Factorization of Measurable Matrix Functions [Reprint 2021 ed.] 9783112471722, 9783112471715

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G. S. Litvinchuk • I. M. Spitkovskii Factorization of Measurable Matrix Functions

Mathematical Research

Mathematische Forschung

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Karl-WeierstraB-lnstitut für Mathematik

Band 37 Factorization of Measurable Matrix Functions by C . S. Litvinchuk and I. M. Spitkovskii

Factorization of Measurable Matrix Functions

by Georgi Semenovich Litvinchuk and llya Matveyevich Spitkovskii edited by Georg Heinig with a foreword by Bernd Silbermann

Akademie-Verlag Berlin 1987

Autoren: Prof. Dr.

Dr.

Xlya

Georgi

Semenovich

Matveyevich

Litvinchuk

Spitkovskil

Akademie

der

Wissenschaften

der

Institut

für

Hydromechanik,

Odessa

USSR

Herausgeber: Doz.

Dr.

sc.

Technische Sektion

Bernd

ins

Die

Englische:

Universität

dieser

Schriftenreihe

3-05-500368-3

ISSN

0133-3019

Erschienen DDK-1086

im A k a d e n i e - V e r l a g

Berlin,

Leipziger

Akademie-Verlag

Lizenznummer Printed

in

LSV

Dr.

202.

vom O r i g i n a l m a n u s k r i p t

der

Berlin,

Straäo

3-4

1987

100/407/87 Democratic

VEß K o n g r e ß -

Reinhard

Republic und VVerbedruck,

Höppner

1065

Bestellnummer: 04800

üerlin

the Gernan

Gesamtherstellung: Lektor:

werden

reproduziert.

ISBN

(c)

Kiirl-ilarx-Stadt

Mathematik

Titel

Autoren

Heinig Karl-Marx-Stadt

Luderer

Technische Sektion

Georg

Mathematik

Übersetzung Dr.

nat.

Universität

703

737

5

(2182/37)

9273

Oberlungwitz

Foreword In t r e a t i n g mathematical problems, i t i s o f t e n necessary t o f a c t o r i z e matrix-valued ( o r even operator-valued) f u n c t i o n s in a c e r t a i n manner. F a c t o r i z a t i o n s c l o s e l y connected with the notion of Wiener-Hopf f a c t o r i z a t i o n play a prominent r o l e . The development of f a c t o r i z a t i o n theory was, among other t h i n g s , stimulated by needs a r i s i n g i n t h e theory o f s i n g u l a r i n t e g r a l operators and i t s manifold m o d i f i c a t i o n s . Richness and profundity of the r e s u l t s obtained as w e l l as t h e i r s i g n i f i c a n c e f o r the theory of s i n g u l a r i n t e g r a l operators and o t h e r branches of mathematics required e x p l a n a t i o n of t h i s theory in monographs. A f t e r the book " F a c t o r i z a t i o n o f Matrix Functions and S i n g u l a r i n t e g r a l Operators" (Birkhauser Verlag, Basel 1 9 8 1 ) ,

excellently

w r i t t e n by K. CLANCEY and I . GOHBiSRG, the present monograph o f G. S . LITVINCHUK and I . M. SPITKOVSKlY " F a c t o r i z a t i o n of Measurable Matrix Functions" i s now the second work a v a i l a b l e on problems of ffiener-Hopf

factorization.

In addition to problems already discussed i n the book of CLANCEY and GOHBjiRG, i t c o n t a i n s a s e r i e s of more r e c e n t r e s u l t s p a r t l y due t o the authors and of considerable i n t e r e s t . I t should be emphasized t h a t the monograph r e f l e c t s not only the s t a t e of the a r t i n Wiener-Hopf f a c t o r i z a t i o n o f m a t r i x f u n c t i o n s , but i t a l s o f o c u s s e s the r e a d e r ' s a t t e n t i o n on many problems which remain t o be solved. The r i c h b i b l i o graphy containing both t h e o r e t i c a l papers and work on p r a c t i c a l

appli-

c a t i o n s i s a f u r t h e r advantage of the book, which w i l l g r e a t l y b e n e f i t a l l those i n t e r e s t e d in f a c t o r i z a t i o n

theory. B. Silbermann

Preface A series of reviews and original papers on factorization theory, which were read at the Odessa city seminar on boundary value problems and singular integral equations in 1973-77, served as a source of the present book. Furthermore, lectures given by the authors at the Department of Mechanics and Mathematics of the Odessa University have been incorporated. The authors are aware that their contribution to factorization theory, although it is very comprehensive (ranging from existence problems of a factorization, questions of estimates and stability of partial indices, to applications to the theory of the generalized Riemann boundary value problem), looks much more modest than the successes of their predecessors. Nevertheless, the necessity of creating a sufficiently complete review of factorization problems of measurable matrix functions explained from a unified point of view seemed so evident to the authors that they decided to write this booklet. It may be considered complementary to the well-known monographs of N. I. MUSKHELISHVILI and N. P. VBKUA, which are devoted to the classical factorization theory of matrix functions with Holder elements. We wish to emphasize that the subject matter will be discussed along lines meant not only for specialists familiar with the material, but also for readers who first become acquainted with the topic as well as for persons whose interest focusses, above all, on applications. The reader's judgment will show in how far this attempt has been successful. The authors

7

CONTENTS

Introduction Chapter 1. Background information 1.1. Some facts on the geometry of Banach spaces and operator theory in these spaces 1.2. Some information on boundary properties of functions analytic and meromorphic in finitely connected domains 1#3 On the operator of singular integration in spaces of summable functions

11 17 17

Chapter 2. General properties of factorization 2.1. The definition of factorization 2.2. Properties of factorization factors 2.3. The domain of factorability 2.4. Factorization of meromorphic matrix functions and their products 2.5. Comments

55 56 58 63 67

Chapter 3. The criterion of factorability. si-factorization and its basic properties

82

3.1. On solvability of the Riemann boundary value problem with factorable matrix coefficient 3.2. A criterion of factorability 3.3« On the normal solvability of the vector-valued Riemann boundary value problem and the problem associate to it 3.4. Left and right S-factorization. A criterion of simultaneous Fredholmness of the Riemann boundary value problem and the associate problem to it 3.5. ^-factorization of unbounded matrix functions on contouiBof class (R. 3.6. ^-factorization of bounded measurable matrix functions on contours of class S. 3.7. Comments

Chapter 4. ^-factorization of triangular matrix functions and reducible to them 4.1. Existence problems of a ^-factorization of triangular matrix functions 4.2. Effective construction of a ^-factorization and estimates for the partial indices of triangular matrix functions with ^-factorable diagonal elements 4.3. Calculation of partial indices of second-order triangular matrix functions 4.4. ¿¡»-factorization of functionally conmutable matrix functions 4.5« Comments Chapter 5. Some classes of factorable matrix functions 5.1. ^-factorization of matrix functions from classes L± + C CO 5.2. Factorization in Banach algebras of matrix functions 5.3« ^-factorization of piecewisp continuous matrix functions 5.4. Comments

29 45

80

84 93 99 105 109 116 126

132 135 140

147 154 163 169 172 180 195 202

Chapter 6. On the stability of factorization factors 6.1. Stability criterion for partial indices 6.2. On the behaviour of partial indices under small perturbations. The structure of the family of ^-factorable matrix functions 6.3. Several estimates for the partial indices of measurable bounded matrix functions 6.4-. Sufficient conditions for coincidence of partial indices of matrix functions whose Hausdorff set is separated from zero 6.5. On partial indices of matrix functions of classes 6.6. On the stability of the factorization factors G, 6.7. Comments -

211 214 218 223 229 236 240 247

Chapter 7» Factorization on the circle 253 7.1. Definition of factorization on the circle 255 7.2. Factorization of Hermitian matrix functions 258 7.3« Factorization of definite matrix functions 265 7.4. Criteria for existence of $-factorization and 272 coincidence of partial indices associated with the behaviour of the numerical domain 7.5. Definiteness criteria and stability of partial 276 indices of bounded measurable matrix functions 7.6. On partial indices of continuous matrix functions 285 7.7. Comments 288 Chapter 8. Conditions of $-factorability in the space L 5 . Criterion of $-factorability in L a of bounded measurable matrix functions 8.1. Auxiliary results 8.2. Sufficient conditions of $-factorability 8.3. Criterion of $-factorability in L t 8.4. Comments

296

Chapter 9» The generalized Riemann boundary value problem 9.1. Criterion of Fredholmness of the generalized Riemann boundary value problem in the space L^ 9.2. Sufficient conditions of Fredholmness and stability. Estimates for the defect numbers of the generalized Riemann boundary value problem in the space L 2 9.3« The generalized Riemann boundary value problem with continuous coefficients. A stability criterion ' 9.4. Comments

315 316

References Subject index Notation index

340 371 372

10

297 304 307 309

325 333 336

INTRODUCTION The notion of factorization is widely used in different branches of mathematics. The term factorization itself simply means the representation of some mathematical object (number, function, matrix, operator, etc.) as a product of objects of the same type, but having certain additional properties. According to the type of the objects and the character of the problem studied (multiplication of numbers, pointwise multiplication of functions, their composition, etc.) this product can be understood in a different manner. Many mathematical propositions are, in fact, theorems on the existence and properties of a certain factorization. In principle, even the fundamental theorem of arithmetic (on the factorization of natural numbers in prime factors) and the fundamental theorem of algebra (on the polynomial factorization over the complex field) can be considered as examples of factorization theorems. The importance of these theorems needs no comments. As more special examples we mention the representation of matrices and operators in a Hilbert space in the form of a product of isometric by positively definite ones (so-called polar representation), the representation of functions of Hardy classes as a product of an inner and an outer one and its generalization to the case of matrix and operator functions, but also the representation of entire functions as product of other entire functions. Many methods for solving equations and boundary value problems of a different kind are based on the idea of factorization. The schema for using factorization in solving equations may be described in the following ways Let some operator A be represented in the form A = A~1A_.

(1)

Then the equation y = Ax is equivalent to the equation A + y = A_x. For a suitable choice of representation (1), the latter relation turns out to be more convenient for investigation, and sometimes it can be solved explicitly. Simultaneously, the original equation will also be solved. For instance, the representation of a square matrix A with principal minors different from zero as the product of non-singular triangular matrices allows us to reduce the solution of a system of linear equations with matrix A to a triangular system. There exists a continuous analogue of the mentioned result for Fredholm integral operators and a, more general result on the factorization of an operator acting in a v

Hilbert space along a chain of projections as well (GOHBEHG, KREIN [5], BAEKAH', GOHBERG [1,2]). 11

The f a c t o r i z a t i o n of matrix functions defined on some contour, which i s studied in the present book also belongs to f a c t o r i z a t i o n s of t h i s t y p e . I t appeared in connection with the study of systems of singular i n t e g r a l equations and boundary value problems f o r analytic functions» Throughout the book, by a f a c t o r i z a t i o n of the matrix function G defined on a closed contour r , which divides the closed complex plane into parts D + ( 3 0 ) and D " ( 3 » ) , we understand a representation of G in the form G(t) = G+(t) where

G+

A(t) G _ ( t )

(t 6

r),

are boundary values of a matrix function analytic and

non-singular in

D~, and

A ( t ) = diag(t

1

,...,t

a

).

Under s u f f i c i e n t l y general assumptions, the integers K 1 i . . . , K q are uniquely defined (up t o o r d e r ) by the matrix function G and are c a l l e d i t s p a r t i a l indices 1 The f a c t o r i z a t i o n of matrix functions i s the main part in the theory of vector-valued Riemann boundary value problems. I t permits us t o reduce the general case of t h i s problem t o the s o - c a l l e d jump problem* In t h i s aspect, the problem of f a c t o r i z a t i o n has been studied under d i f f e r e n t r e s t r i c t i o n s on the smoothness of the contour r and the matrix function G, s t a r t i n g from the end of the previous century by several generations of mathematicians. The r e s u l t of t h i s work f o r the case of piecewise Holder matrix functions and piecewise smooth contours are summed up in the well-known monographs of MUSKHELISHVTLI [ 1 ] , VEEUA [ 4 ] , in the survey of GAKHOV [ 3 ] and the paper of KHV3DSLIDZB [ 1 ] , Since the l a t e f i f t i e s , the f u r t h e r development of the theory of the Riemann boundary value problem, systems of singular i n t e g r a l equations with Cauchy kernel and systems of Wiener-Hopf equations has stimulated

•O A more complete and exact d e f i n i t i o n of f a c t o r i z a t i o n w i l l be given at the beginning of the second chapter. I n c i d e n t a l l y , we note that t o v e c t o r boundary value problems with a s h i f t ( i n p a r t i c u l a r , t o the Gazemann problem) there correspond d e f i n i t i o n s of f a c t o r i z a t i o n d i f f e r e n t from that one given above. In the present book we do not study f a c t o r i z a t i o n s of such a type, since in the v e c t o r case the theory has not been s u f f i c i e n t l y e l a b o rated h i t h e r t o . The corresponding f a c t o r i z a t i o n s f o r the scalar case can be found in the monograph of LITVINCHUK [ 3 ] .

12

the creation of new directions in the factorization problem. The faotorization of matrix functions with elements from the Wiener algebra (GOHBERG, KKEÍN [3]), the factorization of continuous (SIMONENXO [2]) and piecewise continuous (GOHBERG, KRUPNIK [1], MANDZHAVIDZE [6]) matrix functions has been studied, and a number of results concerning the factorization of measurable bounded matrix functions was obtained (DANILYUK [4,5], SIMONENKO [4,6], DOUGLAS [1]). At the Same time, it became clear that even in case of a continuous non-singular matrix function G a factorization with continuous factors G± does not always exist. On the one hand, this gave rise to the search for new classes of matrix functions admitting a factorization with continuous factors (see BUDYAHU, GOHBERG [3,4]) and, on the other hand, it led to the creation of the notion of factorization in classes 1 < p < oo. In case of continuous non-singular matrix functions, this factorization does not depend upon the choice of p. However, such a dependence holds already for piecewise continuous matrix functions. In this connection, there arises the task of studying the set of parameter values p , for which a given matrix function G can be factored. Moreover, in the Riemann boundary value problem theory in L^ spaces, a situation may emerge where the problem has a finite index, but is not normally solvable. Owing to this, we have to impose additional demands upon the factorization in order to guarantee the normal solvability of the boundary value problem. In the following, a factorization satisfying these requirements will be called a ^-factorization. There were also studies on the enlargement of the class of contours r, for which factorization theorems continue to be valid. The crucial idea of these studies reduces to the development of sufficient conditions for the boundedness of the operator of singular integration on I" (see e.g. KHVEDELIDZB [4]). In connection with the problem of calculating defect numbers and the approximate solution of the vector-valued Riemann boundary value pro blem and singular integral equations, the questions of stability of the factorization factors G + and A as well as of finding estimates for partial indices are not yet well understood, even under the classic assumptions made in VEKUA [4], GAKHOV [3], MUSKHBLISHVILI [1]. In the last 15 years a number of new results has been obtained in this direction (sufficient stability conditions, which can effectively be verified, sign preservation of all partial indices, which is also necessary in some cases (cf. VERBITSKIÍ, KRUPNIK [1,2], SPITKOVSKlf [5, 10], POUSSON [2], RABINDRAIUTHAN [1] and others),). In obtaining these results, the notion and properties of the numerical

13

range (or Hausdorff s e t ) of a matrix (MARCUS, MINC [ 1 ] ) i s used, i . e . questions related to the formula f o r the norm of a Hankel operator (see ADAMYAN, AROV, KREfN [ 1 ] ) and the operator of singular integration (VEBBITSKli, KRUPNIK [ 1 ] , KRUPNIK [ 4 ] ) . We would also l i k e to note that the factorization problem for the spectral density of a stationary random process (ROZAUOV [ 1 ] ) which arose in probability theory, some problems of projection methods f o r the solution of systems of singular integral and convolution type equations (GOHBERG, FELDMAN [ 3 ] ) and the theory of canonical systems of linear differential equations (KBe¥n, MELIK-ADAMYAN [ 1 , 2 ] ) show that the importance of the factorization problem f o r matrix functions i s not limited by i t s role in the theory of the vector-valued Riemann boundaryvalue problem. The factorization problem f o r matrix functions arises also in a natural way in the study of convolution type equations on a f i n i t e interval and on systems of intervals (GANIN [ 1 ] , KAHLOVICH, SPITKOVSKli [ 1 ] , KOMXAK [1—4], NOVOKSHENOV [ 1 ] , PAL'TSEV [2-6]. SPITKOVSKli [ 1 5 ] ) . We emphasize that here(in contrast to the classical case of the h a l f - l i n e ) even scalar equations yield a matrix problem of factorization. A l l this increases interest in the peculiarities of factorization of special type matrix functions (unitary, Heimitian, functionally oommutable f a t e . ) , information on which i s scattered in various papers (GORDIENKO [ 1 ] , GOHBERG, KREfN [ 3 ] , KHEfe, MELIK-ADAMYAN [ 1 , 2 ] , NIKOLAICHUK, SPITKOVSKli [ 1 , 2 ] , CHEBOTAREV [ 1 , 2 ] , SHMUL1 YAW [1,2] and others). At present there do not exist monographs or reviews on the factorization of matrix functions, in which the problems mentioned above would be discussed in a s u f f i c i e n t l y complete manner1K There are only three papers of a general nature by BOJARSKI [ 3 ] , GOHBERG, KREIN [ 3 ] , KHVEDKLIDZE [ 4 ] and the monographs of GOHBERG, FELDMAN [ 3 ] , DANILYUK [ 6 ] , PRbSSDORF [ 1 ] and PR'dSSDORP, MICHLIN [ 1 ] , in which some questions on the factorization of matrix functions are explained. The authors hope that the present book w i l l contribute towards f i l l i n g the gaps existing in the literature concerning this topic. We now intend to b r i e f l y describe the contents of the book. The f i r s t chapter is of auxiliary character. I t contains propositions from operator theory in Banach spaces and information on d i f f e r e n t olasses of analytic functions (mainly on Smirnov classes) necessary f o r understandin g the material contained in the main part of the book. In Chaptor 2 the notions of factorization and the domain of factorability of matrix 1'

While this manuscript was in preparation f o r publication, the mono- graph QLANGBI, GOHBERG [ 3 ] came out. I t s contents i s essentially covered by the prytgnn-h ftp^y,

14

functions are introduced, and t h e i r simplest properties are established* Besides, the f a c t o r i z a t i o n problem f o r matrix functions meromorphic i a D+ or D~" ( i n p a r t i c u l a r , r a t i o n a l matrix f u n c t i o n s ) i s treated t h e r e , and the problem of the change of p a r t i a l indices when being multiplied by matrix functions of the same kind i s studied. The connection between the f a c t o r i z a t i o n problem f o r matrix functions and the vector—valued Riemann boundary value problem i s regarded in Chapter 3 . ^ - f a c t o r i z a t i o n properties resulting from t h i s connection and corresponding propositions of the theory of Fredholm operators are established! Chapter 4 i s devoted to triangular and f u n c t i o n a l l y commutable matrix functions. In t h i s chapter the c r i t e r i o n of $ - f a c t o r a b i l i t y and e s t i mates for p a r t i a l indices (but f o r second-order matrix functions even exact foimulae) are derived. In Chapter 5 necessary and s u f f i c i e n t conditions of $ - f a c t o r a b i l i t y of continuous and piecewise continuous matrix functions, but also f o r matrix functions of the c l a s s L^ + C are established. The f a c t o r i z a tion problem in decomposing algebras of functions including the f a c t o r i z a t i o n problem of Holder matrix functions on a piecewise smooth contour and matrix functions from the Wiener algebra on the c i r c l e w i l l be studied. The c r i t e r i o n f o r s t a b i l i t y of p a r t i a l i n d i c e s , s u f f i c i e n t conditions f o r t h e i r non-positivity, non-negativity, coincidence or equality to zero are explained in Chapter 6 . In addition, we s h a l l i n v e s t i g a t e the behaviour of the f a c t o r i z a t i o n f a c t o r s G+ under small perturbations of the matrix function

G .

