleksandrov [4] P i ~ l l y we propose a generalization of Conjecture SCONJECTURE 3. Let ~_

be a s2mmetric domain in

from an~ product domain il Ix @, ished boundar 2. Let ~r ..

and ~zz

Gc ~-I

different

be its distingu-

be two domains in ~

intersectim~

a proper mappin~ such that for some set r. ~r

Then there exists an automorphism ~

The v e r i f i c a t i o n

and let %

~

of C o n j e c t u r e

of /~

such that

3 would l e a d to a c o n s i d e r a b l e

strengthening of a result on local characterization of automorphisms of classical domains obtained in [2] One can see from the proof of Theorem 3 that Conjecture 3 holds for classical domains of the fourth type. At the same time, it follows from results of [I] and [5~ that Conjecture 3 holds also for the balls /ip, 4 REMARK. After the paper had been submitted the authors became aware of S.Bell's paper [6~ that enables, in combination with C2] , to prove Conjecture I

628

REFERENCES 1. A i e x a n d e r H. Proper holomorphic mappings in ~ o Indiana Univ.MathoJ., 1977, 26, 137-146. 2. T y M a H o B A.E., X e H E E H LM. ~ o K a ~ H a ~ xapaKTep~saE~H aHaJ~TE~eCEEX aBTOMOp~ESMOB E~accENecEEX o6~aoTe~. - ~OE~. AH CCCP, I982, 267, ~ 4, 796-799. 3. A ~ e E c a H ~ p o B A.~. K OCHOBa~ Teopz~ OTHOCETex~HOCT~. -BecTH.I~V, I976, I9, 5-28. 4. A x e mape. 5. R u d ball. 6. B e 1 -

E i 1

c a H ~ p 0 B A.B. CymecTBoBaHEe B R y T p e H H ~ x ~ y R E n ~ B MaTeM.c6., I982, II8, I47-I68. n W. Holomorphic maps that extend to automorphism of a Proc.Amer.Soc., 1981, 81, 429-432. S.R. Proper ho~omorphic mapping between circular domains.

Comm.Math.Helv.,

G.M.HENKIN

(r.M.XEHEHH)

R.G.NOVIKOV

(P.F.HOB~KOB)

1982, 57, 532-538.

CCCP, 117418, MOCEBa, ~eHTpa~BHH~ BEOHOM~Ko--MaTeMaT~HecE~R ZHCT~TyT AH CCCP, y~.Kpac~EoBa, 32

CCCP, 117234, MOCEBa, ~eH~HCEEe rope, MIV, ~ex.-~aT.~u'~yJIBTeT

629 12.4.

ON BIHOLOMORPHY OF HOLOMORPHIC MAPPINGS OP COMPLEX BANACH SPACES

Let ~ be a function holomorphic in a domain ~ ~ 3 c ~. It is well known that if { is univalent, then ~f(~) =~= 0 in ~ . Holomorphio mappings # of domains ~ c ~ for 11,> ~ also possess a similar property: if I : ~ ~ 6 ~ is holm, orphic and one-to-one then ~ Q ~ ) = ~ = 0 at every point $ of~(~)-------(~)},K=I~" ~ is the Jacobi matrix), or equivalently the differential ~ ( ~ ) ~ ~--~-~Q~)~ is an automorphism of ~ , and then, by the implicit function theorem, ~ itself is biholomorphic. Note that the continuity and the injective character of immediately imply that ~ is a homeomorphism because ~ ~= ~_~ 0 and that ~(0) = 4 , Then the restriction of ~ to [ 0~O] extends to a positive definite function on ~ . It follows that for all linear

where

BO ~-e~ B ( [ -. O , -C ~ ] - )

In t h i s case a l l normal extremals

can be easily determined and there exists an extremal measure (at least one). At the same time for pol,~.omial s qY of de~ree 2...these

648

problems still do not have a full solution. The simplest operator is provided by ~ ~ _ ~ l l . ~ , ~C~ . For X ~ the problems are solved (see papers of Boas-Shaeffer, Ahiezer and Meiman). For some complex ~ (in particular those for which the zeros of the symbol ~ satisfy the above mentioned condition) ~ admits (after a proper mormalization) a positive definite extension, so that the norm of ~ ( ~ ) coincides with its spectral radius+ Is it ~ossible to calculate the norm for al~

~C~

? How do

the " E u l e r e q u a t i o n s " l c o k in this case? Note that according to the Krein theorem the extremal measure is unique provided K = [ - O , O ] and the spectral radius is less than the norm. Of course, these problems remain open for polynomials of higher de~ree+ The Bernsteln inequality for fractional derivatives leads to the following PROBLE~. Consider on the function ~ ( ~ ) ~ ( ~ - - I ~ I ) ~