Chapter 7 i s dedicated to the f a c t o r i z a t i o n problem on the unit c i r c l e . Furthermore, the f a c t o r i z a t i o n of Hermitian and unitary matrix functions w i l l be studied. The necessity of several s u f f i c i e n t conditions f o r non-negativity (non-positivity, e q u a l i t y to zero, e t c . ) of p a r t i a l indices established in Chapter 6 w i l l be proved. Chapter 8 i s concerned with s u f f i c i e n t conditions f o r $ - f a c t o r a b i l i t y based on l o c a l principle and r e s u l t s of Chapters 5 - 7 . F i n a l l y , in Chapter 9 the theory developed in Chapters 3 - 8 i s used f o r the study of the generalized Riemann boundary value problem, which has manifold applications in geometry, mechanics, and physics (see BUDYAMJ [ 1 ] , GAKHOV [ 2 ] ) . The l a s t section of each chapter contains information and comments on r e s u l t s , which were not r e f l e c t e d in the main p a r t of the chapter, but

15

axe r e l a t e d to i t in c e r t a i n ways. Some unsolved problems are formulat e d . The f i r s t chapter makes an exception. For the r e a d e r ' s convenience, comments on references accompany every proposition (or a group of c l o s e l y connected propositions). L a s t l y , we want to remark that in the book we do not focus oux a t t e n t i o n on such problems as the f a c t o r i z a t i o n problem r e l a t i v e to a contour lying on a Riemann surface ( c f . L . F . ZVEROVICH [ 2 ] , E . I . ZVEROVICH [ 1 ] , ZVEROVICH, PANCHENKO [ 1 ] , iCVITKO [ 1 ] , KRUGLOV [ 1 ] , PAJUCHENKO [ 1 ] , RODIN [ 1 , 2 ] , RODIN, TURAKULOV [ 1 ] , SKLEZNEV [ 2 ] , KOPPELMAN [ 1 ] , R B h R L [ 1 ] ) or with respect to complex (GOHBERG, FELDMAN [ 1 ] , MAHDZHAVIDZS [ 7 , 8 ] ) and Jordan n o n - r e c t i f i a b l e (among them quasiconformal) contours (see KATS [ 3 ] , SELEZNEV [ 1 - 3 ] ) . Furthermore, we do not deal with the natural generalization of the f a c t o r i z a t i o n problem f o r matrix functions - the question on f a c t o r a b i l i t y of operator functions. Concerning t h i s topic the reader i s referred to Chapter 6 in the monograph of GLANCEY and GOHBERG [ 3 ] and the l i t e r a t u r e c i t e d there, but also to the papers of BUDYANU [ 1 ] , LEITERER [ 1 ] , SERGEEV [ 1 ] , GOHBERG, KAASHOEK, van SCHAGEN [ 1 ] , van der MEE [ 1 , 2 ] , MUCKENHOUPT [ 1 ] , NAKAZI TAKAHIKO [ 1 ] , SUCIU [1].1>We do not dwell on various applicat i o n s of the f a c t o r i z a t i o n problem. We only r e f e r to GUSAK [ 1 , 2 ] , DUKHOVNYI [ 1 ] , MALYSHEV [ 1 ] , ROZAtiOV [ 1 ] , where methods and r e s u l t s of the f a c t o r i z a t i o n theory are applied to p r o b a b i l i t y - t h e o r e t i c problems. Applications to mechanics one can find in the monograph of VOROTICH, ALEKSANDROV, BABESHKO [ 1 ] as well as the papers BABESHKO [ 1 , 2 ] , BELOKOPYTOVA, IVANENKO, FH 1 SHTINSKII [ 1 ] , VOROB'EV [ 1 ] , KULIEV, SADYKHOV [ 1 ] , MOSSAKOVSKII, MISHCHISHIN [ 1 ] , TOLOKONNIKOV, PEN'KOV [ 1 ] , KHRAPKOV [ 1 ] , TSITSKISHVILI [3,4],CHERBPANOV [ 1 ] . Concerning a p p l i c a tions to d i f r a c t i o n problems, see the monograph VIROZUB, MATSAEV [ 1 ] and the a r t i c l e POPOV [ 1 ] , CHAKRABARTI [ 2 ] , HEINS [ 1 , 2 ] , IDEMEN [ 2 J . F i n a l l y , f o r applications of the f a c t o r i z a t i o n theory to the method of inverse problems and other questions of t h e o r e t i c a l physics, the reader i s referred to the monograph NOVIKOV ( e d . ) [ 1 ] and the papers of EMEUS [ 1 ] , ZAKHAROV, SHABAT [ 1 ] , ITS [ 1 ] , KRICHEVER [ 1 , 2 ] , MASKUDOV, VELIEV [ 1 ] , MEUNARGIYA [ 1 ] , NOVOSHENOV [ 2 , 3 ] , TROITSKII [ 1 ] , DTJVACZ, ZELAZNY [ 1 ] , IDEMEN [ 1 ] , MEISTER [ 1 ] , RAJAMAKI [ 1 ] , REYMAN, SEMENOVTIAN—SHANSKY [ 2 ] , SIEWERT, KELLEY, GARCIA [ l ] .

1

> see also GOHBERG, LEITEHER [ 1 - 2 ] .

16

CHAPTER 1.

BACKGROUND INFORMATION

This chapter is of an auxiliary character. Its first section contains facts neoessary for the further explanations concerning lineals of finite oodimension as well as aemi-Fredholm and Fredholm operators in Banach spaces» Most of these statements are well-known and have been published in several recent monographs15However, Theorem 1.12 on the Fredholmness of block operators and the local principle in the theory of Fredholm operators make an exception. The latter was proved in the form we need only in SBtONENKO1 s paper [5]» Therefore, Theorems 1.12 and 1.14 will be given with detailed proofs. The remaining theorems in Section 1.1 we state without proof, referring the reader in every case to available souroes. We do not intend to explain the history of semi— Fredholm and Fredholm operator theory, since this has already been excellently described (see, e.g. KATO [1]). In the second section of the present chapter., above all, known theorems on analytic functions of Smirnov classes are stated, whereas in Section 1.3 properties of singular integral operators related to the results of Section 1.2 are described. Here as well as in Section 1,1, we give proofs only for those statements, which we did not find in an appropriate form (Theorems 1,26, 1.27, 1,29, 1.31, 1.35) or which are published only in several articles (Theorems 1.20, 1.32, 1.33)» Since the genesis of the formation of many,results of the theory of Smirnov olasses has received little attention in modern monographic literature, we give some historic information and comments on the material of Sections 1.2 and '1,3. 1.1,

SOME FACTS ON THE GEOMETRY OF BANACH SPACES AND OPERATOR THEORY IN THESE SPACES

A linear manifold in a Banach space not necessarily closed we shall call a lineal. For a closed lineal we use the term subspace. The dimension of the factor-space lineal C

8|C is said to be the codimension of the

being located in a Banach space codim£,

U

9:

= dim(8|£).

See e.g. SOHBIiRG, KRUPNIK [4], S.KREIN [1], MICHLIN, PROSSDORF [1] 17

THEOREM 1 . 1 ,

Suppose £ 1

and l e t £ t 2.

to be a subspaoe o f the Banach space

be a l i n e a l . Then

dim £

< oo implies t h a t £ «

i s also a subspace. On the other hand, i f X a £1 £

i s a subspace and

i s the range o f a olosed operator, theQ

implies that

8,

dim £

< oo

i s a subspaoe, t o o .

The f i r s t part of the theorem i s an elementary f a c t (we r e f e r t o GOHBEBfc, KRUPNIK [ 4 ] , p, 50 f o r i t s p r o o f ) . The second p a r t i s a consequence o f the closed graph theorem ( s e e , e . g . , PALAIS [ 1 ] , Ch, 7 , l b * 1)* DEFINITION 1 , 1 , with

f(x) = 0

The s e t of a l l elements

for a l l

f

of the dual space

x 6 £ i s c a l l e d the orthogonal complement

to the l i n e a l £ (s. »). Obviously,

i s a subspace of the space

ment to the l i n e a l £

8», The orthogonal comple-

coincides with the orthogonal complement to i t s

closure £ . There i s a natural isomorphism between dual to

and the space

8 j Z , As a c o r o l l a r y of t h i s isomorphism, we mention

THEOREM 1 . 2 ,

For any l i n e a l £ ( c s ) , the dimension of the orthogonal

complement coincides with the oodimension of i t s closure ( i n the sense that e i t h e r both values are i n f i n i t e or both values are f i n i t e , and then they are equal to each o t h e r ) . DEFINITION 1 . 2 .

A subspace £ Q

of the Banach space

complementable. i f there e x i s t s a subspace £ , ( c 8 ) vector

x € 8

8

i s called

such t h a t each

admits a unique representation in the foim

x = xQ + x 1

( x . € £ i f j = 0 , 1 ) . In t h i s case, the subspace i s said to be the U V d i r e c t complement to £ Q (with respect to 8 ) , and the space 8 i s saiH to be decomposed in the d i r e c t sum of i t s subspaces £ 0 (written

8 =

and

+ £1),

The notion of the d i r e c t sum i s transferred to the case of an a r b i t r a r y f i n i t e number of summands in an obvious manner. Every subspace in a Hilbert space can be complemented} one of i t s d i r e c t complements i s the orthogonal complement. In every Banach space not isomorphic to a Hilbert one, there are non-complementable subspaces

18

(LBE,SARASON [ 1 ] ) . Nevertheless, there holds THEOREM 1 , 3 ( S . KRBIN M l . p. 3 1 ) . 390

Let

9

a subspaoe of f i a i t e codimension. Then

be a Banach space and SQ

i s complementable, and

the dimension of a l l i t s d i r e c t complements i s equal to I f , moreover, £ i s a l i n e a l dense in dense in to

9

8 , then the l i n e a l C D 8 Q

S Q , and there e x i s t s a d i r e c t complement to

lying in £

codim 8 Q . is

53Q with r e s p e c t

.

We s h a l l say t h a t a l i n e a r (generally speaking, unbounded) operator a c t s in the Banach space l i e in

8

1

3 , i f i t s domain

geneous equation

A , i . e . , the solution s e t of the homo-

Ax = 0 , w i l l be denoted by

Owing to the l i n e a r i t y of the operator ker A are l i n e a l s in

8 .

By

l i n e a r bounded operators acting in with

im A

>.

The kernel of the operator

and

dom A and image

A

ker A.

A , the s e t s

[8]

dom A , im A

we denote the c l a s s of a l l

8 , the domain of which coincides

8 . This c l a s s of operators supplied with the norm

||A|| = sup {||Axj| J x € 8 , Furthermore, the c l a s s

||x|| < 1 }

[8]

i s i t s e l f a Banach space.

i s a Banach algebra, i . e . such a Banach

space, in which the operation of m u l t i p l i c a t i o n i s defined ( i n the case under study, i t s role plays the composition of operators) c o n s i s t e n t with the l i n e a r structure and possessing the property ||AB|| < ||A|| *||B|] * >. DEFINITION 1 . 3 . operator

A , if

The operator

B

dom B c dom A and

i s referred to as a part of the Ax = Bx

for all

Under the conditions of Definition 1.3» the operator also said to be the r e s t r i c t i o n of DEFINITION 1 . 4 .

The values

A on

15

B

dom B (written

a(A) = dim ker A and

are c a l l e d defect numbers of the operator

x € dom B . i s sometimes B = A| dom

B

)»

0(A) » dim(8|Im~A)

A, and t h e i r difference

Most of the d e f i n i t i o n s and r e s u l t s stated below can be transferred to operators acting from one space into another. However, f o r our aims i t i s s u f f i c i e n t to regard operators actiqgin one space. A short survey of the theory of Banach algebras can be found,for example,in the appendix written by 7.M. TIKHOMIROV to the book KGLMOGOROV, FOMIN [ 1 ] .

19

ind A = oc(a) - 3 ( a )

is the index of the operator

A •

The index is defined, if at least one of the defect numbers is finite. If both are finite, then

A

is called an operator with finite index.

In case the image of the operator

A

is closed, the equation

Ax = y

is usually said to be nonaally solvable. For the sake of brevity, in this case the operator

A

itself is also called normally solvable.

DEFINITION 1.5. A closed normally solvable operator Fredholm iff cc(A) and A

is called

p(A) are finite. In case that at least one

of these numbers is finite, A If a(A) 0

1) the operator

($„-, Fredholm) operator

such that if B €[8] and

A + B

is also a

A acting in 8,

||B|| < e , then

($_-, Fredholm) operator,

2) ind (A+B) = ind A, 3) a(A+B) < a(A),

0(A+B) < p(A).

Theorem 1.4 means the stability of semi-Fredholmness or Fredholmness of the index as well as the semi-stability of defect numbers under perturbations of the operator with a sufficiently small noim. THEORfiM 1.5.

If the operator

holm) operator and the operator also a

A

acting in 8

is a

B e[8] is compact, then

($_-, Fredholm) operator and

($_-, *redA+B

is

ind (A+B) = ind A.

Theorem 1.5 expresses the stability of semi-Fredholmness and Fredholmness and of the index of the operator 20

A

under compact perturbations.

Sometimes Theorems 1 . 4 and 1 . 5 are c a l l e d the f i r s t and second s t a b i l i t y theorem* THEOREM 1 . 6 . and

I f the operators

A and

B acting i n

B i s densely d e f i n e d , then t h e operator

8

are iredholm

BA i s also Fredholm and

ind BA = ind A + ind B. DEFINITION 1 . 6 . (see KATO f l l . p . 212). acting i n

8

and

8*

The operators

A and

B

r e s p e c t i v e l y , are r e f e r r e d t o as ad.loint t o

each o t h e r , i f f o r a l l

z 6 dom A and

f € dom B , there holds

(Bf)(x) = f(Ax). Generally speaking, t h e r e are many operators a c t i n g i n a d j o i n t to

A . But i f the operator

8* , which are

A i s densely d e f i n e d , then a l l

operators a d j o i n t t o i t are p a r t s of one and the same uniquely defined operator,which w i l l be denoted by

A* . The operator

A* i s always

closed. THEOREM 1 . 7 .

Let the operator

l y d e f i n e d . Then

A acting i n

(im a / " = ker A*

l a s t inclusion

and

be closed and dense-

(ker a / " 3 i s i * , where i n the

e q u a l i t y holds i f and only i f the o p e r a t o r

( o r , e q u i v a l e n t l y , A*) COROLLARY 1 . 2 .

The closed and densely defined operator

Fredholm) o p e r a t o r . In t h i s case we have DEFINITION 1 . 7 .

A

i s normally s o l v a b l e .

( $ _ - , Fredholm) operator i f f the operator

The operator

r e g u l a r i z e r of the operator i s compact. I f

8

M e[s]

A*

A is a

is a

a(A) = 0(A*), {3(A) = a(A*). i s called a l e f t (right)

A ( € [ 8 ] ) , i f the operator

UA - I

(AM - I )

M i s simultaneously a l e f t and r i g h t r e g u l a r i z e r , then

i t i s c a l l e d a (two-sided) r e g u l a r i z e r . I t follows from the d e f i n i t i o n t h a t i f an operator has l e f t and r i g h t r e g u l a r i z e r s , then a l l of them are r e g u l a r i z a r a and d i f f e r from each other by compact summands. THEOREM 1 . 8 .

The operator

A 6[8]

l e f t ( r i g h t ) i f and only i f i t i s a im A (ker A)

admits a r e g u l a r i z a t i o n from the ( $ _ - ) operator and t h e subspace

i s complementable. Among a l l operators of c l a s s

[8],

21

the Fredholm operators and only they admit a regularization. A simple consequence of Theorem 1«8 i s THEOREM 1,9« class

[8]

I f the product

i s Eredholm, then

I f in t h i s case

B

is a

BA A

(A

of operators

is a

and

A B

and

B

of the

a $_-operator«

i s a $ _ - ) operator, then

A (respec-

t i v e l y B) i s Fredholm. DEFINITION 1 . 8 .

The f a c t o r algebra of the Banach algebra

[»]

with

respect to the ideal ¥ of compact operators i s called Calkin algebra. According to t h i s d e f i n i t i o n , the elements of the Oalkin algebra are classes of equivalent operators, where two operators said to be equivalent, i f equivalence by A €[8]

A-B

A

B

are

In a l l what f o l l o w s , we denote this

A ~ B. The norm of the image

& of the operator

in the Calkin algebra i s called essential norm of

|A|). In other words:

and

A (written

|A| = inf{||A+T|| 1 T t f ) .

Owing to the closedness of the ideal of compact operators, the Calkin algebra provides a Banach algebra. From theorem 1.8 we deduce that the operator and only i f i t s image inverse of

1

i s iredholm i f

& i s i n v e r t i b l e in the Calkin algebra, and the

i s that element of the Calkin algebra, which consists

of a l l regularizers of from the class

A e[8]

A ( o r , what i s the same, of any other operator

A ).

Next we formulate a simple and well-known result from the theory of Banach algebras, which we need below both in general fonn, and f o r the special case of the Calkin algebra. THEOREM 1.10.

Let

{x^}

be a sequence of i n v e r t i b l e elements of the

Banach algebra £

converging in 3 to x . Then the relation supHxj^ || < oo i s necessary and s u f f i c i e n t f o r the i n v e r t i b i l i t y of k

22

x.

Applying Theorem 1.10 to the Calkin algebra we obtain THEOREM 1.11,

Let

{A^} c [ a ]

converging to the operator

be a sequence of Fredholm operators

A . The operator

only i f there e x i s t s a sequence tors

Ak

such that

i s Fredholm i f and

of regularize is of the opera-

supHB^U < 0 0 .

L e t the Banach space spaces»

{B^}

A

8

9 a 9i + 9 t .

be represented as the direct sum of i t s subThen every operator

A e[9]

may be written in

the form of a block-matrix

(1.1)

A = where A ^

*22 •

i s a l i n e a r bounded operator from

into

( i , d = 1,2').

The equation ( 1 . 1 ) must be understood in the sense that the decomposition

x s x 1 + xa

Ax s y , + y »

(x^j 6 8^)

(y^ 6 9^), where

corresponds to the decomposition y, =

+ Alax2,

y 2 = A a l x^ +

Aa a^a• Supposing that in the representation ( 1 . 1 ) one of the blocks

A^

is

compact we can formulate the c r i t e r i o n of Fredholmness of the operator A

in terms of i t s remaining blocks. For the sake of definiteness, we

assume the block THEOREM 1.12. A €[9]

A 12

Let in the representation ( 1 . 1 ) of the operator

the block

operator

A

to be compact.

A 12

be compact. Then f o r the Ji'redholmness of the

i t i s necessary and s u f f i c i e n t that the following condi-

tions holdt 1)

An

i s a $_-operator,

2)

Aaa

i s a $ + ~operator,

3)

Y 1 = dim

4-)

Ya = codim C t < 00, where .£2 = i m A 2 2 + A 2 1

< oo, where

= ker A n D (x€8., : A 2 1 x € im A 2 2 } ker A 1 1 .

I f the conditions 1 ) - 4 ) are s a t i s f i e d , then the index of the operator

A

i s calculated by the fomiula

(1.2)

ind A a oc(Aaa) - P ( A , , ) + Yi - Ya. Moreover, i f

A 1 t = 0, then

a(A) = a(A tl ) + Y1 1 », M

=

+ y

(1.3) 23

Proof.

Owing t o Theorem 1.5 i t s u f f i c e s t o study the case

A 1 t = 0.

F i r s t l e t us assume t h a t the conditions 1 ) - 4 ) are f u l f i l l e d . We show that of

A 8

i s Fredholm and ( 1 . 2 ) holds. L e t onto

along

82

and

P°

P,

denote the p r o j e c t i o n

i t s r e s t r i c t i o n on

leer A. I t

is

e a s i l y v e r i f i e d that im P? -- Li Since

ker A|ker P °

and

ker P ° = ker A 1 1 (

i s isomorphic t o

im P?, t h i s i m p l i e s

oc(A) < oo

and dim ker A|ker P? = a ( A ) - a ( A n ) = dim ^

1

= Yi*

T h e r e f o r e , the f i r s t r e l a t i o n ( 1 . 3 ) i s proved. Next we prove that

im A

has f i n i t e codimension and ( 1 . 3 ) holds. By

assumption, there e x i s t d i r e c t complements M2

to £

of

im A

i n 82« We show that

2

in 8. L e t

Mi

M = Mi + M2

y € im A D M. Since and, on the other hand,

From t h i s we g e t

y € im A fl M,. In view of

Taking now

y = 0. Thus

y = y i + y2

im A n

in 8i

and

i s a d i r e c t complement

P n im A = im A n , we have

y , = P i y € im A i i

y e C t n Ma, i . e .

to

y i € M1# Henoe = im A D ! ! ,

y^ = 0. t h i s means

im A 0 M = { 0 } .

(y H € 8.= , j = 1 , 2 ) u u

a r b i t r a r i l y , we have a

decomposition 7i = A11X1

+ f»

y2 = AjiUi + A2ax2 + f t , where

u, € ker A , 1 f

f ^ € M..,

x.. 6 8^ (d = 1 , 2 ) .

vVe have, furthermore, a decomposition AaiXi = A21u!, + A 2 2 x 2 + f a with

ui 6 ker A n , x 2 € 8 a , f i 6 M a . Putting

x = Xi+Ui-Ui + x 2 - x 2

we obtain Ax = y i - f , + y 2 - f 2 + f i , which i s We proved

y = Ax + f , where

f = f i + f 2 - f' 2 € M.

8 = im A + M. In p a r t i c u l a r , t h i s implies that

f i n i t e codimension equal t o

im A

f 3 ( A i i ) + Y 2 « Since by Theorem 1.1

has a im A

i s c l o s e d , the second r e l a t i o n ( 1 . 3 ) r e s u l t s . The index formula ( 1 . 2 ) i s a consequence of ( 1 . 3 ) * 24

Now l e t us assume that R i s a regidarlrer of An

and

Rta

A i s Fredholm and

[

Hi 1

Ria |

®ai

®2a J

A* Then, obviously,

Rn

i s a r i g h t r e g u l i z e r of

a l e f t regularizer of A 2 t . Hence 1 ) and 2) are f u l f i l l e d .

Since, a s proved above,

ker A|ker A 2 2

Yi < oo. Further, the f i n i t e n e s s of codimension of £ , = im A n 8 t

i s isomorphic to CM we have

(3(A)

implies the f i n i t e n e s s of the

in 8 « . Hence 3) and 4 ) are s a t i s f i e d ,

too, and the theorom i s proved.

=

Theorem 1.12 i s formulated in SPITKOVSKlf [11] (see a l s o SPITKOVSKI* [ 1 2 ] , where t h i s theorem was presented with p r o o f ) . Formulae ( 1 . 2 ) allow f o r obtaining in case s u r j e c t i v i t y of the operator

A i a a 0, a c r i t e r i o n of i n a c t i v i t y and A. This r e s u l t was obtained independently

by STOROZH [ 1 ] . I t i s omitted here, since we do not need i t . Of course, various special cases of Theorem 1*12 were known previously. One of them important f o r the following i s the case that some of the diagonal blocks of the operator

A i s Fredholm. This case can be investigated

(even without compactness assumption f o r

Ai») with the help of

Frobenius' formulae ( s e e KIRILLOV, GVISHIANI [ 1 ] p. 85 as well as PETROV [ 1 ] ) . However, we need only the case i f

Ai a

i s compact.

Thus, we obtain the corresponding r e s u l t as a consequence of Theorem 1 . 1 2 . COROLLARY 1 . 3 .

Let in the representation ( 1 . 1 ) of the operator

A

the block

be compact, and l e t one of the diagonal blocks

At ,

and

A 22

A1t

be Fredholm. Then f o r the Fredholmness of the operator

A

i t i s necessary and s u f f i c i e n t that the second block i s a l s o Fredholm. I f t h i s condition i s f u l f i l l e d , then ind A = ind An + ind A 2 2 . I f , in addition,

A 12 = °>

then

a(A) = a ( A t i ) + a ( A l 2 ) - m, p(A) = p ( A n ) + P(A 2 1 ) - m, where

m = dim(A 21 ker An)|&m A cx D A 21 ker A n ) .

In f a c t , i f e . g . the operator

A4i

i s Fredholm, then conditions 1 ) 25

and 3 ) of %eorem 1 . 1 2 are automatically s a t i s f i e d , and thanks t o the f i n i t e dimension of the l i n e a l t o the condition that

A aa

A t 1 ker A , , ,

condition 4 ) i s equivalent

i s a 4_-operator.

The v a l i d i t y of the formulae f o r the index and the defect numbers follows from the corresponding formulae of Theorem 1 . 1 2 , taking i n t o account that f o r Fredholm

An

and

A aa

we have

dim(ker An D £ 1 ) = dim ker An - m, dim(8 a |£ a ) = dim(8 a |im A aa ) - m. In view of Corollary 1 . 3 , the condition of Fredholmness of the diagonal 1 blocks of the operator ( 1 . 1 ) with compact blocks outside the diagonal i s s u f f i c i e n t f o r the Fredholmness of the operator

A i t s e l f . In the

general case, t h i s condition i 3 not necessary. A simple example i s provided by the operator of two-sided s h i f t of i n f i n i t e m u l t i p l i c i t y acting in a Hilbert space f j . This operator i s i n v e r t i b l e , but,decomposing ^ in a suitable manner in an orthogonal sum, i t can be represented in the form(1.1), where

At a = 0

and

An

i s the operator of one-

sided s h i f t with an i n f i n i t e dimensional kernel. In Chapter 4 we shall consider some examples in d e t a i l , which i l l u s t r a t e the same s i t u a t i o n , but are d i r e c t l y related to the subject of the present book. QOROLIiARY 1 . 4 .

Let the Banach space

8

be represented as the d i r e c t

sum of i t s 3ubspaces operator Pj

( j = 1 , . . . , k ) in such a way that the u commutes up to a compact summand with any operator

A €[8]

projecting

8

on

8^

p a r a l l e l to the d i r e c t sum of the remain-

ing subspaces«

APuS - PuHA . Then f o r the Fredholmness of i s necessary and s u f f i c i e n t that the operators A - • = 0J u 3-i k ered in 8.= be Fredholm. In t h i s case ind A = £ ind "

Corollary 1 . 4 can be proved by induction on

k

A

it

consid-

0=1

with the help of

Theorem 1 . 1 2 . However, we do not intend to v e r i f y t h i s a s s e r t i o n , since the r e s u l t of Corollary 1 . 4 can be e a s i l y obtained d i r e c t l y and i s wellknown (see e . g . GOHBiiHG, KBiSIH [ 3 ] ) . As a r u l e , in the present book the role of the Banach space played by the space 26

L?(r)

(briefly

L^)

8

will be

of n-dimensional vector

functions given on the rectifiable contour is summable in the pth power

V , each component of which

(1 ^ p < »). in addition to the general

theorems given above, we need the following propositions connected with the specific nature of the space L^ . THEOREM 1.13 (M. RIBSZ-TORIH)» subspace

L^

If the operator

A«[L° ] maps the

(1 < p^ < p

P e(Pi»P03> to® operator

< oo) into itself, then, for all n A| Q belongs to the class [it-]. L P

The proof of this well-known theorem can be found, for instance, in KRASNOSEL1 SKli et al. [1]. Now we are going to explain the local principle of the theory of $redholm operators by changing the abstract local principle for Banach algebras described in Ch. 12 of the book GOHBERG, KRUPNIK [4] in an appropriate manner. The formulation of the local principle is premised by several definitions. We denote by by

R

C(r) (=C) the space of continuous functions on

the lineal of rational functions with poles off

DEFINITION 1.9.

The operator

T, and

T .

A e[Lp] is called an operator of

local type, if it commutes up to a compact summand with the operator of multiplication by an arbitrary function of class Since the lineal

R

is dense in

R .

0 with respect to the uniform norm

(see WALSH [1] p. 67), local type operators possess, for any the property

aA - Aal € if .

We denote the class of continuous functions on

r equal to one in some

neighbourhood (depending on the function) of the point DEFINITION 1.10. at the point a € 0t

a € C,

The operators

A, B

tQ(e r) , if, for every

such that

DEFINITION 1.11.

t € r by 0^..

are said to be equivalent e > 0,

one can find a function

|(A-B)al| < e , |a(A-B)| < e.

The operator B 1 (Br respectively) e[L°] is referre

ed to as a left (right) local regal arizer of the operator

A €[lp]

at the point

such that

t Q € r , if there exists a function

a €

al - B-L Aal £ ¿f (al - aABr 6 f ).

27

An operator point

t

A having a l e f t and a right l o c a l regularizer a t some

6 T will be c a l l e d l o c a l l y ffredholm at t h i s point,

THEOREM 1,14 ( I . B , SIMOKEHKO). Let t o r , I f we can, f o r every Iredholm at the point

t

point, then the operator Proof,

A e[L°]

be a l o c a l type opera-

t 6 T , specify an operator and equivalent to the operator

locally

A at this

A i s Fredholm,

Above a l l , we show that the existence of a right l o c a l regula-

r i z e r of the operator the operator

Aj. a t the point

t € r

and i t s equivalence to

A a t t h i s point imply that the operator

a l o c a l regulorization from the right at the point use that the operator Let

Aj.

Br

point

A a l s o admits

t . Here we do not

A i s of l o c a l type.

be a right l o c a l regularizer of the operator t

and

a1

be some function from

a t the

C^. s a t i s f y i n g the condition

a.,I - a^A^Bj, 6 cf , Due to the equivalence of the operators t , there e x i s t s a function

at the point

a2 G

A and

Aj.

with

< ||Br|| ^ , In other words, i t i3 possible to choose an operator

T €f

in such a way that

HaaCA-Aj.) + T|| < ||B r ||~ 1 , But then

||a a (A-A b )B r + TBr|| < 1 , The operator

K = I + a a (A-A^.)B r + TB r

by Theorem 1 , 5 , the operator Let

B0

i s Predholm, I + a^A-A^B

, and

a^. to be a non-negative function of c l a s s C^ equal to zero

hose points

Then

I + a a (A-A t )B r

be some regularizer of the operator

suppose at

i s i n v e r t i b l e , and consequently

t € T f o r which e i t h e r

a^ = a^a-i = a^a»

and, therefore,

a^t) £ 1

or

a 2 ( x ) £ 1,

a^I - a t AB p B o =

= a t I - a t ( I + (A-A t )B p )B 0 + a t ( l - A,jBp)B0 = a t ( l - ( l + a 2 (A-A f c )B r )B o ) + a^Cail - a 1 A t B r )B Q , Taking into account the compactness of the operators

I - ( I + a2(A-Aj.)B r )B o

the operator B

t

=

r o

B B

point

i s

a^.1 - a t AB r B 0 a

ri

and

a i l - a i A ^ j , , we deduce that

i a also compact, i . e . , the operator

S h t l o c a l regularizer of the operator

A at the

t ,

Using that

A i s an operator of l o c a l type, we show now how to con-

s t r u c t a "global" right regularizer from i t s right l o c a l ones.

28

The neighbourhoods

U^

of the points

t € r

in which the functions

Gi_ are p o s i t i v e form an open covering of the contour r . We choose N N a f i n i t e subcovering {a,. }.= ,, from i t . Then the function a = E &+. ^ d D_n d=1 d i s p o s i t i v e at a l l points of the contour r . Hence the operator N . B = ( E ( L B + )a I i s well-defined. I t i s just the desired r e g u l a r i d=i ^ d z e r . Indeed, since A i s of l o c a l type, we have Aa^. I ~ a^ A and, d i) therefore, Aaj. B^ ~ a^ AB^ • But we have a^. AB^ ~ a^. I » which d d d d d d d implies ACEat B t ) a " 1 I = £ ( A i t B t ) a " 1 I ~ E (eu. l ) a ~ 1 I = I . d d d d d The existence of a l e f t regularizer of the operator

A

i s proved i n a

similar way (of course, from i t s existence we conclude that the r i g h t regularizer constructed above i s also a l e f t one). Owing to Theorem 1.8, there results the Fredholmness of

A .

=

We remark that the proof of the local principle described here can be l i t e r a l l y transferred to the case of operators acting in where

X

Lp(x) »

i s a compact Hausdorff space with measure.

This general case was considered in the fundamental work of the theory of l o c a l type operators by SIMONENKO [ 5 ] , where f o r the f i r s t time the notion of such operators was introduced and the l o c a l principle was proved. 1.2.

SOME INFORMATION ON BOUNDARY PROPERTIES OF FUNCTIONS ANALYTIC AND MEROMORPHIC IN FINITELY CONNECTED DOMAINS

Let us begin with the consideration of functions analytic in the unit disk

A = {£ * |£| < 1} . '-Che proofs of the results on such functions

formulated here can be found in Ch. 2 of the book of PRIVALOV [ 2 ] and in Ch. 5 of HOFFMAN [ 1 ] . DEFINITION 1.12.

Functions of the form

f ( * \ z ) = exp(iv + ^

/ ¿ g ^ " 5 i n k(e)de) , —u e - z

(1.7)

29

where

v

i s a r e a l constant and

k

a non-negative function Kith

summable logarithm, are said to be outer. Before formulating the theorem about properties of outer functions, we r e c a l l t h a t we understand by angular boundary values ( o r l i m i t s ) of the function

f

when the point THEOREM 1 . 1 5 .

a t the points z

tends to

t

t

of a contour the l i m i t

lim f ( z ) , z -* t along paths non-tangent to the contour.

Functions of the form ( 1 . 7 ) are analytic in

have a . e . on the unit c i r c l e angular boundary values

A and

f^e^(eie)

f u l f i l l i n g the condition

| f ^ e \ e i e ) | = k ( 6 ) . Every a n a l y t i c func-

tion not having zeros in

A with a continuous bounded logarithm i s

an outer function. The c l a s s of a l l functions analytic and bounded in

A i s denoted by

Hg . According to Fatou's theorem, the angular boundary values e x i s t a . e . on the c i r c l e f o r any function from DEFINITION 1 . 1 3 .

A function from

H^ .

H^ , the angular boundary values

of which are a.e* equal to one, taken absolutely, i s c a l l e d inner. In p a r t i c u l a r , Blaschke products, i . e . functions of the form ,

B(z) = z 1

oo a t a, - z n ~7 ^ _ , k=1 |ak| 1 - afcz

are inner functions, where a sequence of points from

1

i s a non-negative integer and

A with

product vanishes a t the points

a^,

{a^}

is

£ ( l - | a^.1 ) < » . a Blaschke and, in case

1 > 0 , at

z = 0

and only at these p o i n t s . An example of an inner function not having zeros in

A i s provided by

the function S ( z ) = exp(- ^ where

|i i s a singular ( i . e . concentrated on a s e t of Lebesgue measure

zero) measure on THEOREM 1 . 1 6 . form 30

/ *n(e)) , —u e —z

. Such a function i s referred to as singular. Any inner function

f i * ) a ABS , where

B

f^^

can be represented in the

i s a Blaschke product,

S

i s a singular

f u n c t i o n and

\

i s a constant

(|X| = 1 ) . Such a r e p r e s e n t a t i o n

is

unique. DEFINITION 1.14-,

All f u n c t i o n s r e p r e s e n t a b l e i n the form o f a

q u o t i e n t o f a f u n c t i o n from THEOREM 1 . 1 7 , class

D

A function

Hw f

t o an o u t e r f u n c t i o n form the c l a s s D.

not i d e n t i c a l l y zero belongs t o t h e

i f and only i f i t can be r e p r e s e n t e d i n t h e form o f a

product o f an o u t e r by an i n n e r f u n c t i o n « Such a r e p r e s e n t a t i o n u n i q u e . Outer f u n c t i o n s and only t h e y belong t o t h e c l a s s g e t h e r with t h e i r r e c i p r o c a l

Hp,

'^he f u n c t i o n

0 < p < 00 , sup

D have a n g u l a r boundary v a l u e s

circle.

DEFINITION 1 . 1 5 « class

to-

ones.

I t i s evident t h a t functions of c l a s s a . e . on the u n i t

D

is

f

analytic in

A

belongs t o t h e

if

/^If(reie)|p

a e < 00 .

r < 1 -it

We note t h a t , f o r any f u n c t i o n

f

(1.8)

analytic in

A ,

M ^>(r) =

/ I f C r e 1 ® ) ! 1 * d6 i s a non-decreasing f u n c t i o n o f r . Consequently, -n c o n d i t i o n ( 1 . 8 ) i s e q u i v a l e n t t o l i m M .p(r) < 00 o r , what i s t h e r-»1-0 p > 1 same,

sup M f ( r i r ) < 00 , where

{rk}

i s an a r b i t r a r y sequence o f

p o s i t i v e s c a l a r s tending to one from below. The c l a s s e s

Hp

with

0 < p < 00 a r e c a l l e d Ifardy c l a s s e s . The r e l a -

t i o n between Hardy c l a s s e s and the c l a s s

D i s e x p r e s s e d by the f o l l o w -

ing theorem. THEOREM 1 . 1 8 ( P . Y a . POLUBARINOVA-KOCHINA). A

belongs t o the c l a s s

the f u n c t i o n

f(k)

Hp

COROLLARY 1 . 5 ( V . I , SkURHOV). then

f

i f and only i f

defined i n f € D

and

made up o f i t s angular boundary v a l u e s on the

c i r c l e belongs t o the c l a s s

p^ > p ,

( 0 < p < 00)

A function

Lp . If

f 6 Hp

and

L^

f o r some

f e

L e t J 0 be a simply connected domain i n t h e extended complex p l a n e , boundary o f which i s a c l o s e d r e c t i f i a b l e Jordan curve

the

T .