[-+,4]

o(>0

+ The problem is ~o find

If 06 ~ ~ then ~ is even and is convex on [ 0 , ~ ] , and therefore coincides on [-|~ ~] with a restriction of a positive definite function by the Polya theorem. For O~ < ~ T becomes concave on [0,|] and moreover ~ cannot be extended to a positive definite function on ~ . Indeed, if ~ is positive definite then - ~ " is a positive definite distribution. At the same time, --~" is nonnegative, locally integrable on a neighbourhood of zero and non, integrable on a left neighbourhood of the point ~ = | . A positive definite function cannot satisfy this list of properties. The best known estimate of the norm for O~ C (0,4) is 2(4+0()~~ It is evidently not exact but it is asymptotically exact for o~-+-0 and o~ +-~ | . It should be noticed that in the space of trigonometric polynomials of degree ~ PP~ the norm of the operator of fractional differentiation coincides with its spectral rsdius for o~ >I o~0 , where oCo ~ o ~ o(P~)< ~. Another example is related to the family { ~ of functions definite.

Consider the family for

teristic function

~

o~ > +

.

Since every charac-

satisfies the unequality l~(~)Im~ ~(~+I~(2~)I),

649

there are no positive definite extensions for c ~ 2 Consider now the case ~ < o ( < 2 ° The following idea has been suggested by A.V.Romanov Extend ~ ( ~ ) to (|,2) by the formula

v(1 ÷ Extend now the obtained function on ( 0 , ~ o an even periodic function of period 4 keeping the same notation ~ for this function. We have

k

~

,t

where the sum is taken over odd positive integers. It is easily verified (integration by parts) that ~ K and

0

are of the same sign. Clearly ~I > 0 Hence ~ is positive definite if

and

for

k

s.

@

(cf. [4], Ch.V, Sec.2,29). The ~unction ~(c&) decreases on (4,2)and ~(4)=oo ~(Z)< 0 . Therefore the equation ~ ( O C ) ~ 0 has a unique solution oCo ~ ( ~, ~) (Romanov's number). The function ~ is positive definite on [-I,~] if C ~ o C 0 . At the same time slightly modifying the arguments from [5], Tho4.5.2 one can easily show that for 0 < o C I < o C ~ < Z the function

4-

is positive definite on R . Hence ~4 is positive definite on [-4~4] if so i s ~ z . The "separation point" ~0 is clearly ~ o ~ Is it true that main open,

~o > o ~ @

? For C ~ > ~

the above problems re-

650 PROBLemS IN THE MULTIDIMENSIONAL CASE. For ~ ,except the case when the norm of ~ ( ] ) ) coincides with its spectral radius, very few cases of exact calculation of ~(D)~ and discription of extremals are known. The GENERAL PROB L E M h e r e is to obtain proper generalizations of Boas-Schaeffer's and Ahiezer-Meiman's theorems, i e, to obtain "Euler's equations" at least for real functionals. Our problems concern concrete particular cases; however it seems that the solution of these problems may throw a light on the problem as a whole. If ~ : ~ - ' ~ ~ is a linear form then the operator with the symbol ~ I~ is hermitian on B ( ~ ) and hence its norm coincides with the spectral radius. This again will be the case for some operators with the symbol of the form (p o ~)I K where p is a polynomial. The following simple converse statement is true . If is a polynomial and

for every symmetric convex star

K then ~ : a linear form and p is a polynomial. Does the ~ similar converse statement hold when

p o~

where ~ ia ranges over

the balls? Let

where p ~ |

and ~ C ~ ) = - ~

~

•

The operator ~ ( ~ )

is

obviously the Laplacian ~ .4 The norm of ~ coincides with the spectral radius in the following cases: n : ~ ~ : ~,p=|~ ~ , p : ° ° . The proof is based on the following well known fact: if ~ is a probability measure then { ~ " I ~ ( ~ ) ~ = ~} is a subgroup. The case e : £ turns out to be the most pathological and perhaps the most in~eresting. We have IIAIIB(Ka : ~ . In this case extremal measure is not unique and it would be interesting, to describe all extremal measures (notice that they form a compact convex set). The problem of calculation of the norm can be reduced to the one-dimensional case for operators of the form ~(~) in ~(~2) . It is possible to calculate the norms by operators with linear symbol in the space ~ (K2) explicitly, For example, the norm of Cauchy-Riema~u ope-

651 rator equals 2 and its normal extremal is unique. Namely,

However, for the operators of the second order the things are is a positive , the spectral and normal extremals

more complicated. If the symbol *'(~) -- ( A ~,~) real quadratic form then radius of Z(D) coincides wi~h that of ~ are of the form

II (D)II

where

10~1 i s the norm,

~ ~ ~

A

t~! ~

At the same time nothin~ i s known about ext.remals ...and norm of the operator

0~+ "@==

in

B (Kz)

for

~=Z.