31

DEFINITION 1.16.

We shall say that a function

i s an element of the class

D

rk

analytic in &

) , 0 < p < oo , i f i t i s possible

to f i n d an expanding sequence of domains boundaries

f

with r e c t i f i a b l e

such that:

U r k C £> ,

2)

u f i = i ) k K sup J |f(t)|P|dt| < co . krk

3)

The classes

Ep(-© )

Smirnov classes LAVBENT'EV. In curves disk A

are called Smirnov classes. This d e f i n i t i o n of

E^O© ) SMIHNOV1

was suggested by M.V. KELDYSH and M.A. s original definition i t was required that the

are images of a c i r c l e under a conformal mapping of the onto the domain

. The definitions of KELDYSH/LAVRSNT'ev

and SMIHNOV are equivalent (concerning this topic, see DANILYUK [ 6 ] p. 91, GQLUZIN [ 1 ] p. 423 and PRIVALOV [ 2 ] p.. 203). Consequently, the following c r i t e r i o n of the membership of functions to Smirnov classes holds. THEOHBM 1.19.

'•'•'he function

cp defined in a simply connected

domain £) belongs to the class = q> « 0)

i f and only i f the function

belongs to the Hardy class

H^ . Here by

u

a func-

tion i s denoted, which r e a l i z e s the conformal mapping of the disk onto the domain £) , and root of

u)1

continuous in

A

i s an arbitrary branch of the pth A .

Below we shall be concerned with functions analytic, generally speaking, in a multiply connected domain J0 + * I t i s assumed that this domain i s obtained from a bounded simply connected domain JB + (0) pairwise d i s j o i n t simply connected domains £

+(o)

lying inside i t . The boundaries of the domains

by removal of

( j = 1,...,m-1) ( j = 0,...,m-l)

are supposed to be Jordan r e c t i f i a b l e curves. We denote by 2) " ( j ) the m-1 complement to S ( d ) U r ( j ) . The contour r = U r ( j ) i s assumed to be oriented in such a manner that at an anticlockwise revolution along i t the domain

32

i s on the l e f t .

LEMMA 1.1»

Every f u n c t i o n

f

+

a n a l y t i c i n t h e domain

admits

a unique r e p r e s e n t a t i o n * ( • ) = " e V 0U ) d=o where t h e f u n c t i o n £. u Proof.

Set

« € z>+ ,

are a n a l y t i c i n J2)~( j )

£,(2) = J cm

v

J

£(t)

T.-a

which i s o r i e n t e d i n the same way as t h a t i t borders t o g e t h e r with i n 2)

+

(1.9)

i a a n a l y t i c i n £ ) + ( 0 ) , and the f u n c t i o n s

£

(j = 1,...,m-l)

,

,

and vanish a t i n f i n i t y .

where the contour

r ( j ) , i s chosen i n such a manner

r(j)

a doubly connected domain l y i n g

and not containing the p o i n t

z . Clearly,

y., _

b u t , i n view of Cauchy's i n t e g r a l theorem, the value depend on the choice of

Yj

a

•

y . „ (c.©*"), ~(k) UJZ> U.S (k) , i . e . i n the e n t i r e complex p l a n e . Since, i n a d d i t i o n ,

f ^ i 0 0 ) = 0 , we can conclude t h a t

L i o u v i l l e ' s theorem. But then we have tion

fQ

f „0 ( z ) = -

m-1 E f>(z) = 0 k=1 K

for

f k = 0 , by

z

. Thus, the f u n c -

i s a l s o i d e n t i c a l l y equal t o zero i n J 0 + ( O ) . This proves t h e

uniqueness of the r e p r e s e n t a t i o n i n the form ( 1 . 9 ) f o r Hence IAmma 1.1 has been proved.

f(z) = 0 .

=

In what f o l l o w s , we need a g e n e r a l i z a t i o n of the Smirnov c l a s s e s t o the case of m u l t i p l y connected domains. DEFINITION 1.17 (KHAVINSOH h i ) . m-connected domain JB

+

The f u n c t i o n

belongs t o c l a s s

£

a n a l y t i c i n the

+

if

Ep(«S ) (0 < p < oo)

and only i f t h e r e e x i s t s an expanding sequence of m-connected 33

domains

s a t i s f y i n g c o n d i t i o n s 1 ) - 3 ) of D e f i n i t i o n 1 . 1 6 and,

moreover, the requirement 4-) t h e l e n g t h s

| r k | of the c o n t o u r s

THfiOBBM 1 , 2 0 (Decomposition t h e o r e m ) . the m-connected domain £>

+

rk

a r e equibounded.

The f u n c t i o n

belongs t o the c l a s s

f

analytic in

B^(«© + )

i f and

only i f t h e f u n c t i o n s

f • from t h e r e p r e s e n t a t i o n ( 1 . 9 ) f u l f i l t h e u f „o €p E ( i & ( 0 ) ) , f ,J e E pj . S ~ ( d ) ) , j = 1 , . . . , m - 1 . +

conditions Proof.

f € Ep (,8 + ) . We r e p r e s e n t t h e boundaiy

Let

o c c u r r i n g i n D e f i n i t i o n 1 . 1 7 i n the form rk(d)

rk

r k of the domain m-1 = U I ^ Q ) , where j=o

i s a c l o s e d c u r v e , which b o r d e r s t o g e t h e r w i t h

connected domain l y i n g i n «0

+

r(d)

a doubly

. According t o e q u a t i o n ( 1 . 9 ) and

Minkowski's i n t e g r a l i n e q u a l i t y , t h e f o l l o w i n g r e l a t i o n holds» 1

f | f d ( T ) p | d T | < (( E I lf1(T)|P|dT|)SSETO7 1 r kK( d ) 3 i?d r K ( d ) k 1 + (

S

rk(d)

if(T)|P|dT|)5SE^P^)ma3C(l'P)

Denoting by f^

(1.10)

•

t h e upper bound of t h e a b s o l u t e v a l u e of the f u n c t i o n

i n the domain

f ^ , on t h e s t r e n g t h of t h e maximum modulus p r i n c i -

p l e , we can claim t h a t i t i s a l s o an upper bound f o r

|f("E)| f o r

T € r k ( d ) (d + 1 ) . T h e r e f o r e ,

Moreover,

/

|f(-c)|

rk(d) t o the c l a s s

p

|
describing a confonnal mapping of the disk on a domain with r e c t i f i a b l e boundary, always belongs to the c l a s s

H.,

(see PRIVALOV [ 2 ] p. 173,

DMILYUK [ 6 ] p . 88, GOLUZIN [ 1 ] ) . Now condition 3 ) can be r e a d i l y v e r i f i e d in the same manner as i t was done above f o r the function f J• . Consequently £ € Ep(JiD + ) , which completes the proof of the theorem. = In TUMABKIN, KHAV1NS0N [ 1 ] i t was shown t h a t the omission of condit i o n 4 ) in Definition 1.17 does not lead to an extension of the notion of c l a s s e s

Kp

in f i n i t e l y connected domains ( f o r a simply connected

domain t h i s f a c t i s a consequence of Theorem 1 . 1 9 and the r e l a t i o n o)' 6 H 1 ) . In other words, Definition 1 . 1 6 can be used f o r a r b i t r a r y f i n i t e l y connected domains1 This allowed f o r considering Smirnov c l a s s e s in Jordan domains with n o n - r e c t i f i a b l e boundary. For such domains tjie decomposition theorem does not hold as i t was c l a r i f i e d in TUMARKIN, KHAVINSON [ 2 , 3 ] . She c l a s s of a l l functions analytic and bounded in some domain 2) w i l l be denoted by

2^(2) ) .

In what follows, we s h a l l simply write whereas

E~ P

Bp

instead of

E p (JD + ) ,

denotes the c l a s s of functions defined on the s e t m-1

ST => JEf*(0) u^y,, J © + ( o ) ) , the r e s t r i c t i o n of which on £>"(0) r e s p e c t i v e l y ) i s a function of c l a s s class

c o n s i s t s of functions from

Ep(je) " ( 0 ) ) E~

(£>+( j )

(E p (.& + ( j ) ) ) . The

vanishing at i n f i n i t y .

I t i s c l e a r from the proof of Theorem 1 . 2 0 that the system of domains^ -

Note t h a t in the monograph DUREN [ 1 l the problem of description o f multiply conneoted domains f o r which Definitions 1 . 1 6 and 1.17 are equivalent i s referred to as an open one. 35

occurring i n D e f i n i t i o n 1.17 can be chosen as one and the same f o r a l l functions o f a Smirnov c l a s s . Hence, applying Minkowski's and H&lder's i n t e g r a l i n e q u a l i t i e s in a standard manner, we can e a s i l y derive the following result. THEOREM 1.21. 2)

f o r any

1) pif

The c l a s s e s

1

and

B~

are l i n e a l s ,

p a €(0,oo] , Pi < p* , the i n c l u s i o n s

h°ld' ®Pi 2 BPa 3) i f f € B^ (S^)

P = (PT 1 + p T

B*, E~

r

1

,

g e

(E^)

, then

E^ o E^

,

f g 6 B j ( E ~ ) , where

.

More profound p r o p e r t i e s of functions from Smirnov c l a s s e s are gathered below i n Theorems 1.22, domain JD

+

1.24, 1.25. For the case

of a simply connected

, the arguments of these theorems can be found i n the mono-

graphs GdLUZIN [ 1 ] , DMILYUK [ 6 ] , PRIVALOV [ 2 ] . With the help of the decomposition theorem 1.20, these theorems can be t r a n s f e r r e d t o the more general case of a f i n i t e l y connected domain i n an obvious manner 1 > f which j u s t i f i e s the r e f e r e n c e s t o GOLUZIN [ 1 ] , DANHYUK [ 6 ] , PRIVALOV [ 2 ] b e f o r e the formulations o f Theorems 1.22, 1.24, 1 . 2 5 . THEORiiM 1-.22 ( s e e PRIVALOV \2~\ p . 204. DANILYUK T6l p . 91. GOLUZIN M l P» 4 2 2 ) . p > 0 , has l i m i t s

f+(t)

almost a l l p o i n t s on

T

of the c l a s s

The f u n c t i o n

f u n c t i o n of c l a s s cide with

f

f+

We denote by i n the form

M^ ( =

( f ~ ) , which i s d e f i n e d L^ .

defined a . e . on

L * ( L ~ , L ^ ) , 0 < p < co , i f \ + / — Bp

( B p , E^)

a t almost a l l +

f

B* ( E ~ ) ,

along a l l non-tangent paths at

t 6 r . The f u n c t i o n

belong t o the c l a s s

36

(f~(t))

f

by these l i m i t s , belongs t o the c l a s s

DEFINITION 1.18.

-

Every f u n c t i o n

r

we assume t o

there e x i s t s a

the boundary values of which c o i n -

t 6 r

.

+ + R)

f = f, + f,

the c l a s s of functions

, where

f< 6 if"

and

f

representable

f,€ R .

Concerning t h i s t o p i c , see UMARKIN, KHAVTNSON [ 3 ] , where Theorem 1.24 i s formulated ( i n p a r t i c u l a r , on p . 7 3 ) f o r the case of a f i n i t e l y connected domain.

+ + If" + R , If" + C axe introduced in a similar way* SomeP P + + times i t w i l l be convenient to use the notations L~ = U L~ , + + P > 0 p 1T= U i C . p > 0 P Beginning with Chapter 2, we s h a l l , as a r u l e , not deal with s c a l a r + + functions from c l a s s e s , L^ , e t c . , but with n-dimensional vecThe c l a s s e s

t o r functions and

oxa matrix functions, the elements of which belong

to these c l a s s e s . We denote by

X°

the space of n-dimensional v e c t o r

functions, the components of which belong to some function c l a s s ( i n the case

X = L^ , t h i s notation was already used in Section 1*1)•

I t would be natural to denote t&s c l a s s of elements from

X

X by

nxn matrix functions with

• However, we renounce t h i s notation, since

i t i s too troublesome. Instead, the c l a s s of such matrix funotions i s simply denoted by

X . From the context i t w i l l be always c l e a r whether

we speak about a s c a l a r or a matrix function* + Vector functions of the c l a s s (E^ + R) can be reproduced from t h e i r l i m i t s on tions

r

in a unique manner. L e t us show t h i s . I f the vector func+ and f t from the c l a s s (S^ + R) have angular l i m i t s

f1

coinciding a . e . on

r , then t h e i r difference f a f , - f , i s also a + vector function from (Ei^ + R) and has angular l i m i t s equal to zero a . e . on

r . But in view of the following uniqueness theorem, t h i s i s

possible only f o r

f = 0 .

THEOREM 1 . 2 3 (PRIVALOV f2l p. 292, GQLUZIN M l p. 4 1 3 ) .

A function

meromorphic in a domain with r e c t i f i a b l e boundary, which has singular l i m i t s equal to zero on a s e t of p o s i t i v e l i n e a r measure, i s i d e n t i c a l l y equal to zero. Taking into+ account the +i n a c t i v i t y of the correspondence between + + c l a s s e s B~" + R and If" (respectively between E and L J), we some— r

Jr

^

Jr

jr

times shall i d + e n t i f y vector functions from (E~p + R ) a with vector func* t i o a s from foimed by t h e i r l i m i t s . + a + In p a r t i c u l a r , i f f e ( ^ ) , under f ( z ) , z € SB ~ we s h a l l simply understand the value a t the point

z

of t h a t vector function o f c l a s s

(E~ + R ) a , the boundary value of which i s

f . In connection with t h i s , 37

note that the i d e n t i f i c a t i o n of functions of class ing functions of class

L*

H^

with corresponds

on the c i r c l e i s in general use. For the

l a t t e r class even the notation

Hp (or

H*)

i s preserved (see HOFFMAN

[ 1 ] ) , For the sake of uniformity, the class L~ on the c i r c l e i s denoted by

H~ . Clearly, a l l the assertions of Theorem 1.21 may be transferred + + n ± n ±n

from the classes the product

fg

to classes

+ R)

, (Lp)

and

(M p )

, if

i s defined componentwise.

In the classic case of functions analytic in a domain and continuous up to the boundary, the values in the domain are expressed v i a boundary ones with the aid of Cauchy's i n t e g r a l . This method cannot be transferred d i r e c t l y to the case of classes

+

for

p < 1 , because a

i s not summable. However, f o r functions from the boundary function + classes

B1 , Oauchy1s integral remains to be a natural means of con-

tinuation from a contour to a domain. THEOREM 1.24 (PRIVALOV f2l pp. 202. 206, DANILYUK T6l p. 133, SQLUZIN f l l pp. 420-423)» The vector function f defined on belongs to r

(l«t)n

J f (t)

=0

((i" .

In accordance with Theorems 1.24 and 1.22, any vector function the class

(Lp) Q

((£~)a)

»

P > 1

l i e s in the subspace of

of a l l functions satisfying (1.11) and, f o r

f

consisting

p = 1 , coincides with i t .

Corollary 1.5 shows that f o r the c i r c l e the coincidence holds f o r

38

of

p > 1 , too* To c h a r a c t e r i z e the s i t u a t i o n i n g e n e r a l , we introduce another d e f i n i t i o n « DEFINITION 1.19 (aee PRIVALOV [2~\ p . 250. DANILYPK f6~l p . 90)» A simply connected domain Jb i s c a l l e d Smirnov. i f t h e d e r i v a t i v e of the f u n c t i o n

u> , which maps

A conformally on £> , i s an o u t e r

function. A closed Jordan curve f

w i l l be c a l l e d Smirnov (curve of c l a s s C),

i f each of the domains i n which T decomposes the extended complex plane i s Smirnov. The c l a s s of Smirnov ourves i s so wide t h a t i t i s r a t h e r d i f f i c u l t t o c o n s t r u c t examples of ourves not belonging t o t h i s c l a s s . ? o r t h e f i r s t time, such an example was b u i l t by M.V. KBLDYSH and M.A. LAVEENT1 EV. Their c o n s t r u c t i o n , with some s i m p l i f i c a t i o n s , i s described i n PRIVALOV [2] pp. 229-250. ^n a n a l y t i c d e s c r i p t i o n of a l l non-Smirnov curves the i n t e r e s t e d reader can f i n d i n DUREN, SHAPIRO, SHIELDS [ 4 ] . Various s u f f i c i e n t conditions of membership of curves t o t h e Smirnov c l a s s are e s t a b l i s h e d i n PRIVALOV [ 2 ] pp. 250-257» I n p a r t i c u l a r , one can f i n d there LAVRENT1 EV1 s r e s u l t t h a t a curve i s Smirnov as soon as

(1.12) Here along

s(t1,t2)

i s the distance between p o i n t s

t1

and

tt

measured

T .

THEOREM 1.&5 (PRIVALOV [ 2 ] pp. 26^-266. DANILYUK T6l pp. 92-95). I f the contour

r c o n s i s t s only of Smirnov curves, then, f o r

0 < Pi < Pa < « , the e q u a l i t i e s ) Q = (Lp ) n H we have

f € (L*)

a

( L p f ) n =* ( i ^ ) n H

and

are v a l i d . E s p e c i a l l y , i n case t h a t ((£~)

a

r e s p e c t i v e l y ) i f and only i f

p > 1

f €

and

condition (1.11) i s f u l f i l l e d . I f a t l e a s t one component of t h e contour

r

such t h a t

i s not a Smirnov curve, then there e x i s t s a f u n c t i o n f € L i (£7) » b u t

f f L* (£~)

Due t o Theorem 1.25, i n t h e case of a contour curves, the l i n e a l s

(L*)

n

and

(L~)

n

for all r

f € L00

p > 1.

c o n s i s t i n g of Smirnov

are subspaoes i n

L a . In f a o t , 39

this assertion i s true f o r every r e c t i f i a b l e contour

r , however, the

corresponding argument requires some preliminary constructions, which we are now going to deal with* We shall i d e n t i f y the space q. = p/(p - 1 ) )

f =

with the dual to --

E e L^ , 0=1 3 0 p

g =

L^

form (1.13) ( i . e . satisfying the condition - D)

(1.13)

E g^, e . 3=1 3 0 , Lp r ja1 J where

n f =» E f -¡ 6 h 3=1 3 3

and

e., 3

denotes the j th unit vector.

THEOREM 1.26.

The orthogonal ( i n the sense of the form ( 1 . 1 3 ) )

complement to

(Lpn

Proof. If

((L~)a)

coincides with

(L*)a

((L~)Q) .

Obviously, i t i s s u f f i c i e n t to prove the statement f o r

f € L* , g € L* , then we deduce

f g € h*

and

= 0 for all , «— k g e Lp ( L p ) . Setting g ( t ) = t , where k i s a non-negative (negative) integer, we obtain + means that i e l i

r

/ f ( t ) t k d t = 0 . According to Theorem 1.24, this (Li) .

The following considerations are f i r s t made f o r the case of a simply connected domain £) + . Suppose that the function formal mapping of the unit disk onto T = W(G)

40

the condition

/ f(T)g(x)dr = 0

r

CD realizes a con-

. A f t e r substituting can be rewritten as

/ *(»(£)) i-^Ts) s(»(5)) P V+(0))

the boundary value on

r

fQ

i s the

.

taking as

g

of functions from the class

Ep(J2T( j ) )

and acting by the scheme described, we can state that

i s in f a c t the boundary value of a function from By Theorem 1.20, this just means that

f^

•

f € L* . Thus, also in case of

a multiply connected domain the orthogonal complement to L* i s the + o_ L^ • Analogously, the assertion concerning the classes L^ , oclass . L^

allow to be transferred to the case of multiply connected domains.^

COROLLARY 1.6. For any p €[1,oo] , the sets spaces in Lp . Indeed, f o r

p €(1,oo] , ( L p )

((L~) )

( L p a , (L~) a

are sub-

is the orthogonal complement to

some lineal and i s therefore a subspace. The closedness of the classes (Lt)a

and

(li7) n

THEOREM 1.27.

already mentioned. Let the vector functions

coincide a.e. on some arc

6(wtf)a

and

€ (M7)a

y c r . Then each of these vector functions

i s the analytical continuation of the other through

y .

+

Proof.

By definition of the classes

tions

= cp+ + r 1 f

In i t s turn,

r^ = r^

+

=_9~ + r a

with ±

- r^ , where

r^

with poles concentrated in £> functions

and

functions classes

and

(l7)D

1

then i t contains

.

£ ® as a proper

p a r t . However, f o r our p u r p o s e s i t i s c o n v e n i e n t t o c o n s i d e r t h e operators

S , P

and

Q i n t h e spaces

L^ , assuming t h e i r domains of

d e f i n i t i o n t o be t h e l i n e a l X, ^ , i r r e s p e c t i v e of t h e n a t u r e of t h e contour

r . H e n c e f o r t h we s h a l l do s o .

THEOREM 1 . 3 1 . 1)

The o p e r a t o r s

S, P

c l o s e d and, f o r 2)

the operators

and

Q studied i a

(1 < p < oo)

are

p < co , d e n s e l y d e f i n e d , S (P, Q)

and

- S (Q, P )

c o n s i d e r e d i n t h e spaces

L^ and (1 < p < oo, q = p / ( p - 1 ) ) , r e s p e c t i v e l y a r e a d j o i n t f H t o each o t h e r i n t h e sense of D e f i n i t i o n 1 . 6 , 3)

4)

the operator

S

idempotent:

2

S

f o r the function

i s i n v o l u t o r y , and

P

2

= I|

a s well as

Q

are

2

, P = P , Q* = Q , *P 9 6 - i 2 p the e q u a l i t y

( S 9 ) 1 = 2S(fSq>) - +

Thus

< Sf,g

-

... > x j ^

,

€ M* , and the number o f zeros of

det

c a l c u l a t e d with regard to t h e i r m u l t i p l i c i t y i s equal t o

G ^

d•

Obviously, such r e p r e s e n t a t i o n s e x i s t . We s h a l l obtain one o f them by introdmcing the s c a l a r r a t i o n a l f u n c t i o n and p o l e s o f which are concentrated i n

f

of index

P+(G)

the zeros

«8+ , where we suppose t h a t

fG e L^ . Furtheimore, we s e t = fG , where

A

« ( t )

= t'P+C

,

G

W(z) =

N i s the number o f zeros of the function

I ,

f(z)

det(fG)

i n Jb

In v i r t u e of the f a c t t h a t t h i s f u n c t i o n has no p o l e s i n «S

+

(2.17) +

.

, N may

be c a l c u l a t e d as f o l l o w s : N = ind + det ( f G ) = n ind f + i n d + d e t G = n P + ( G ) + ind + det G . Now we d e s c r i b e the procedure of "zero separation" of

det G ^ ^ , which

permits, f o r a given r e p r e s e n t a t i o n ( 2 . 1 6 ) o f the m a t r i x f u n c t i o n a r e p r e s e n t a t i o n o f p r e c i s e l y t h i s type but with an index by one. S t a r t i n g from the r e p r e s e n t a t i o n ( 2 . 1 6 ) of formulae ( 2 . 1 7 ) f o r

j = Ii

r

G

for

j = 0 , i.e.,

smaller

defined by

and applying t h i s procedure

get a r e p r e s e n t a t i o n ( 2 . 1 6 ) o f of t h i s m a t r i x f u n c t i o n i n

G

j

G ,

N times, we

a factorization

L^ .

fhe procedure mentioned c o n s i s t s in the f o l l o w i n g ; For a given m a t r i x

function

G ^ ^ from the r e p r e s e n t a t i o n ( 2 . 1 6 ) and an a r b i t r a r i l y chosen

zero

of the f u n c t i o n

z„o function By

70

wSj

U

det G+^ ^ i n as i n d i c a t e d in Lemma 2 . 1 .

\...

^

,' we c o n s t r u c t a m a t r i x

we denote the t u p l e obtained by ordering the

sequence

....»«¡¡¿j »

+ 1 , •^i»***»*^

i a

a

aon-increasing

manner. I t i 3 c l e a r that such an ordering can be carried out with the help of one transposition permuting the k th element of the sequence with the 1 th one ( l < k ) . By

T^)

we

denote the matrix obtained

from the unit matrix a f t e r permutation of the columns with the same numbers. Then, s e t t i n g

= diag[t H *

= T 1/2

and p i q » / ( p i + q 2 ) > 1 / 2 , +1 thus, the elements of the matrix functions 7T are summable with e x -

ponent 1/2 . Consequently, t h e i r poles lying on I f , in p a r t i c u l a r ,

1

T must be simple.

i s the triangular polynomial matrix function

occurring in the converse statement, then a l l roots of i t s diagonal 79

elements must be simple. The number of different roots of therefore, between

- [-|*

(1)

V a ]

- x^ 2 ^|/n)

greater than or equal to For

- J

2

det Y

lies,

(i .e. the smallest integer and

|x( 1 ) -

n = 1 , the first value is precisely equal to

-

2.5. COMMENTS The factorization of the type (2.1) as well as the notions of partial and total indices, under classical assumptions concerning the matrix function

G

and the contour

r , where introduced in the paper of

MUSKHKLISHVILI and VEKUA [1] (see also the monographs N. VEKUA [4], MUSKHELISHVIL1 [1]). Factorization in

was first studied by PRIVALOV [1], for a review

of subsequent results, see KHVEDELIDZE [1]. The right-factorization was presented for the first time in GOHBERG, KHEIN [ 3 ] . It would seem that the first detailed study of the relations between the factorization of one and the same matrix function for different values of the parameter

p

is accomplished in the present book.

Theorems 2.1 and 2.3 were formulated in SPITKOVSKII [2]. At the same place the notion of the factorability domain was introduced. Corollary 2.1 is usually called the invariance theorem for partial indices (see N. VEKUA [4], GAKHOV [3], MUSKHELISHVILI [1], SIMONENKO [4]), Theorem 2.2 on the connections between factorizations with one and the same total index was actually proved in MUSKHELISHVILI, VEKUA [1] (see also GOHBERG [4]). For the factorization of power functions, see VEJOJA [4], GAKHOV [ 3 ] , GOHBERG, KRUPNIK [4]'. The method of "splitting off zeros™ was first applied to the factorization problem by GAKHOV [1-3]« The factorization problem for rational matrix functions (Theorem 2.9) was solved by N.P. VBKUA [1] (see also N. VBKUA [4]), and for matrix functions of the type (2.11) with Holder factors

A^

(up to T) by GAKHOV. In a spezial case, the latter problem

was studied by N. VEKUA, KVESELAVA [1,2] and SHERMAN [1]. The general

80

Theorem 2.10 has been proved here, i n p r i n c i p l e , by the same method as was used by GAKHOV f o r the r e s u l t mentioned above* Note t h a t a somewhat d i f f e r e n t procedure of constructing a f a c t o r i z a t i o n based on the preceding t r a n s i t i o n t o a matrix f u n c t i o n with constant determinant and with trigonometric polynomials as elements, has been r e c e n t l y proposed i n JONCKHEEHE, DELSARTE [ 1 ] , This a p p l i e s , i n p a r t i c u l a r , t o r a t i o n a l matrix f u n c t i o n s with z e r o partial In Ch.

indices. 1

of

CEEJABOBX,

GOHBERG

[3]

formulae are presented which express

the p a r t i a l i n d i c e s of r a t i o n a l matrix functions by s p e c t r a l data of the m a t r i x polynomial corresponding t o i t . For more r e s u l t s in t h i s d i r e c t i o n we r e f e r t o

BART,

GOHBERG,

KAASHOEK

of a matrix f u n c t i o n of the kind ( 2 . 1 1 ) , i f l i n e and

det

A+

[1-3].

The f a c t o r i z a t i o n

the contour

are e n t i r e f u n c t i o n s , was considered in

r

i s the r e a l HURD [ 1 ]

in

connection with a p p l i c a t i o n s t o problems of determining a s t a t i o n a r y temperature regime. Theorems 2.6 - 2.8 and 2.11 are proved in SPITKOVSKII [ 1 3 ] ( s e e also SPITKOVSKli [ 2 ] ) . The estimates of the p a r t i a l i n d i c e s e s t a b l i s h e d i n «/

r

n

these theorems g e n e r a l i z e r e s u l t s o f NIKOLAIOHUK |_1J obtained by him f o r the c l a s s i c a l case by the same method of "separating z e r o s " . The formulae f o r the t o t a l index stated in these theorems are analogues of the c l a s s i c a l formula of MUSKHELISHVILI ( s e e VEKUA [ 4 ] , MUSKHELISHVILI [ 1 ] ) , according t o which the t o t a l index of a non-degenerate Holder matrix f u n c t i o n i s equal t o the index of i t s determinant. For M u s k h e l i s h v i l i ' s formula and i t s extensions t o other c l a s s e s , we r e f e r t o Chapter 5 . Hemark that the formula f o r the t o t a l index and the n o n - n e g a t i v i t y of the p a r t i a l i n d i c e s of a matrix f u n c t i o n a n a l y t i c i n ^ uous i n «P + U r )

+

(and c o n t i n -

are a l s o presented in the a r t i c l e of GHEBOTAREV and

GAKHOV [ 1 ] . Theorem 2.12 i s published f o r the f i r s t t i m e .

81

CHAPTER 3. THE CRITERION OF FACTORABILITY. «-FACTORIZATION AMD ITS BASIC PROPERTIES la this chapter we shall study relationships between properties of the Riemann boundary value problem for a piecewise analytic vector function and the factorability of its coefficient, i.e. the matrix function

G .

It is well-known that in the classical case, if we seek for a solution of the Riemann problem with a Holder matrix G , the factorability of G

is equivalent to the Fredholmness of the boundary value problem.

The transition to the solution of the Riemann problem with measurable +

matrix function

G

in the Smirnov classes B~ leads to a new quality.

It turns out that the factorability of G , generally speaking, does not imply the Fredholmness of the corresponding boundary value problem. In fact, the vector-valued Riemann boundary value problem with a factorable matrix function classes L^

G

and the associate problem considered in the

and L^ , respectively, have finite defect numbers, and

the indices of the problems are opposite. However, these problems are, in general, not normally solvable, i.e. their images can be not closed. Nevertheless, these images are in a sense well-situated. More strictly speaking, the following weakened closedness condition is fulfilled! the image of the Riemann boundary value problem (and of the problem associate to it) contains all those rational vector functions belonging to its closure. Moreover, the factorability of the matrix coefficient of a Riemann problem in

is equivalent to the property described

above, a property intermediate between finiteness of the defect numbers and Fredholmness. This fact is the main result of Sections 3.1 and 3*2. In Section 3«3 it will be further stated that to guarantee the normal solvability and, thus, the fredholmness of the Riemann problem with factorable coefficient

G , certain two operators K

with the help of the factorization factors of G well as the projectors

P

and

Q

and K 1

composed

in a certain way as

have to be bounded in

. In order

to make the notions of Fredholmness of a boundary value problem and factorability of its coefficient equivalent, it is natural to confine the definition of factorization given in Ch. 2 by adding the requirement 82

of boundedness of the mentioned operators

K

and

K^

ia the space

. This special type of factorization ensuring the Fredholmness of the corresponding Riemann problem will be introduced in Section 3*4and called S-factorization. In what follows we shall almost always have to deal with just ^-factorization. The results of Sections 3.1 - 3«4 will be fonnulated and proved under the only suppositions that the contour is rectifiable. The matrix function G

is unbounded in general. In Section 3.5, the contour r

is,

in addition, assumed to belong to the class Si , on which, by the very definition, the operator of singular integration spaces Lp , 1 < p < oo . The condition the closed operator P + GQ matrix coefficient

1>

S

is bounded in the

r 6 % permits us to consider

instead of the Riemann problem with

G • This enables us to utilize the powerful appa-

ratus of operator theory for investigation. Here we shall demonstrate the necessity of the condition for P + GQ

G

€ L^

to be a $_-operator and the criterion of $-factorability

resulting from this fact. Finally, under the condition

G € L^ , in Section 3,6 we change over

from the examination of closed, generally speaking, unbounded operators to the study of bounded ones. It will be established that the problem of $-factorability of a matrix function

G

defined on a composite

contour can be reduced to the corresponding problem for its restrictions on the connected components of the contour»

$-factorability is

preserved and of local character if the images of the contours are smooth enough. In 3,7 comments for further reading will be given.

Incidentally, note that the equation (P+GQ)

]. 83

3.1.

ON SOLVABILITY OF THE RIEMANN BOUNDARY VALUE PROBLEM WITH FACTORABLE MATRIX COEFFICIENT

Let

S

be a m a t r i x f u n c t i o n of a t h o r d e r and

t o r f u n c t i o n both d e f i n e d on

g

a n-dimensional v e c -

r . The v e c t o r - v a l u e d Riemann boundary

value -problem i s s t a t e d i n t h e f o l l o w i n g way«

is a vector function, the j th component

is identically zero, if than or equal to

c. of which u and a polynomial of order less

K. > 0 9 u — k. - 1 otherwise. u

The general solution of the homogeneous problem (3-1) is deteimined by formula (3.6) with

(p* = cp~ = 0 . In other words, the " + "-component of

the solution of the homogeneous problem (3.1) is a linear combination (with polynomial coefficients of a certain degree) of the columns of the matrix function

G + , and its "-"-component is the combination —1 —"1

(with opposite coefficients) of the columns of

G_

with this, the matrix function taking the values and

G~^(z)

for

z

. In connection G+(z)

for

z

€«&+

is called a canonical matrix of the

problem (3.1). Analogously, the general solution of the homogeneous problem (3.2) is determined by formula (3«&' ) in which ty* =

= 0 .

It is easy to see that the pairs of vector functions (G + (z)

ea ,

- Gl 1 (z) A ~ 1 ( z ) z k e^}

(k = O.....-1+Xj,

89

where

j ranges over those indices for which x^ > 0) form a basis in

the space of solutions of the homogeneous problem (3.1)• In 'Hie space of solutions of the homogeneous problem (3.2) a basis is generated by pairs of the form where

{(G|)~1(z)zke.. , -Gl(z) A(z)zke.j} (k = 0,...,-1-x;J

j ranges over those indices for which x^ < 0).

From this and Theorem 3.1 we deduce the following result« COROLLARY 3.1» Let the matrix function G admit a factorization (2.1) in Lp . Then 1) the dimensions of the kernels of problems (3.1) and (3.2) can be calculated via the formulae n a, = E max{xj,o} , j=1

J

n aa ® ¡C max{- x.,0} j d=1

J

(3.8)

2) for the solvability of problem (3.1) it is necessary that, for all

d = 1,...,n with x. < 0 , the conditions u / (G^CtJsCtJJj, tkdt = 0 , k = 0,...,-xd-1

(3.9)

are satisfied; 3) for the solvability of problem (3.2) it is necessary that, for all

j = 1,...,n with x. > 0 , the conditions J

1 r/((G:r (t)h(t)).

tkdt = o ,

k = -x^.,.,-1

(3.91)

are valid. According to formulae (3.8), for n > 1 , both defect numbers a., and aa

may be different from zero, so that assertion 3) of theorem 3.1

cannot be transferred to the vector case (for n = 1 , foimula (3.8) in close agreement with assertion 3) of Theorem 3.1 shows that If the vector function G^g

= 0).

lies in ¿C!? (in view of assertion 1)

of Theorem 3.2 only this case is of interest), then condition (3.9) may be rewritten as follows« for those

j for which x. < 0 , the j th u component of the vector function Q(G~ g) has a zero of order not less than 1 - Xj at infinity.

90

The equivalence of the l a t t e r condition with condition ( 3 . 9 ) i s an immediate consequence of the following proposition. LEMMA 3.1.

*or a function

0)

and therefore, S (Q 1 , and x(t) = 1 otherwise). 1

®hen we have cp 6 and, by assumption, x®-i = B

) - zATB(zkcp) = cAe^ . Since the left-hand side of the

latter equation is the difference of two vector functions from we conclude from this that for

100

,

Ae^ e L^ . The relation obtained is valid

j = 1,...,n , which means

A €L

.

2)

Under the conditions

form

1 , ~

€ L^ . Now we choose the interval

Y = {e i6 J 2 1 ~ 2s -1 < | < 2 3 _ 2 s -1} . Then / k(6)pd6 = u(2 2 - 23 + s p .2 1 - 2 8 ) , / k(8)_qde a h(2 2 - 2s +s~ 4 .2 1 ~ 2s ) Y Y and -1- ( / k(e) p de) 1/p ( s k(er^e) 1 /«! = IYI Y Y =

2 ^ 1

.

(22-2S +

aP>2 1-2s ) 1/ P(2 2-2s + S -Q. 2 1-2 S ) 1/

c o i n c i d e with the images o f

Tp(G')

and

the

have the same property, which proves

Lemma 3.5« THEOREM 3 . 1 6 . class X

The m a t r i x f u n c t i o n

i s ^-factorable in

i s tfredholm i n the space kernel of operator indices of

G 6 L^

defined on a contour o f

i f and only i f the operator

P + GQ

L^ . In t h i s c a s e , the dimension o f the

P + GQ c o i n c i d e s with t h e sum o f p o s i t i v e p a r t i a l

G , the codimension o f the image i s opposite t o the sum

o f negative p a r t i a l i n d i c e s , and the index i s equal to the t o t a l index of

G . The operator

K + K1

(where

K

and

K1

are defined

by formulae ( 3 « 1 6 ) ) i s a two-sided r e g u l a x i z e r f o r the o p e r a t o r P + GQ . Proof.