REFERENCES I, A x a e s e p

H.H. ~ e ~ a z

no ~ e o p u

annposcmMa~a.

Moc~m,

HaT-

rm, 1965. 2. r o p a H E,A. HepsBeHc~a BepHmTe2HS C TOq~Z s p e ~ Teopaa ouepSTOpOB. -- BeOTH.XspBE.yH-TS, ~ 205. I I I ~ F J ~ S MaTeaaTz~m a MexaHME8, m~n.~5. - XaD~EOB, B ~ 8 m~oaa, HS~-BO XaI~E.~H--Ta, 1980,

77-105. 3. r o p ~ H E.A., H o!o B ~ ~ a o C.~. 3KcTpe~ma~ HeEoToI~X ~i~eloe~z~a~x one~aTopoB. - ~ o ~ no T e o p H onep~TOpOB B ~HE-~oHaa~x nI~OTImHOTBaX, MZ~OE, 4-11 ~ a s 1982. Tesac~ ~oz~., 48-49. 4. Z y g m u n d

A.,

Trigonometric Series, vol.l. Cambr Univ. Press,

1959. 5. L u k a c s London,

E .

Characteristic functions, 2 n d e d . ,

Griffin,

1970.

E.A. GOB.IN

CCCP, 117288, MOCEBa,

(E.A.rOPm~) MezaEz~o-~a TemaT ~ e C~a~ ~ a ~ T e T

652 13.8. old

ALGEBRAIC EQUATIONS WITH COEFPICIENTS IN COMMUTATIVE BANACH ALGEBRAS AND SOME RELATED PROBLEMS

The proposed questions have arisen on the seminar of V.Ya.Lin and the author on Banach Algebras and Complex Analysis at the Moscow State University. In what follows A is a commutative Banach algebra (over C ) with unity and connected maximal ideal space M A . ~or ~ ~ , denotes the Gelfand transform of • . A polynomial p ( ~ = ~ + ~ ~'~ + •+ ~ ~ ~ £ is said to be s e p a r a b I e if its discriminant ~ is in+ vertible(i.e, for every ~ in M~ the roots of ~ + ~4 (~) + +~(~) are simple); ~ is said to be c o m p 1 e t e I e r e d u c i b 1 e if it can be expanded into a product of polynomials of degree one. The algebra is called w e a k 1 y a 1 g e b r a i c a 1 1 y c 1 o s e d if all separable polynomials of degree greater than one are reducible over it. In many cases there exist simple (necessary and sufficient) criteria for all separable polynomials of a fixed degree Wv to be completely reducible. A criterion for A = C(~) , with a finite cell complex ~ , consists in triviality of all homomorphiams ~4(X) , B(~), B(~) being the At%in braid grQup with threads [I]. If (and only if) ~.< 4 this is equivalent to ~ ( ~, ~ ) = 0 (which is formally weaker). The criterion fits aS a s u f f i c i e n t one for arbitrary arcwise connected locally arcwise connected spaces ~ . It can be deduced from the implicit function theorem for commutative Banach algebras that if the polynomial with coefficients ~ is reducible over ~ ( ~ A ) then the same holds for the original polynomlal p over ~ . On the other hand (cf, [2], [3]) for arbitrary integers ~ , ~ , 4< k % ~ < co there exists a pair of uniform algebras A c B , with the same maximal ideal space, such that ~/~ = 4 , all separable polynomials of degree % ~ are reducible over A , but there exists an irreducible (over ~ ) separable polynomial of degree ~ . WE INDICATE A CONSTRUCTION OP SUCH A PAIR. Let G k be the collection of all separable polynomials ~(~)= A k +~4~ k'~ + ,.. t ~ k with complex coefficients ~'I~''" ~ E k , endowed with the complex structure induced by the natur~l embedding into ~ k ~ (~4....,~k)" Define ~ as the intersection of G k , the submsmifold {E 4 = 0 ,

653

is a finite complex. The algebra ~ is the uniform closure on of polynomials in Z~,,,,Z k and ~ consists of all functions in B with an appropriate directional derivative at an appropriate point equal to zero. With the parameters properly chosen, (A,B) is a pair we are looking for (the proof uses the fact that the set of holomorphic functions on an algebraic manifold which do not take values 0 and S is finite, as well as some elementary facts of Morse theory and Montel theory of normal families that enable to control the Galois group). Do there exist examples of the same nature with

A

weakl~al-

~ebraicall~ closed? We do not even know any example in which A is weakly algebraically closed and C ( M A) is not. A refinement of the construction described in ~4~ and [ 5B may turn out to be sufficient. If X is an arbitrary compact space such that the division by 6 is possible in H ~ ( ~ Z ) then all separable polynomials of degree 3 are reducible over C(~) . The situation is more complicated for polynomials of degree 4: there exists a metrizable compact space of dimension two such that ~ ~ ~ ) = 0 but some separable polynomial of degree 4 is irreducible over C (~) [6~. on the other hand, the condition that all elements of ~ (~,~) are divisible by ~! is necessary and sufficient for all separable polynomials of degree ~~ to be completely reducible, provided ~ is a homogeneous space of a connected compact group (and in some other cases). These type's results are of interest, e.g., for the investigation of polynomials with almost periodic coefficients. Is it possible to describe

"al!" spaces

~

(mot necessaril Y

~om~ct) for which the problem of compl~$e reducibi~t~ oTer

C(X)

of the separable pol.ynomials can be solved in terms of one-dimensional cohomolo~ies? In particular, is the condition

~

(~,2) = 0

sufficient in the case of a (com~act~ hqmogeneous space of a oommected Lie ~roup?

(Note that the answer is affirmative for the homo-

geneous spaces of c o m p 1 e x Lie groups and for the polynomials with h o 1 o m o r p h i c coefficients ~9~). Though the question of complete reducibility of separable pmlynemials in its full generality seems to be transcendental, there is

654 an encouraging classical model, i.e. the polynomials with holomorphic coefficients on Stein (in particular algebraic) manifolds. Note that the kmown sufficient conditions t9] for holomorphic polynomials are essentally weaker than in general case. The peculiarity of holomorphic function algebras is revealed in a very simple situation. Consider the union of ~ copies of the annualus ~ Z : ~-4 < I~ ~ < ~ 1 identified at the point ~ = ~ . It can be shown that a separable polynomial of prime degree ~ with coefficients holomorphic on these space, and with discriminant ~=~ is reducible if ~ >, ~0 (~, ~) , primarity of Yv being essential for Y~>/~ [I0]. If ~=~ , ~v can be arbitrary [2], and we denote by ~0(~) the corresponding least possible constant. Now if is even then ~o(~) = ~ , and so the holomorphity assumption is superfluous. However ~o ( ~ ) ~ C ( ~ ~ if ~ and ~ are odd, with C ( k ) ~ for k ~ 5 . At the same time ~o(~) -< C ~ for all • . These results, as well as the fact that ~o(?)~/P--.~ as p tends to infinity along the set of prime numbers, have been proved in [I0~. Nevertheless the exact asymptotic of ~o(p) re-mains ~ o w n ,

it is ur...known even whethe r

~@ (p)--~ oo

a..ss ~-~oo

.

If ~ is a finite cell complex with H 4 ( ~ , ~ ) = 0 then each completely reducible separable polynomial over C (~) is homotopic in the class of all such polynomials to one with constant coeffinie n t s (the reason is that ~'~$CG):0 for ~>~ ). Let X:ivIA and consider a polynomial completely reducible over ~ . Is it p ossib!e to realize the homotop.y within the class of pQlynomials over

~ ?

Such a possibility is equivalent, as a matter of fact, to 13dthe holomorphic contractibility of the universal covering space ~ for ~ . It is known [li] that ~ : C ~ ~ V ~-'k , ~-~ being a bounded domain of holomorphy in C~ homeomorphic to a cell [12]. In ~ there are contractible but non-holomorphically contractible domains [12~, though examples of bounded domains of such a sort seem to be ,~n~own (that m i ~ t be an additional reason to study the above question). Evidently ~5 = ~£ x ~ is holomorphically contractible. Is the same true for

~

with

~

?

There are some reasons to consider also transcendental equations ~W)=0 , where IRA-* ~ is a Lorch holomorphic mapping (i.e. ~ is Fr~chet differentiable and its derivative is an opera-

655

tor of multiplication by an element of A ). In [13] the cases when equations of this form reduce to albebraic ones have been treated (in this sence the standard implicit function theorem is nothing but a reduction to a linear equation). A systematic investigation of such trancendental equations is likely to be important. This might require to invent various classes of Artin braids with an infinite set of threads.