Comparing a s s e r t i o n s 1 ) and 2 ) of Lemma 3 . 5 , we f i n d t h a t the

images o f the operators

P + GQ

(in

L^)

and

Q + G1P

(in

are

c l o s e d only simultaneously. Due to Theorem 3 . 1 , the i n d i c e s o f t h e s e operators sire a l s o f i n i t e only simultaneously. Consequently,

the

Fredholmness of one of these operators i s e q u i v a l e n t to the iredholmnesa o f the o t h e r . With regard to C o r o l l a r y 3.4-, t h i s enables us t o derive 117

the criterion of 4-factorability mentioned la the theorem. Taking Theorem 3.1 into consideration, the formulae for the defeot numbers of the operator

P + GQ

are obtained from relations (3«8).

Now we evaluate the product

K + K1

by

P + GQt

(K+Kn )(P+GQ) = (G~1 A ~ 1 < * v 1 + G + P&; 1 )(P + G + A 0 , we choose a matrix function

(G. .)!?"""] =„ ••-i) i f 0 '

by l a s s than

nant

of which s a t i s f i e s the r e l a t i o n

a(t)

s/2

it

G € L 00 .

d i f f e r i n g from

with r e s p e c t t o the noxm, the determia

€ L^

(by the induction

hypothesis, t h i s can always be done). Setting

G ±n = G ± a , G a j = GQ;j ( i , j = 1 , . . . , n - 1 ) , G a a ( t ) =

e ( t ) + G a a ( t ) , where |«(t)| = e/2 , arg C ( t ) = arg(det G ( t ) / a ( t ) ) , we have | det G ( t ) | = | det G ( t ) + a ( t ) « ( t ) | = | det G(t)| + | a ( t ) e ( t ) | > | | a ( t ) | . Since

—1 a € L^ , the matrix function

At the same time, THEOREM 3 . 1 8 . then Proof.

If

B~1

Let

AP + BQ s > 0

G a l s o belongs to c l a s s

||G - G|| < e , which proves Lemma 3 . 6 .

A" 1 ,

e x i s t s an

A

A , B € LM

and

AP + BQ

L ro .

=

i s a 4 + - or $ _ - o p e r a t o r ,

€ L^ . be a $ _ - o p e r a t o r . Due t o Theorem 1 . 4 ,

such t h a t i f

||A - A.JI < s , then

there

A^P + BQ i s a l s o

a $ _ - o p e r a t o r . In accordance with Lemma 3 . 6 , we may assume t h a t Ai

€ Lm . But then, along with operator

aT^CA^ + BQ) = P + a71BQ —1 the matrix function Therefore,

B

i s a l s o Fredholm. In view of Theorem 3 . 1 3 , —1

Ai

B~ 1 (= B ^ A . ^ 1 )

Furthermore, f o r

B € L^ ,

A1P + BQ , the operator

inverse to

A^ B

also belongs t o Q+ B

AP

belongs to the c l a s s L^ . Lw .

i s a «^-operator t o g e t h e r with

AP + BQ . Using Theorem 3 . 1 3 again ( t h a t p a r t r e l a t e d t o operators of the kind Q + GP), we observe t h a t A" 1 € L00 . If

AP + BQ

P + Ai BQ

A B € L^ . From t h i s we g e t

i s a $ + - o p e r a t o r f then arguing as above, we conclude t h a t

i s a # + -operator # In v i r t u e of Lemma 3*5 and Theorem 3«1,

Q + (Ai B) P

i s then a $ _ - o p e r a t o r .

The f u r t h e r argument i s t r i v i a l .

=

121

THEOREM 3.19. The operator

Let

A

and

AP + BQ

B

be

nxn

matrix function of class

is Fredholm in

and the matrix function

G = A

B

if and only if

is «-factorable in

case the kernel dimension of the operator sum of positive partial indices of

AP + BQ

A

-1

L^.

€ L^

L p . In this

is equal to the

G , and the codimension of the

image is opposite to the sum of negative partial indices. —1 Proof.

The necessity of the condition

A

€ L^

follows from the

preceding theorem. If this condition applies, then in view of the equality

AP + BQ = A(P + G Q ) , the operators

AP + BQ

and

P + GQ

are Fredholm only simultaneously and have the same defect numbers. To complete the proof, we have to apply Theorem 3.16.

=

Notice that the following condition will also be necessary and sufficient for the Fredholmness of the operator AP + BQ in the space n —1 Lp t B € Loo and the matrix function B A is «-factorable from the right in

Lp .

Theorems 3«16 and 3.161

serve as a chain link, with the help of which

results and methods of the theory of linear bounded operators can be used for the study of problems related to «-factorization. In particular, the stability property of Fredholmness under small perturbations (Theorem 1.4) leads to the next result. THEOREM 3.20.

If the matrix function

the left (right) in tion as

F

G € L^

, then, for some

is «-factorable from

e > 0 , the matrix func-

is also «-factorable from the left (right) in

||F - G|| < e , where the total indices of

F

and

L p , as soon G

coincide.

THEOREM 3.21. 1)

Let

T

and

L

be two simple smooth closed contours, and assume 0

to be a diffeomorphism of the contour function

G € L 00 „

(right) in in

122

Lp

Lp

defined on iff

T

L

onto

T . The matrix

is «-factorable from the left

GQ = G o 0

defined on

from the left (right), if

and from the right (left), if

0

first case the total index of

GQ

0

L

is «-factorable

remains the orientation,

changes the orientation. In the is equal to the total index

of

G , and in the second case it is opposite to it.

2)

Assume

r

curves

r• , and let

to be a closed contour consisting of

tion defined on

G

G^ , j = 0,...,m-1

matrix function defined on

in

G

Tj

L

and

P

Gq

GQ

is ^-factorable

Uj , j = 1,...,m-1 , are u

^-factorable from the right (left) in

of

G .

is ^-factorable from the left (right)

from the left (right) in

G

we denote the

and coinciding on it with

if and only if the matrix function

total index of

connected

be a measurable bounded matrix func-

r . By

'•Che matrix function

m

L^ . In this case the

is equal to the left (right) total index

minus the right (left) total indices of

G^t

0 = 1,...,m-1 . Proof. Lp(L)

1)

We introduce the operator

Lp(L)

Fredholm iff the operator ( W G W SQ

- 1

)

and

L°(r)

mapping from

by the rule (Wcp)(t) = 0

and

A^

are

t(€ r ) . a r b i t r a r i l y , we choose an arc

||G(t) - Gt(x)|| < ellQH""1

a way that

A

G w i l l be

G and

owing to the e - l o c a l equivalence of denote a function continuous on takes values from the i n t e r v a l neighbourhood of the point

y

a . e . on

r

Gt

y(s t )

in such

( t h i s i s possible at

t ) . By

a

we

which i s equal t o zero outside

[0,1]

on

Y > aid equals

1

y •

in some

t . Then we have

laCA-A^I < ||a(A-Aj.)|| = ||a(G-Gt)Q|| < ||a(G-Gt)||.||Q|| = ||Q|| ess sup||a(T)(G(T)-G t (T))|| < M - e s s sup||G(x)-Gt(T)|| < e . t€Y

Together with

« r

A

and

A^. , the operator

A - Aj. i s of l o c a l type.

As mentioned in Section 1.1, f o r any function the operator

f(A-A^) - (A-Afc)fl

f

continuous on

T ,

i s compact, In p a r t i c u l a r , we have

| a(A-A^.)| = |(A-A t )aI| and, thus,

KA-A^all < e.

Due to D e f i n i t i o n 1.10, the operators the poitrt

*

A

t , which proves Theorem 3*22.

and

A^

are equivalent at

=

A simple example of matrix functions e - l o c a l l y equivalent at a certain point

t

y i e l d those matrix functions which coincide a*e. in a neigh-

bourhood of t h i s point. Now we mention the e - l o c a l principle c o r r e sponding to t h i s special case. COROLLARY 3.9.

L e t the contour

T

be covered by the system

of open arcs. I f one can, f o r given

{y^}

G , s p e c i f y a c o l l e c t i o n of

matrix functions Gh(t) = G(t) u

G • € L „ which are ^ - f a c t o r a b l e in L such that G » p a . e . on Y-; » then G i s also ^-factorable in L_ • J

P

S u f f i c i e n t conditions of $ - f a c t o r a b i l i t y obtained v i a the e - l o c a l p r i n c i p l e w i l l be derived in Chapter 8. 3.7.

COMMENTS

The material explained in Section 3.1 - 3.5 i s , in p r i n c i p l e , taken from SPITKOVSKII [13,14]. The results of Section 3.1 generalize the

126

classical solution scheme for the Riemann boundary value problem

based

[3],

on factorization of its matrix coefficient (VEKUA [4], GAKHOV MUSKHELISHVILI [1], KHVEDSLIDZB [4]). Assertion 3 ) of Theorem 3»1 f under different assumptions on

G

ensur-

ing, however, its factorability, was mentioned in GOHBERG [1], SIMONBNKO [1,4], Its validity for an arbitrary function of class defined on the circle was, for arbitrary

L^

p = 2 , proved in COBURN [1] and, for

p €(1,oo) , in GOHBERG, KRUPNIK [ 3 ] .

The proof conducted

here is essentially that of COBURN [1], In connection with Theorem 3.2 we still refer to MEUNARGIYA [3], where the general solution of problem (3*1) is obtained in a foisn containing the values of the unknown vector function and its derivatives explicitly. Arguments close to those used in the proof of Theorem 3»4 the reader can find in GOHBERG, FjSLDMAN [ 3 ]

as well as in IDEMEN [ 2 ] ,

where the

case of rational matrix functions given on the circle is examined, but also in HEINIG [1], where a similar approach was used for the inversion N of operators A = a + £ b.Sc H generalizing (from case N = 1 ) opera0=1 3 a tors of the type AP + BG (see also HEINIG/ROST [1]). Theorems 3-5, 3.15, 3.15 1

are explained in SPITKOVSKII [6], The original

variant of Theorems 3.15 and 3.15 1

can be found in SPITKOVSKli

In the same paper the interpolation property of

[2].

factorability

(Theorem 3 . 9 ) was mentioned. In connection with this, we refer to the article KUCHMENT [1], in which the interpolation property of Fredholmnefis with the same index for an arbitrary linear operator acting and bounded in an interpolation family of Banach spaces was proved. The notion of ^-factorization of a bounded measurable matrix function in

Lp

is due to I.B. SIMONBNKO (for

p = 2, in SIMONENKO [3,4], for

the general case, in SIMONENKO [6]. L e m m a 3.5 and Theorems 3.16, 3.21, 3.22 are also due to him. In the case

n = 1 , the notion of ^-factorization, in principle can be

found also in DANILOTIC [1,2] and WIDOivi [1]. 127

The term "^-factorization" was proposed in SPITKOVSKII [2,13] and motivated by Theorem 3,16, according to which the operator

P + GQ

is

Fredholm, i.e., it is a ^-operator if and only if the matrix function G is SB-factorable. Notice that operators of the type

T^F) and

T^(F)

introduced at the

beginning of Section 3.6 are known in the literature as Toeplitz operators, and the matrix function

F

is usually called the symbol of the

corresponding operator. In this way, Lemma 3«5 allows us to reformulate all the assertions about operators of the kind

P +

and

Q + G'p

in terms of Toeplitz operators and vice versa. The results of some authors are

formulated for Toeplitz operators. Henceforth, we shall

refer to such results having in mind their analogues for the operators P + GQ

and

Q + G'P . y/

Theorems 3.6, 3.6' and 3.11 formulated in SPITKOVSKII [6] are a natural extension of Theorem 3*16 to the case of a rectifiable contour and a matrix function not necessarily bounded, whereas Theorems 3«13 and 3»14 are generalizations of corresponding results of SlivlOHENKO [4,6] from the case

G 6 L^ , r € %

Note that the requirement

renouncing the restriction

G € L^.

r { X can here also be omitted

(see SPITKOVSKII [14]). Actually, Theorem 3.13 in the scalar case with p = 2

was proved in WIDOM [1] and, for real-valued functions, in

HARTMAN, WINTHBR[1]. The boundedness criterion (3.21) for the operator with weight

e

S

in the space

L^

has been derived for the circle in HUNT, tíUCKBNHOUPT,

WHEBDEN [1] and, for a smooth contour, in KOKILASHVILI [1]. In the same paper the sufficiency of condition (3.21) in case of piecewise smooth curves was mentioned, whereas the necessity in case of piecewise Lyapunov curves was established in KRUPNIK [5]. Theorem 3«17 (on the $-factorability of power functions) is an auxiliary result for the theory of ^-factorization of piecewise continuous functions, with which we shall be concerned in Chapter 5. Appropriate references will also be given there. 128

Lemmas 3.4-, 3.6 and TheorenB 3.18, 3.19 are borrowed from KRUPNIK [3], Finally, we insert some remarks on results not mentioned in the present chapter, but being directly related to it. First of all, we refer to NYAGA [1] and DZHVAESHEiSHVILI [1], where the study of the scalar Riemann boundary value problem on a rectifiable +

contour

T

in wider classes than

L1

has been initiated (in particu-

lar, in classes of functions which can be represented as a differentiated Cauchy integral). Under sufficiently general assumptions on the coefficient

G , the scalar Riemann boundary value problem in classes

different from

L

was studied in ANTONTSEV [1], KATS [1-3]. The

papers of DANILOV [1], ZAPUSKLALOVA, KATS [1], KATS [3] and SEIFULLABV [1] are devoted to the investigation of new effects occurring in the theory of the scalar Riemann boundary value problem under weakened conditions on the contour Operators of the type

r .

T j (F) = y F | i m j ) , where ?

(bounded) projector in some Banach space

8

and

is an arbitrary F

is an arbitrary

operator from [s] , are an abstract analogue of Toeplitz operators. In particular, Wiener-Hopf operators, i.e. integral operators on the half-line with a kernel depending on the difference of the arguments can be represented in the form T y(F)

Ty (F) . Therefore, the operators

are referred to as generalized Toeplitz operators or general-

ized Wiener-Hopf operators. The general theory of such operators was created in DEVINATZ, SHINBROT [1]. Especially, (under some additional restrictions) the validity of the abstract analogue of Theorem 3.13 is shown there, namely, the necessity of the invertibility condition of the operator

F

for

Ty(F)

to be a

or a $_-operator. For more

results in this direction we refer to the recent monograph of SPECK [1]. Incidentally, note that the theoiy of systems of Wiener-Hopf equations is equivalent to the theory of operators

P + GQ

and

ty + G'P . This equivalence can be established with the help of the Fourier transform} for more detail concerning this topic, see GOHBERG, KRiiiN [3], GOHBERG, FELDMAN [3].

129

In Ch. 8 o f the book GOHBERG, FELDi.lAN [ 3 ] a l s o o t h e r types o f systems o f i n t e g r a l equations are enumerated, the theory of which may be e r e c t e d with the aid of r e s u l t s r e l a t e d to operators studied i n t h e p r e s e n t c h a p t e r . In p a r t i c u l a r , the examination of operators of the foim AP + BQ has been suggested by the study of systems of s o - c a l l e d paired i n t e g r a l e q u a t i o n s . A v a r i e t y of types of i n t e g r a l e q a t i o n s and p r o blems of mathematical physics r e d u c i b l e to the Riemann boundary value problem were discussed i n the monograph by GAKHOV and CHERSKli [ 1 ] ( s e e a l s o Chapters 1 - 7 i n GOHBERG, FELDMAN [3]). V a r i a n t s o f the l o c a l p r i n c i p l e (Theorem 3 . 2 2 ) can be found i n CLANCEY, GOHBiiRG [ 1 ] and GLANCKY, GOSSELIN [ 1 ] , In CLANCEY, GOHBBRG [ 1 ] the notion of l o c a l f a c t o r a b i l i t y o f m a t r i x f u n c t i o n s was introduced ( a matrix function a t the p o i n t

t

G € L^

and m a t r i x f u n c t i o n s

on

A+SA~1

r

i s called locally factorable

€ r , i f t h e r e e x i s t an open a r c

t h i s p o i n t , domains i l

operator

defined on

+

c«8

+

containing

the boundary of which c o n t a i n s +

A + € Ep(£2 )

i s bounded in

Y £ T

such t h a t

Lp(y) , and

-1

A+

-

Y »

±

€ E ^ ( Q " " ) , the

G(t) = A+(t)A_(T)

a.e.

y)» Furthermore, t h e r e was proved t h a t the ^ - f a c t o r a b i l i t y o f

G € L^

i s equivalent t o i t s l o c a l f a c t o r a b i l i t y a t every p o i n t o f the

contour. In GLANOEY, GOHBERG [ 2 ] i t i s e s t a b l i s h e d t h a t i n the s c a l a r case f o r a Lyapunov contour tion tor

G

a t the point

P + GQ

tQ

r , the l o c a l f a c t o r a b i l i t y of the f u n c i m p l i e s the l o c a l Fredholmness o f the opera-

a t t h i s p o i n t . For

p = 2 , the converse a s s e r t i o n

is

also t r u e . In GLAWCEY, GOHBERG [ 2 ] i t was mentioned t h a t f o r

p 4= 2

the question

of v a l i d i t y o f the converse a s s e r t i o n i s s t i l l open. This problem has been solved by I . B . SE.IONENKO, who i n a s e r i e s o f papers (8IM0NENK0 [ 7 - 1 1 ] ) s u c c e s s i v e l y renounced the r e s t r i c t i o n s on

p , n

and

T

and,

f i n a l l y , proved the equivalence o f l o c a l Fredholmness o f the operator P + G 0 ) , i t i s easy

t o deduce the r e l a t i o n s -

(f)r|G+(«)r4

for all finite +

r .

Since

and

G + € Ep

the f u n c t i o n

< (ln|G+(z)|)r
• (k .) and (k\) V ( k . ) i f f the tuple Proof.