REFERENCES I. r o p ~ R E.A., ~ E H B.A. Aaredpa~ecE~e ypaBHeHY~I C Henpepm~G~ Eos~eHTaM~ E HeEoTopHe Bonpocw aare6pam~ecEo~ Teopn~ Eoc. -MaTeM.cd., I969, 78, 4, 579-610. 2. r o p ~ H E.A., ~ ~ H B.A. 0 cenapadex~m~x n o ~ o M a x Ha~ Eom~yTaTZBm~M~ daaaxoB~M~ axre6paM~. -~oEa.AH CCCP, 1974, 218,

3, 505-508. 3. r o p n H E.A. ro~oMop~HHe ~ y ~ E L ~ ~a ax2edpa~ecEoM MHOrOOd-p a s ~ ~ IIp~IBO~MOCT]~ ceHapade~H~x n~n~OMOB Ha~ HeEoTOp~M~ EOM-~ . ~ y T a T I ~ d a H a x o B ~ a~redps~. - B EH.: Tes~cH ~OF~.7-~ BceCO~BHO~ TOH.I{OH~., MI~HCE, 1977, 55. 4. r o p E H E.A., K a p a x a H ~ H M.H. HecEoa~Eo saMeqaHN~ od ~ r e 6 p a x HelIpepRBHMX ~ y H I ~ Ha ~ O K ~ H O C M S H O M EOMIIaETe. - B m~.: Tes~cH ~oF~. 7-~ Bceco~sHO~ TOII.KOH~., MHHCE, 1977, 56. 5. K a p a x a H a H ~.H. 0d a~edpax Henpep~BHMX ~ y ~ E L ~ Ha ~O-ESJIBHO C M S H O M EOMIIaETe. -- ~HEI~.aHaJI. E ero n p ~ . , 1978, 12, 2, 93-94. 6. J~ ~ H B.A. 0 IIOX~HOMaX ~eTBepTo~ CTelIeHI~ Ha~ a~redpo~ Henpep~mm~x @ym~n~. - ~ m : ~ . a ~ s . ~ . ~ ero r r p ~ . , 1974, 8, 4, 89-90. 7. 3 ~o s E H D.B. A~e6pa~ec~'~e ypa~Rem~:~ c Henpep~mm~m EOS~H-LV~eHTaM~ Ha O~H0pO~H~X npocTpaHcTBaX.- BecTHnE M~Y, oep.MaT.Mex., 1972, ~ I, 51-53. 8. 8 ~ s ~ H D.B., ~I ~ H B,~, HepasBe~B~eHH~e axredpa~ecE~e p a c ~ p e H ~ EOMMyTaTEBHRX daHaXOBMX a~redp. - f~aTeM.c6., 1973, 91, 3, 402,-420. 9. /l ~ H B.~I. AJmeOpo~m~e (~yHEs~a H roaoMop~m~e SaeMeHTH ZDMO-mon~ecE~x r10ynn EOM~e~c~oro M~o~oodpas~. -~oEx.AH CCCP, 1971, 201, I, 28-31. I0. 8 I0 3 I~ H ~0.B. HenpHBo~w~e cenapade~H~e nom~HOM~ c rO~OMOIX~-HBMH E O S ~ I ~ e H T a M I ~ Ha HeEoTOpOM I¢~lacce EOMII~eEOHRX IIpOCTpaHOTB° -MaTeM.Cd., 1977, 102, 4, 159--591.

656

II. K a ~ ~ M a H W.H. ro~oMop~Ha~ y H ~ e p c a : ~ a a ~ H ~ H B ~ npocTpaHCTBa nO~G~IOMOB des EpaTHRX EOpHe~. -- ~ . aHaJI. E ere n p ~ . , 1975, 9, I, 71. 12. H i r c h o w i t z A. A p r o p o s de principe d'0ka.- C.R.Aca~. sci. Paris, 1971, 272, ATS2-A794. IS. r o p ~ E.A., CaH~ e c Eapxoc @ep ~a~e c. 0 ~pa~c~e~eHm:~x ypaBHeRm~X B z o ~ a T ~ B H ~ X 6a~axoB~x a~edpax. -~y~.aRa~. ~ ePo ~ p ~ . , I977, II, I, 63-64. E.A.GORIN (E,A.IDPMH)

CCCP, 117284, MOcEBa ~ e ~ H c E E e ropH MOCEOBCE~rocy~apCTBeH~ YHHBepCZTeT Mexam~o--MaTeMaT~ecE~$BEy~TeT

COW~S~TARY BY THE AUTHOR

Bounded contractible but non-holomorphically main of holomorphy in C ~ have been constructed questions, including that of contractibility of space ~ , seem to rest open. A aetailed exposition of a par~ of ~ 131 can

contractible doin K14]. All other the Teichmuller be found in ~15].

REI~ERENCE S 14. 3 a ~ ~ e H 6 e p r M.r., Jl H H B.~I. 0 rOHOMOp~Ho He CT~:r~BaeM~x o ~ a ~ m : e H H ~ X o6~aCTHX rO~OMOp(~HOCS .-- %oF~.AH CCCP, 1979, 249, ~ 2, 281-285. 15. F e r n ~ n d e z C. S a n c h e z , G o r i n E.A. Variante del teorema de la funcio~n implicita en ~lgebras de Banach conmutativas. - Revista Ciencias Matem~ticas (Univ. de I~ Habana, Cuba), 1983, 3, N I, 77-89.

657 13.9. ola

HOLOMORPHIC MAPPINGS OF S O ~ SPACES CONNECTED WITH ALGEBRAIC ~VNCTIO~S

I. For any integer ~ , = ~ > ~ + ~4A~'~+ . • . + % ~ , and let consider the polynomial p(~) . Then ~ is a polynomial in ~ (~) be the discriminant of ° ,''',~ and the sets G ~ = { ~ :

p

=

uc N { z : z, = o } ,

5G~, = { ~ : ~

= o,

~(z)

are non-singular irreducible affine algebraic manifolds, isomorphic %o G ~ X ~ . The restriction ~ = ~ I~

= t.}

oGp~

being

:

@

G~ " C* == C \ [0} is a locally trivial holomorphic fibering with the fiber S G ~ . These three manifolds play an important role in the theories of algebraic functions and of algebraic equations over function algebras. Each of the manifolds is ~ ( ~4, 4) for its fundamental group ~4 , ~4 (G~) and ~4 CG~) being both isomorphic to the Artin braid group ~(~) with ~ threads and ~4 ( S G ~ ) i being isomorphic to the commutator subgroup of ~(~), denoted B (~) ([I],[~). ~ and ~ p -cohomologies of G~ are k~own [I], [ ~ , [ ~ . However, our knowledge of analytic properties of , ~ ~ G ~ essential for some problems of the theory of algebraic functions is less than satisfactory ( ~ ] - [ I ~ ) . We propose several conjectures concerning holomorphic mappings of Go and ~ G ~ Some of them have arisen (and all have been discussed) on the Seminar of E.A.Gorin and the author on Banach Algebras and Analytic Functions at the Moscow State University. 2. A group homomorphism H - - ~ H ~ is called a b e 1 i a n (reap. i n t e g e r ) if its image is an abelian subgroup of H~ {reap. a subgroup isomo~hic to Z or { 0 } }. For comple~ spaces X and Y , C(X,~)__ Ho~ C X , Y ) and H 0 ~ * C X , Y ) stand for the sets of, respectively, continuous, holomorphic and n o n - c o n s t a n t holomorphic mappings from X to ~ . A mapping ~ C ( o G~)is said to be s p 1 i t t a b 1 e if there is ~ ~ C ~ , ~) such that is homotopie to ~ ° ~ ° ~, ~ ' ~ - - - - ~ * being the standard mapping defined above; ~ is splittable if and only if the induced homomorphism ~, :~(~) ~ ~4 < ~ ) ~4 C G °~) ~ ~(~) is integer. There exists a simple explicit " description of splittable elements of H0~ , G~) [6].

658 CONJECTLhgJ~ I. Let ~ > 4 ~EH0~CG

,~)is

and ~ @ ~

. Then

splittable; (b) H 0 ~ * ( ~ ,

(a) ever~ SE n ) =

2.

It is easy to see that (b) implies (a). Let ~ (~) be the union of four increasing arithmetic progressions with the same difference ~ (~-~) and ~hose first members are According to [6], if and F ~ ( ~ ) then all ~ 0 in H ~ CG~o ,G~)_ are splittable. A complete description of all non-spllttable ~ in H0~(G; ,G~) has been also given in [6]. If ~ >~ and ~ < ~ , there are only trivial homomorphisms from ~/(~) to ~(~) [11]. Thus for such and ~ all elements of C ( ~ , G~) are splittable and all elements of g (S G~ ~ ~ G~) are contractible. The last assertion implies rather easily that H0~*(~G~ ~ ~ G ~ ) ~ ~ . It is proved in [10] that for ~ ~ ~ each # ~ Ho~G~, ~G~) is biholomorphic and has the form ~ (~,...~ ~ ) = ~ 6 2 ~ , 63~3,.. , ~ ) with 6~(~-0 = ~ A useful technical device in the 3, Let C** = ~ \ {0, ~} topic we are discussing is provided by explicit descriptions of all functions ~ E mo~ * (X, C'*) for some algebraic manifolds associated with g~ , ~ and ~ ( [6], [8],[9], [10] ). This has led to the questions and results discussed in this section. Let ~ be the class of all connected non-singular affine algebraic manifolds. For every X ~ the cardi~lity ~(X) of mo~ ( X ~ C ) i s f i n i te (E.A.Gorin).Besides, if "H~>'I¢I,G~ [~.(X),'lJ ("6(X) is the rank of the cohomology group "~ (~) for any d i S t i n c t points ~ ~ • . , , ~,14.,Ig;(~ . Using these two assertions, it is not difficult to prove that, given X ~ and ~>,~ , the set m0~ ~ ~X, ~ ) is finite. In particular, for every ~,~>/~ the set H o ~ ( ~ G ~ , ~G~) is finite. Let Top,X) be the class of all y in ~ homeomorphic to X ; it is plausible that for an,~