(k!.) 0

hold simultaneously

i s some permutation of the tuple

(k.) . 0

F i r s t of a l l , we intend to explain what do conditions ( 4 . 8 ) and

( 4 . 8 ' ) mean f o r some concrete values of

k = max k . , then 0 max{k.,~k,0} = 0 f o r a l l j = 1 , . . . , n . Consequently, inequality ( 4 . 8 ) with J n t h i s choice of k i s equivalent to E max{k'--k,0} < O . In turn, the 0=1

k . If

J

l a t t e r inequality i s s a t i s f i e d i f and only i f (d=1,• • • , n ) , i . e . i f 142

max{k'.-k,0} = 0 0 max k . < max k. . In t h i s way, condition ( 4 . 8 ) J 0 d 3 1

with

k = max k. is equivalent to max k!, < max k. •

i

i

d

i

~

A

3

If the latter inequality is fulfilled, then for k > max k^ , the equalities min{kj-ktO} = k^-k , min{k!j-k,o} = kj. -k are true and, therefore, to say that condition (4.81) is valid amounts n n to saying that E (k,-k) < E (k'.-k) . Obviously, the inequality ob0=1 3 ~ J=1 3 tained is satisfied iff n a . E k < E k' . j=1 « 3=1 « Analogously,one can readily verify that, for 1

k = min k^ , condition

1

(4.8 ) is satisfied iff min k^ < min k . . When the latter inequality J

J

is fulfilled, then, for k < min kj , condition (4.8) is equivalent to J a n "" E k, > E k' . 3=1 3 3=1 0 Thus, if the conditions (4.8) and (4.8') ara fulfilled for and

k = max kH

k = min k. u , then relations (4.9) are true, which proves statement

1) of the lemma. Furthermore, if the relations (4.9) are valid, it follows from what was proved above that conditions (4.8) and (4.81) are satisfied for k k, > ... > k ^ , k, > k, >..•>

= min kj ,

= min k|j , and if

> ^3^=1 '

( k j ) ^ , then in view of relations (4.9), But then, obviously,

( k ^ ) ^ >- (k!j)^=1

and

. In this

k ^

(k^)^ >

=

. • 143

Due to the induction hypothesis, the tuple mutation of the tuple

( k Jj)j = -| provides a p e r -

( k . ) ^ , . . Reordering these tuples in a non-

increasing manner, we thus deduce that they coincide. Consequently, the i n i t i a l tuples proved.

(k-j)^

and

coincide, t o o . Lemma 4.1 i s

=

THEOREM 4.7.

L e t the diagonal elements of the matrix function

^-factorable in

G be

L^ , and assume the index of the j th diagonal e l e -

ment to be equal to of G+

k- . Then the tuple ( k . ) of p a r t i a l indices u J G i s majorized by the tuple ( k . ) . The f a c t o r i z a t i o n factors 0 and A can be obtained with the help of a f i n i t e number of a l g e -

braic operations and solutions of scalar Riemann problems. Proof.

According to Corollary 3.1» the dimension of the kernel of pron blem ( 3 . 1 ) with matrix c o e f f i c i e n t G i s equal to a = £ max{*.,,0} , 0 3=1 and the dimension of the kernel of the scalar problem with the c o e f f i cient G -• i s equal to a- = max{k.,0} . On the strength of Theorem 4.2, JJ

J

J

n n we conclude that E max{K.,o} < E max{k.,0} . 0 0 3=1 0=1 n n n Analogously, - E min{n.,0} < - E min{k.,0} , i . e . E min{n.,0} > J d 3 0=1 ~ 0=1 0=1 ~ n _k > E min{k-,0} . Multiplying G by t ~ , i t s p a r t i a l indices and the

~

J

0=1

indices of the diagonal elements decrease by

k . Consequently, the

inequalities proved continue to be v a l i d , substituting in them K^-k

and

k^

by

k^-k . Thus

G., = GD

D ( t ) = d i a g [ t k l , . . . »t15®] . The

i s triangular along with

elements are ^-factorable in

by

(k^) > ( x ^ ) .

Now we introduce the matrix function —1 matrix function

*. 0

G , i t s diagonal

L^ , and t h e i r indices are equal to zero.

According to Theorem 4.6, the p a r t i a l indices of the matrix function G i are equal to zero and i t s f a c t o r i z a t i o n factors

and

G^ ^

are

constructed with the help of a f i n i t e number of algebraic operations and the solutions of scalar Riemann problems. Using Theorem 2.7» we see that the matrix function

G can be factored with the same degree of

e f f e c t i v e n e s s . The proof o f Theorem 4.7 i s complete,

144

a

From Theorem 4.7 and statement 1 ) of Lemma 4.1 we obtain the following result, COROIiLARY 4.3.

The p a r t i a l indices of a triangular matrix function G

with ^-factorable diagonal elements are situated between the smallest and l a r g e s t index of the diagonal elements, and the t o t a l index i s equal to the sum of the diagonal elements indices. We denote by

(i• •

the order of the zero of the ( i , a ) th entry of the r

matrix function

\

(G^ ')""''

(introduced in the proof of Theorem 4 . 7 ) at

i n f i n i t y . Note that a l l entries outside the diagonal of at

oo , so that

'Ac shall write

n• H > 1

for

^

vanish

i ^ j .

ji.f y. = oo , i f the ( i , j ) th entry of ( G ^ 1 ) ) - 1

is identi-

c a l l y zero. THEOREM 4.8.

The tuple of p a r t i a l indices of a lower (upper) triangu-

l a r matrix function coincides (up to a permutation) with the tuple of indices of i t s diagonal elements i f and only i f k^ - k ± < for i > d ( i < o) . Proof.

(4.10)

We focus our attention on the case of a lower triangular matrix

function. I f condition (4.10) i s f u l f i l l e d , then, setting A = D ,

G_ = D^G^^D , we get a f a c t o r i z a t i o n of

G+ = G^1 ^ ,

G . Thus we have

only to v e r i f y the a n a l y t i c i t y of the entries outside the diagonals of the matrix functions

G_

and

G_

teed by the f a c t that the ( i , o ) i s equal to zero, i f point

z = oo , i f

at the point

z = oo . I t i s guaran-

entry of the matrix function

i < j , and has a zero of order

i > j , and the ( i , o ) th entry of

from i t v i a multiplication by

H. . G_

at the i s obtained

z^ 3 ""^ 1 .

How, l e t condition (4,10) f a i l to be f u l f i l l e d . Suppose, at f i r s t , that k, - k ± < |i±1 Setting

(i=1

n-1) , k 1 > 0

kQ +

< 0 .

n1 = n-1 , n2 = 1 , we represent the matrix function

the form ( 4 . 4 ) . In this case the f a c t o r i z a t i o n of which in + (t)t Gnn ^ ) = Gnn v '

By

and

G2 L^

i s the r i g h t lower entry

G

GQn

in of

G,

we write down in the following way:

aG~

( t )' *. nn^

k . ^

u

( i < j ) , then the p a r t i a l indices coincide with the i n -

dices of the diagonal elements. In order to prove t h i s proposition, we have to use Theorem 4 . 8 and the f a c t that 4.3.

n,J. HJ > 1

for

i I j .

2

CALCULATION OF PARTIAL INDICES OF SECOND-OHDER TRIANGULAR MATRIX FUNCTIONS

In case of second-order triangular matrix functions, the question of c a l c u l a t i n g the p a r t i a l indices admits a complete solution. For the sake 147

of d e f i n i t e n e s s , we study lower triangular matrix functions, i . e . mat r i x functions of the form ( 4 . 4 ) with.

n1 = n2 = 1 .

F i r s t of a l l , consider the oase of ^ - f a c t o r a b l e diagonal elements ( i n accordance with Theorem 4.3, in t h i s case the matrix function

G

S - f a c t o r a b l e ) . Denoting the indices of the functions

G2

k1

and

k2

G^

and

by

r e s p e c t i v e l y , in view of Theorem 4.7 and Corollary 4 . 4 , we

may only claim t h e f o l l o w i n g « the p a r t i a l indices of k.,

and

is

k2 , i f

k.,-k2 < 1 ,

k,-k 2 > 1 , where

X

and are equal to

G coincide with

ki-\

i s an integer situated between

and 0

k 2 +* , i f

and [ ^ ( k ^ k j ) ] .

For the exact calculation of the p a r t i a l indices of a matrix function G of the form ( 4 . 4 ) , we propose the f o l l o w i n g algorithm. As in the proof of Theorem 4.7, we construct matrix functions

such that

G= G ^

The function

^ D

1

.

$ ( z ) = a~(z)/|7(z)

order of the zero of function Then

l i e s in the class

$

at

= Qq(z) + R1(z)/$(z),

$- n 1 .

, Q2 = H 2 - m , • • • . I f the process stops

, qn =

at some stage, i . e .

.

i s a polynomial of degree

Acting in t h i s way, we get a sequence of polynomials of degrees

m > )i

ni +

k1

and

k2 . I f

n

o

,

n

o

+

> k,-k 2 , then the p a r t i a l indices of

n G

kt+l^ .

The f i r s t statement of the theorem f o l l o w s from Theorem 4 . 8 , $

has the same order at

00 as the lower l e f t element of the

< are

matrix function

(G^1

.If

nQ < k 1 -k 2 , but

> k.,-k a , then

we obtain a f a c t o r i z a t i o n of G by setting k«—k»—u " —u 0 5T(t)R1(t) §7(t)Q0(t)t Gjt) = | -k1+ka+u I » a~(t)t 0 5l(t)t ^ / ka+H, t - 0

In case that

H0+Hi
k 1 -k a , we put

/1

/t * ^

\0

1 /U,

1 I

o

lT(t)R1(t)t»A^ Gjt)

s7(t)k'+k«+tX(t)

^

=; ' gT(t)R2(t)tk-"-k^i

etc.

0

sKiOt-^Ci-^Ctte^t))/

=

COROLLARY 4.5. indices

k.,

every integer

Let

and

G 1f G2

ka

(€ L ^ )

be « - f a c t o r a b l e functions with

r e s p e c t i v e l y , and suppose

k 1 > ka . Then f o r

\ 6 ( 0 , [ ^ ( k 1 - k a ) ] ) , one can choose a function

H € L^

in such a manner that the p a r t i a l indices of the matrix function ( 4 . 4 ) are equal to

and

ka+X . — + k

Proof. where

We intend to look f o r functions $ € E7

H

and has a zero of order

z is oo . The condition

H € L^

obviously, f u l f i l l e d i f f

of the foim = k 1 -k a -A.

i s equivalent to

$ = §7 1 •

150

f i n i t e n e s s o f t h e v a l u e p., imply t h a t t h e o p e r a t o r ) i s Fredholm, i . e . , f u n c t i o n G., i a ^ - f a c t o r a b l e i n L^ , which c o n t r a d i c t s t h e assumption o f t h e t h e o r e m . Thus

0 1 = 0 . The e q u a l i t y

at = 0

may

be p r o v e d a n a l o g o u s l y . I n t h i s way, t h e c o n d i t i o n s d e s c r i b e d i n t h e f o r m u l a t i o n o f Theorem 4 . 1 0 a r e a c t u a l l y n e c e s s a r y and s u f f i c i e n t f o r t h e $ - f a c t o r a b i l i t y o f

G •

I n a d d i t i o n , from t h e e q u a l i t i e s

(4.5),

0, = a, = 0

and t h e r e l a t i o n s

t h e f o l l o w i n g formulae f o r t h e d e f e c t numbers o f t h e Riemann boundary v a l u e problem w i t h m a t r i x c o e f f i c i e n t a = Ya >

G

result»

P = Y-i •

Comparing t h e s e f o r m u l a e w i t h t h e e x p r e s s i o n s o f t h e d e f e c t numbers v i a p a r t i a l i n d i c e s e s t a b l i s h e d i n Theorem 3 . 1 6 , we s e e t h a t t h e f o l l o w i n g cases are

1)

possiblet

Hi>0,

Then

x2 < 0 .

Yi = - x a

, y 2 = x-i » i . e . ,

l e s s of the values

2)

x1 > O ,

Then

Yi = 0 ,

Since then

3) Then for

x2

YiYa

and

formulae ( 4 . 1 1 ) a r e f u l f i l l e d

Yi + Ya •

x2 < 0 . y 2 = x , + x 2 > 1 , where

x 2 < *y(x.,+x2) = Ya/2

i s an i n t e g e r , we o b t a i n t h a t

x 2 = 0 , which means

x, < 0 ,

regard-

x 2 < [Ya/2]

.

and, i f

Ya = 1 »

x., = Ya •

x2 < 0 .

Ya = 0 > Yi = ~ ( H i + H a ) > 1 . As i n c a s e 2 ) , we c a n e s t a b l i s h Yi = 1 » t h e e q u a l i t i e s

x2 = -1

and

I n summary, i f e i t h e r o f t h e c o n d i t i o n s fulfilled,

x1 = 0

YiYa 4= 0

hold. and

then f o r m u l a e ( 4 . 1 1 ) a r e t r u e . T h e r e f o r e ,

remains t o be examined, namely, when

Y1+Y2 < 1

o n l y one

Yt = 0 , Ya > 1 • GQ(t) = t_5jG(t)

k = [Ya/2] . The p a r t i a l i n d i c e s

and

3 x,-k

= x2-k

m a t r i x f u n c t i o n meet t h e c o n d i t i o n s o f c a s e 1 ) . I n f a c t , was proved above, and

is

situation

To t h i s end, we i n t r o d u c e t h e m a t r i x f u n c t i o n

x2 < k

that,

x , > ^ ( x 1 + x 2 ) = Ya/2 > k .

the

with of

this

inequality

Consequently

= dim M q , where M q = {cp € k e r T q ( t ~ ^ G 1 ) : Qt-kHcp € IM T Q ( t _ k G 2 ) } I n t h i s way,

x 1 = k + dim M Q , x 2 = Y a - k ~ dim M q

.

, and t h e p r o o f

of

151

the theorem will be perfected as sooa as we establish that cides with the subspace generated in The relation

cp € ker Tq(t~^G 1 )

M

and

may be rewritten in the form t~kG1cp 6

q> 6 ker TQ(G1 ) . In turn, if k

Qt'^PG^ = 0 . If further,

Qt_kH£p s Qt'^jf , where

G1

€ ker TQ(G 1 ) , then

cp € ker T ^ t " " 1 ^ )

Qt^^q» = Qt~ PG 1 9 . Consequently, and

coin-

by conditions (4.13),

€ Lp • If the latter condition applies, then QG.,

,=1

f

D = S*CS , where

C i s the matrix defined by

formula ( 4 . 1 9 ) . I t s ( r , s ) th entry i s determined by the formula ^¿[(l-l)(s-1)-(r-l)(k-l)] 1 ( E c, = n " . e rs ln> k J- E-

E o n+l-k l - Qq> €

6 fl++ ft" * ft i s v a l i d . Thus, 3 ) ==> 4 ) . Vice versa, i f this case

SA — c A, then also

PR c L*

consideration that

f o r any T

PA — c A . Since ft — c L0„0 .-we have in '

p < oo . Hence

FA c A fl L * . Taking i n t o

i s a Smirnov contour, we obtain from Theorem 1.25

that ftfTLp = &0L* = ft+. In other words, FA e f t A n a l o g o u s l y , QR c ft". Since the converse inclusions are obvious, the implication 4 ) ==> 3 ) i s v a l i d . I f 3 ) i s s a t i s f i e d , then, f o r every

a € ft , we have

a = Pa + Qa € ft+ + ft" , i . e . 3 ) ==> 1 ) . Implication 5 ) ==> 4-)

is 183

evident, thus, i t remains to prove the implication 4 ) aa> 5)* In view of 4 ) , the operator

S

i s defined everywhere on ft . Now we want to

show that i t i s closed« i f Pa k = all

+ S)a k

a k -» a ,

-» ¿(a+b) ,

b=1,2,the

Sak -» b

Qak = ¿ ( 1 - S)a fc -» iy(a-b) . Since, f o r

inclusions

Pa k € ft+ , Qafc € ft" hold (here we

use the implication 4 ) ==> 3 ) proved above) and in ft , then Hence,

a+b €

ft+

( a k , a , b € ft), then

ft+,

ft"

are closed

, a-b € ft" .

S(a+b) = a+b , S(a-b) = b-a

and

Ba a S(^(a+b) + iy(a-b))

= ¿(a+b) + ^(b—a) = b . Due to the closed graph theorem, the operator S i s bounded in ft • Thus Lemma 5.5 i s completely proved* In the space

=

ftQ

of u—dimensional vector functions with components n from ft we introduce a norm, setting Bf|| _ = E ||f J L , where a ft jd 3 H Q f s £ f 0jeJ • . To any matrix function A 6 ft we assign the operator of 0=1 multiplication by t h i s matrix function in ftn . As i t can be easily v e r i f i e d , the norm of this operator i s determined by the formula ||A|| = with

A * ( a ^ ) ^

max

(5.8)

.

In the sequel we suppose that the (L 00 ,R)-algebra ft i s decomposing. According to Lemma 5.5, this supposition i s equivalent to the boundedness of

S

viewed as an operator in

ft

and, thus, also in fln •

Consequently, i t plays here the same role as the assumption r € 1 in Section 3,5 and 3.6. Up to the end of the present section (with the exception of Theorems 5.12 and 5.13), the l a t t e r condition ( i . e . r e * . ) can be replaced by the weaker requirement r € € used in the proof of the previous lemma. Every nxn matrix function bounded operators P + GQ numbers of these operators a(P+GQ) < P(Q+G'P)

G € ft can be associated with the linear and Q + G'P acting in ftQ . The defect axe connected by the relations , P(P+GQ) > a(q+G'p) (5.9)

as in case when the operators P + GQ and spaces Lp and L^ , respectively.

Q + G1 P

are studied in the

The relations ( 5 . 9 ) are a simple consequence of the f a c t that the kernel

184

of the operator P + GQ Q

vector functions on fl

(Q + G'P) exactly coincides with the set of orthogonal (in the sense of the bilinear form

(1.13)) to the image of the operator

Q + G*P

(P + GQ) . In its turn,

this assertion can be proved in the same way as the corresponding result from Theorem 3.1. As in Chapter 3» in addition to the operators P + GQ and it is convenient to introduce the operators

T^(G) and

via formulae (3.22). Since in the case in question im Q = (fl~)a t these operators act in

(R+)a

and

Q + G'P ,

Tp(G') defined

im P = (ft+)a and (fl~)n > respectively.

Assertion 1) of Lemma 3*5 on the relationships between defect numbers of the operators P + GQ

and

T^(G) as well as Q + G'P

and

Tp(G')

can be transferred (together with its proof) without any changes to the situation under study. However, assertion 2) of this lemma is no longer valid, since the operators

QGQ and PG'P

do not act in spaces dual

to each other (as it was the case in Chapter 3)» rather in one and the same space ftn . Therefore, nothwithstanding that formulae (5*9) are available, we cannot a priori assert that the operators 1

Q + G P

P + GQ

and

are Fredholm only simultaneously.

THEOREM 5.6. Let ft be a decomposing (L^jRj-algebra, and let

G be

a matrix function of n th order with entries from ft . The matrix function P + GQ

G and

is factorable in ft if and only if the operators Q + G'P

regarded in fln are Fredholm and their indices

are opposite. In addition, the value sum of positive partial indices and

a(P + GQ) coincides with the 0(P + GQ) is opposite to the

sum of negative partial indices of G . Proof. Let operators

G

be factorable in ft . Due to conditions (5»7)« the

G+PG~1 , GIQ(GI)"1 , G~1 A~ 1 QG~ 1

and

(G^)~1 A"1P(Gl)""1

are bounded in flQ . '-^he same calculations which where carried out in the proof of Theorem 3.16 show that the operator G~1 A~1QlG71+G+I>G+1 is a two-sided regularizar of the operator P + GQ . In the same way one can deduce that

(G^)~1 A^PC&l)""1 + G^Q(G^)"1

is a two-sided

regularizer for Q + G'P . Therefore, according to Theorem 1.8, the

185

Fredholmness of the operators be proved oa coaditioa that

P + GQ and G

acting in

can be f a c t o r e d in

Furthermore, in case of a matrix function solutions of the homogeneous equation + formulae ( 3 . 6 ) ( f o r

Q + G'p

will

A .