H4(X,Z)) thenFlo~,'~(.X,l~\{~,...,}~})

~

~

the function

~:To~ (X)--'~'+

is bounded. I even

do not know any example disproving the following stronger CONJECTURE: there exists a function such that l ~ ( X ) ~ ~)('("(.X).) A function

~: 2 + - " 2 +

~: ~ + - - - - ~ +

for all X {p..jI~ • with ~ < ~ ) ~< ~4 ~ , ~>~.

° ~ ° Hog(G~,X¢), ~ HogCX¢, G~).

CONJECTURE4, (a) Let ~4 ~ I f ~0) i£

~ ! ~ ~ $

for every

t,teK.

DEFINITION. A compact set K ( ~ C ) is r e g U I a r if there exists a mapping ~K: ~ + ~ K ) enjoying the following property: for every '5 > 0 and for every# S-analytic on there exist a function ~ analytic in ~K ($) and a set W ,

W ~ u(K)

such that W ~

I~KC~,),

#IW= odlW.

The set

(1)

.-4

S-- {j : j -- 4,£,...1U[o] is not regular. Indeed, putting

< 0

{- (j-'+ (j+1)-I) j-4,~,. ,.

fl

we see that and j but lytic.

~( U $)

is S-analytic on S contains no set where

QUESTION. Is every plane ¢ontinuum(i,e. s#t) regular?

for all ~values of a 1 1 ~ are anaa q0mpact connected

671 This question related to the theory of analytic continuation probably can be reformulated as a problem of the plane topology. Its appearance in the chapter devoted to spaces of analytic functions [the first edition of the collection is meant -Ed.] seems natural because of the following theorem, a by-product of a description of the dual of the space ~ ( ~ ) of a l l fvalctions analytic on K . THEOREM. Let Bore! measure on K e~K A ~\ K

, with

~

a reRular compact set and ~ such that

~(e)=

0

C~(K\e)

~

K

a positive for ever~

• Then every function

~

e

,

anal2tic in

i,s, representable b2 th e f o l l o w i ~ formula

A +,o

(~)~)0

,r,+,-,-.m

beinR a sequence of

"*+

'

L~(~)-functions and

,,++,.,+II'/"" L+(~) = 0

This theorem was proved in [ I ] . The regularity of K leads to a definition of the topology of ~ ( ~ ) explicitly involving convergence radii of germs of functions analytic on ~ . Unfortunately, the regularity assumption was omitted in the statement of the Theorem as given in [1] (though this assumption was essentially msed in the proof - see [I], the beginning of p.125). The compact K was supposed to be nothing but a continuum. A psychological ground (but not an excuse) of this omission is the problem the author was really interested in (and has solved in [I] ), namely, the question put by V.V.Golubev ([2], p.111): is the formula (2) valid for every function ~ analytic in ~ \ ~ provided K is a rectifiable simple arc and ~ is Lebesgue measure (the arclength) on K ? The regularity of a simple arc (and of every 1 o c a 1 1 y - c o n n e c t e d plane compact set) can be proved very easily, see e.g., [ ~ , p.146. The Theorem reappeared in [4] and [ ~ and was generalized to a multidimension~l situation in [ ~ . It was used in [6] as an illustration of a principle in the theory of Hilbert scales.

672

We have not much to add to our QUESTION and to the Theorem. The local-connectedness is not necessary for the regularity: the closure of the graph of the function ~ - ~ ~ , t ~ ~0~ ~] is regular. The definition of the regularity admits a natural multidimensional generalization. A non-regular continu~n in C ~ was constructed in K7]. The regularity is essential for the possibility to ~epresent functions by Golubev series (2): a function analytic in ~ \ S (see (])) and with a simple pole of residue one at every point j-~ (] = ~ , ~ o o , ) is not representable by a series (2). Non-trivial examples of functions analytic off an everywhere discontinuous plane compactum and not

representable by a Golubev series (2) were given in ~8]. REFERENCES

I.

2. S.

X a B ~ H B.H. 0 ~ a2a~or p ~ a ~opaHa. - B ~ , : " H c c ~ e ~ O B 2 ~ no CoBpeMeHH~M npo6~eMaM T e o p ~ ~ y ~ z ~ EOMn~eKcHoro nepeMeHHoro ". M., {~sMaT~s, I96I, I2I-I3I. r o x y 6 e B B.B. 0 ~ H o s H a ~ e a H s J m T ~ e c ~ e ~yHzny~. ABTOMOp~-~ e ~yHKL~H. M., ~ESMaTI~g8, 1961. T p y T H e B B.M. 06 O~HOM a~axore p ~ a ~opaHa ~ ~ MHOr~X E o ~ e E c ~ x nepeMesH~x, r O ~ O M O p ~ Ha C ~ B H O JG~He~Ho BH-nyEm~K M~omec~Bax. - B C6.~rOJIOMOp~HNe ~yHEI~H~ ~g~OISLX E O ~ e E C H R X

4. 8. 6.

nepeMe~"o KpacHo~pcE, H~ CO AH CCCP, 1972, I39-152. B a e r n s t e i n A. II. Representation of holomorphic functions B a e tions M ~ T

by boundary integrals.-Trans. Amer.Math. Soc., 1971, S 69,27-37. r n s t e i n A. If. A representation theorem for funcholomorphic off the real axis. - ibid. ]972,165, 159-165. ~K P ~ H B.C., X e H ~ Z H r.M. ~ [ ~ H e ~ e s a ~ a ~ Eown-

~IeECHOI~O a H ~ s a . 7.

-Ycnexz

Ma~eM.HayE, 1971, 26, 4, 93--152.

Z a m e R. Extendibility, boundedness and sequential convergence in spaces of holomorphic functions. - Pacif.J.Math., 1975, 57, N 2, 619-628.

8.

B ~ T y m z E H A.P. 06 o~Ho~ sa~a~e ~a~xya. - HSB.AH CCCP, cep.MaTeM., 1964, 28, ~ 4, 745--756. V. P. HAVIN

(B.n.XAB~H)

CCCP, 198904, ~eE~Hrps~ HeTepro~, F~6xHoTe~Ha~ n~omaA&, 2 ~eHHHzpa~cE~A rocy~apcTBeEH~ yH~BepcxTeT, MaTe~aT~zo-~4exa~m~ ec E ~ ~aEy~TeT

673

* * *

CO~ENTARY

BY THE AUTHOR

The answer to the above QUESTION is YES. It was given in [9] and [10] . Thus the word "regular" in the statement of the Theorem can be replaced by "connected"

(as was asserted in 11] ).

REFERENCES

9, B a p ~ o ~ o M e e B H8

8 I ~ OK!08CTHOOTB.

A.JI. AHa~aTa~ecKoe n p o ~ o ~ e H ~ e c K O H T g H ~ y ~ -- 3 8 1 ~ C K ~ H S ~ q H . C e M ~ H . ~ 0 ~ , 1981, I13, 2 7 -

40. 10. R o g e r s

J.T.,

Z a m e

W.R. Extension of analytic functions

and the topology in spaces of analytic functions. Math.J.,

1982, 31, N 6, 809-818.

- Indiana Univ.

674 8.3. old

THE VANISHING INTERIOR OF THE SPECTRU~ Let A

and B be complex unital Banaoh algebras and let , then i% is well known that

~4cB

6'ACx ) --,

an~ S ~ ( ~

B,,B(x)

~

(~

,

where ~ ( ~ ; is the spectrum of ~ relative to ~ and ~gA(X) is its boundary. Taking ~ %0 be the unital Banach algebra generated by X , in this context we say that x i s n o n-t r i v i -

a 1

i~

~

~(~

~

*

. ~ilov DJ has proved that if

~ ~.

~-

is permanently singular in ~ (i.e. ~ - X is not inver%ible in any superalgebra ~ of ~ ) if, and only if, ~ - - x is an approximate ~e~ ~sor (AZ~ o~' A i.e. i~ ~ , ~ A , ~ ' ~ , l l = ' ~ , s u c h -~hat,8,,~(A-:~--,-O

(I~, > ~,).

Let ~ ~ ~ ~ 0 ~ ~ ~ 0.

(~ a~})

be a sequence in ~

denotes a sequence of complex numbers) is a Banach algebra

under the norm N~

~ ~ ~ II ~--- E 0

~d

with ~0~---1 and ~t.t~