G f a c t o r a b l e in

(P+GQ)

x Q . Then there exists a

6 > 0

such that

215

every matrix function

G1

from the 6-neighbourhood of

G can be

factored with the same t o t a l index and p a r t i a l indices lying in [*q»H-1 ] • Proof«

-X

Consider the matrix function

t

G ( t ) . I t s p a r t i a l indices

are non-negative. Due to the previous theorem, there e x i s t s a such that a l l matrix functions from the 6^neighbourhood of

6Q > 0 t~* n G(-t)

aire factorable with the same t o t a l index and have non-negative p a r t i a l indices« Analogously, a l l matrix functions from a certain 6-neighbourhood of 6 =

t"*iG(t)

min{6a||t"*>t,l||

have

have non-positive p a r t i a l indices. Set , S^lt""* 1 !!} . Then, provided that

| | t ^ G , - t ^ G l l < 6,

that the matrix function

and

t"-HnG


k,

and

k 1 =...=1^ > k 1 + «j,

k, = k\ •

k a < k^ . Then

2i (k',+1 )-(kJ l -1) > 2 . Therefore, there e x i s t numbers

m,l < m , such that 220

(k..)!?_,j y

k Q < k^ . I f , f o r instance,

Consequently, we shall assume that ki~ka

a j: 1 , we con-

1

and

k t - k ^ j < 1 , ki-k^ > 2 ,

Consider the tuple

obtained from

the elementary operation f o r

with the aid of

r = 1 , s = m and show that

>• (k!j)|j=1 . We restrict ourselves to the v e r i f i c a t i o n of inequality ( 4 . 8 ) in which k^ i s replaced by k ^ . n For k > k, , we have E max{k^-k,0} = 0 , f o r k < k^, 0=1 E max{k^1^-k,0} > E max{kUk,0} . Finally, i f

kL < k < k', , k, > k .

then kj. > k ^ > k^-1 > k j , where k^ > k!j f o r 3=1,...,1, so that n n n n s E max{k',-k,0} < Z max{k,-k,0} - 1 < Z max{k,-k,0} -1 = E max{ky ; -k,0>. J 3 3 3 0=1 ~ d=1 ~ d=1 0=1 Repeating these considerations, after tuple

d (< l )

steps we come to the

( k j j d ) ) ° a 1 > C k |j)j = i t wbere either

or

< k,

(if

(k.)!? =1

?

(k^)jj ^ ^ (kj)j ( k ! , ) ^ , because

Lemma 6.1 i s proved. THEOREM 6.5.

d< l)

N of elementary operations one can for which either

. In any case, as shown above, then

(if

d = l ) . From this we conclude that after the

application of a f i n i t e number obtain a tuple

> kQ

k£ N ) = k^

-» ( k j ) J = 1 .

( k ^

?

(k^))^

or

But

.

=

Let the matrix function

G be factorable, and assume

the tuple of i t s partial indices to be then in an arbitrary neighbourhood of

(*.j)° a• (*!j J ^ p G there exists a matrix func-

tion the tuple of partial indices of which i s

(*!j)jj_,| .

Proof. The assertion i s evident i f (x'.) = ( x . ) . I f (x 1 .) ± ( x . ) , — — J d j j then, according to Lemma 6.1, there exist tuples (xS ')!?_«, , k=0,...,N / \ /• \ u d=I ' such that

= *-j t

x

j

=

*Jj

011(3

'bl:ie (k"*""1) ^

tuple i s obtained

from the k th one with the aid of one elementary operation. Arguing as in the proof of the necessity in Theorem 6.3, we conclude that in any neighbourhood of the given (factorable) matrix function there exists a matrix function whose tuple of partial indices i s obtained from the tuple of partial indices of the i n i t i a l one with the help of one elementary operation previously chosen. Specifying e/2-neighbourhood of

G a matrix function

s > 0 , we choose in the G., with the tuple of partial 221

( H J 1 ' ) ^ , in the e/4-neighbourhood of

indices G»

the tuple of partial indices of which is

After all, we obtain a matrix function G N G

a matrix funotion

with the tuple of partial indices

(x|.

and so on.

from the e-neighbourhood of , which proves Theorem

6.5. s COROLLARY 6.1. In any neighbourhood of a factorable matrix function there exists a matrix function with a stable tuple of partial indices» With the help of Theorem 3*7'

all results derived in the present sec-

tion can be transferred to the case of right-factorization. COROLLARY 6.2. In any neighbourhood of a matrix function factorable from the left and the right there exists a matrix function with a stable tuple of left as well as right indices. Proof. For given e > 0 , we take in the e/2-neighbourhood of the original matrix function G

a matrix function G 1

with a stable tuple

of partial indices (due to Corollary 6.1, this is always possible). If e is sufficiently small, then the matrix function G, will be factorable not only from the left but also from the right. In view of the stability of its left indices there is a 6-neighbourhood of G., to which belong only matrix functions with the same tuple of partial indioes. Setting 6 1 = min{6,e/2} and using an analogue of Corollary 6.1 related to right-factorization, we find in the 6.,-neighbourhood of G,

a matrix function G 2

with a stable tuple of right indices.

This is just the desired one. = The results of the present section admit a topological interpretation. Denote by

of

partial indices of which is

factorable matrix functions1' the tuple of (x.j. The representation

$ =

U

\

yields a decomposition of the class of all factorable matrix functions into disjoint non-empty sets, where« 1) if

^ c clos ®(Mi ^ , »J «) 2) if (x^) } (xj.) , then ^ n clos = 0 . _______________ 0 0 1) Recall that in this section the factorization is always understood as ^-factorization. 222

(Hj) y

(x!j) , then

Properties 1 ) and 2 ) are reformulations of Theorem 6 . 5 and 6 . 4 , respect i v e l y . Theorem 6 . 3 means that 3)

the s e t

i s open i f and only i f the tuple

( H -j)

satisfies

condition ( 6 . 2 ) . Since a tuple

( k . ) having the property ( 6 . 2 ) i s comparable with a l l u tuples with the same sum and i s minimal ( i n the sense of the ordering then property 1 ) means, e s p e c i a l l y , that 4) *x -

under the condition ( 6 . 2 ) , the s e t u

•

f a i l s to have property ( 6 . 2 ) , then the s e t

l i e s in the complement (with respect to are open and d i s j o i n t ,

^

$ H ) o f an open dense s e t .

Therefore, i t i s nowhere dense. Because, f o r d i f f e r e n t $x

i a

M , the s e t s

^ i s nowhere dense in t h e i r union

e i t h e r . Thus! 5)

if

x 1 - x Q > 2 , then

^ i s a nowhere dense subset of

$ .

F i n a l l y , from property 2 ) we can conclude that 6)

the s e t s

^

and

®(x'.)

8X6

separated i f f the tuples

(kL) are not comparable. u With Theorem 3*20 in mind we can imagine the s e t

(*-j)

and

9

as the union o f a

countable number of separated open s e t s ® x ' s divide into an open s e t nowhere dense s e t

6.3.

*(x,

)

(

gn ^ . E

. Now we can every of the n * ^ ( E = * « *•» - * n < 1 ) and a 3=1 x f 2) .

V =

ESTIMATES FOR THE PARTIAL INDICES 0? MEASURABLE BOUNDED MATRIX FUNCTIONS

In Section 6.1 i t was shown that inequality ( 6 . 2 ) i s a necessary and s u f f i c i e n t condition of s t a b i l i t y of p a r t i a l i n d i c e s . However, i f we want to use t h i s c r i t e r i o n , we must know the value of the l a r g e s t and smallest p a r t i a l index. Therefore, we are i n t e r e s t e d in a s s e r t i o n s l a which besides 4 - f a c t o r a b i l i t y of a c e r t a i n matrix function also e s t i mates of i t s p a r t i a l indices are e s t a b l i s h e d . 223

Sufficient c r i t e r i a of non-negativity and non-positivity of partial indices are also of some significance, since the defect numbers of the vector-valued Riemann boundary value problem with matrix coefficient G are stable i f and only i f a l l i t s partial indices have the same sign. The present section i s devoted to a result of precisely such a type. Owing to Theorem 6.2, every nxn matrix function which i s ^-factorable in

Lp

with zero partial indices has a neighbourhood entirely con-

sisting of matrix functions which have the same property. In particular, the matrix function identically equal to bourhood. Denote by

&p > n (r)

I

possesses such a neigh-

the largest possible value of the radius

of such a neighbourhood. In other words, ftp

a(r)

is a number such

that, f o r

||I - G0|| < 6 p > a (r) , the operator function

P + GQQ

(6.6)

i s invertible and there exists an nxn matrix

F f o r which

||I - F|| = 6

(r)

and the operator

P + FQ

i s non-invertible. We denote by LHHHA. 6.2. ———— — 1)

Ap the interval with ends

)

the numbers

3)

6p,n(r)

r -

2||S||Ln ||Q||~2 and 2 " V 1 + IISII'

q = p/(p-1) . p,a_ ( r )

are true:

are estimates from below f o r P

6 p f a ( r ) < Sin u/max{p,q} j

, then

6q,n(r)

6rfB(r) > 6p>a(r)

»

1) The estimate

r € Ap | in particular,

for

= max{6 p t n (r): 1 < p < » } Proof.

and

The following assertions related to

6p > a (r) | i f 2

p

.

_(r) > follows from the contracting ~ Lp mapping principle, since, f o r ||I - G|| < ||Q|| „ , the operator P + GQ = I + (0 - I)Q

P'Q

d i f f e r s from the unity operator by a summand with norm

smaller than one. To prove the second estimate from below f o r we consider an arbitrary matrix function

G from the

neighbourhood of the identity matrix function. Then 224

6p^Q(r) ,

2||S||/(1+11811*)-

||I - G|| < for some G, =

a > ||S||Q L P 1

a* - 1

G

-22 1 + o1

(6.7)

(> 1) . We introduce the matrix functions G = (l-Gi)(l+Gi)~1 . Obviously, G

and

and

G., are

^-factorable only simultaneously and the tuple of their partial indices coincide. Moreover, due to condition (6.7) and the identity i

+ Gi

= -f2!_ (i + £!±1 (G-I)) , a1- 1

the matrix function

2a2

I + G1

is invertible in L^ . Consequently, the

invertibility if the operator P + G,Q = ^(l+GS)(l+G 1) is equivalent A to the invertibility of the operator

I + GS . In this way, the desired

estimate will be obtained, if we clarify that the inequality

||G|| < a"1

holds, provided that condition (6.7) isA valid. In other words, we have to show that the matrix function Y = oG will be contracting as soon + g * (I-G) is contracting. But 2o are connected via the relation

as the matrix function I

X =1

Y = (I - eX)~1(X - el) , where

e = a

X

and

(6,8)

. It remains to use the well-known fact from matrix

theory that, for any

e €[0,1] , transformation (6.8) maps a contrac-

tion into a contraction. The latter can easily been checked by showing that the spectrum of

Y

lies inside the unit disk.

Finally, let the condition continuous function cos^/p + Clearly,

f

T € "R be fulfilled. Consider a piecewise

given on r

sin 2rc/p and

cos it/p - ¿j sin 2rc/p with p = max{p,q} .

||f - 1|| = sin it/p . Furthermore, at the half of the points

of discontinuity of the function 2n/p

which takes only the two values

a

f

the argument jump is equal to

and at the other half it equals

the discontinuities of

f

2it - 2rc/p = 2u/min{p,q} • If at

there exist the tangents to the contour

r

(which can always be effected), then we are able to use Theorem 5.15, according to which this function is not ^-factorable in Hence 6

p

(and L^).

6 ^(r) < sin u/p . From here and the obvious relation rI ' < 6p ^ r ) we deduce the inequality to be proved. 225

2)

I t ia not hard to see that the inverse transforation to ( 6 . 8 ) i s

this transformation i t s e l f * Therefore, i t maps the 1-neighbourhood of the zero matrix function onto i t s e l f (and not only into i t s e l f as mentioned above)» Moreover, i t is a one-to-one mapping. Setting I, s I + si ,

X1 = ( I - eX)' ,

from ( 6 . 8 ) we observe that, f o r a r b i -

trary s < 1 , the transformation

X, = (1 - e'XA)'1-1 maps the e-neighbourhood of the identity matrix function one-to-one onto i t s e l f . At the same time, Theorem 3«?' and Corollary 3«6 imply that the matrix function

X1

i s «-factorable in

indices i f and only i f the matrix function with zero partial indices. Thus, f o r some

Y*

Lp

with zero partial

i s «-factorable in

L

q.

e < 1 , the ©^neighbourhood

of the identity matrix function consists completely of matrix functions which are «-factorable in

Lp

with zero partial indices f o r

i f and only i f they possess this property f o r the evident relation

6r

Q(r)

< 1

r = p

r = q . This aa well as

(which i s true f o r a l l

r €(1,»))

imply assertion 2) of the lemma. 3)

Now we consider the

function

6„ p,a_(r)-neighbourhood of the identity matrix

I . According to assertion 2 ) , every matrix function

G from

this neighbourhood i s «-factorable with zero partial indices not only in

LJT

but also in

function

L*A . Owing to Theorem 3«9 and 2.2, the matrix

G is «-factorable with zero partial indices in a l l

Lr

with

r € Ap . Hence, the 6p^a(r)-neighbourhood of the identity matrix function is a part of i t s 6 r

n (T)-neighbourhood,

i.e.

a(r)

< 6p

n(r)

( r € A^) . Since, f o r a l l p €(1,co) , the interval A contains the point r = 2 , we deduce that the l a s t of the relations to be proved i s also correct. THEOREM — — 6.6.

= Let the matrix functions

G and

X.+ € L00

be such that (6.9)

Then» 1)

if

X+ € M+ ,

and, f o r a l l 226

6 L* , X_ € L~ , lT 1 €

r e A^ , the partial iv-indices of

, then

Ap c

«(G)

G are non-negative,

2)

if

X+ €

, ^

and, for all Proof»

€
|/(t) « l k d • d ^ The function ty does not vanish on T . Indeed, i f Yt ^ Yt

then, by assumption, the eigenvalues of of

At

d

and A^ k

a . e . on

Yt

d

n k

¥ &«

^ i n t ek r s e c t i o n F(*t_) l i e in the

•

Therefore,

A+. fl A+. i 0 • Since neither of the disks A+. and A. td td Tc k contains zero, then the interval connecting t h e i r centres z- and z,_K d does not contadn zero e i t h e r . Consequently, zero does not belong to the interval

•

Now we are going to show t h a t the matrix function G^ = \|/F s a t i s f i e s « o the condition ( 6 . 6 ) . For t h i s purpose, note t h a t , i f the eigenvalues of

F(t)

Zj. F ( t )

l i e in the disk l i e in the disk

6 s sin a/2 At

and d matrices

A+. , then the eigenvalues of the matrix d AQ with centre a t the point 1 and radius

( > sin a H / 2 ) . I f the eigenvalues of

F(t)

l i e both in

At

, then we recognize t h a t the eigenvalues of both the * Z j ^ F ( t ) and z ^ F ( t ) l i e in AQ and, t h e r e f o r e , also of a l l 231

matrices of the form the matrix

GQ(t)

£F(t)

l i e in

with AQ

£ €

a . e . on

. Hence, the eigenvalues of T , and those of

the disk

{zs|z| < 6} . Taking into account that

matrices

I - GQ(t)

are normal together with

I - GQ(t)

6 < 6_ n ( r ) P, Q

in

and the

F ( t ) , we find that

condition ( 6 . 6 ) i s f u l f i l l e d . In view of statement 3) of Lemma 6.2, the matrix function factorable in

Lr

for all

r € Ap

F

is 4-

and i t s p a r t i a l indices are equal

to zero. Applying now Lemma 6.3 with matrix function

GQ

X = ^

, we conclude that the

i s also ^-factorable in

L p , r € A^ and i t s p a r t i a l

indices axe equal to each other and to the index of the function Thus Theorem 6.10 i s proved. COROLLARY 6.4.

=

Leb the matrix

U(t)

more, suppose that, f o r any point arc

Y-fc ( 3 t )

of the contour

r

be unitary a . e . on

and a sector

v < 2 arcsin 6 P »na( r )

of the matrices

l i e in

interval

A^

S^

U

S^. with vertex at the

such that a l l eigenvalues

f o r almost a l l

t € y^ . Then the

i s e n t i r e l y contained in some component of the domain

of § - f a c t o r a b i l i t y of the matrix function indices of

T . Further-

t € T , we are able to indicate an

origin of apex angle U(t)

x •

U

and a l l p a r t i a l p -

are equal to each other.

In f a c t , a unitary matrix i s a normal matrix the eigenvalues of which have absolute value 1. Therefore, the f a c t that the eigenvalues of the matrices

U ( t ) , t € y^

in the intersection

l i e in the sector

implies that they l i e

of this sector with the unit c i r c l e . I t r e -

mains to note that the arc scribed in the sector

S^

may be included into a disk

A^

in-

S^. . A f t e r that, Theorem 6.10 w i l l be applied, s

The normality condition occurring in Theorem 6.10 i s very i n c i s i v e . Thus, we t r y to f i n d an analogue of these propositions related to matrix functions which do not necessarily take normal values. In looking f o r such an analogue, the following considerations are useful. The eigenvalues of the normal matrix

G(t)

belong to some disk i f and

only i f the convex hull of the set of eigenvalues of

G(t)

belongs to

this disk. I t i s well-known that the convex hull of the s e t of eigen-

232

values of the normal matrix H(A) = {

E

A = (a^)? coincides with the s e t -L» J— I

a i 1 5 j 1 « £ I g J * = 1} .

(6.10)

However, formula ( 6 . 1 0 ) continues to make sense f o r a matrix

A not

necessarily normal. DEFINITION 6 . 2 .

'¿'he s e t

H(A)

HausdorfC s e t ) of the matrix For any matrix

A , H(A)

i s called the numerical domain (or

A .

i s a convex compact subset of the complex

plane containing a l l eigenvalues of the matrix

A (GLAZMAN, LYUBICH

[ 1 ] , MARCUS, MING [ 1 ] ) . I t i s quite natural to suspect that there e x i s t s u f f i c i e n t conditions of $ - f a c t o r a b i l i t y of the matrix function

G and estimates f o r i t s

p a r t i a l indices formulated in terms of H(G). DEFINITION 6 . 3 .

By

d(A)

part of points from

we denote the smallest value of the real

H(A) .

I t i s well-known ( c f . MARCUS, MINC [ 1 ] ) t h a t d(A) value of the Hermitian matrix LEMMA 6.4» tion

Then for Proof.

i s the smallest eigen

Re A = tj(A+A*).

Assume the matrix function

G f 1M

to s a t i s f y the condi-

ess i n f d(G(t)) > l/l-M ( r ) ||G|| . p , a ter Ap £ $(G)

and a l l p a r t i a l r - i n d i c e s of

(6.11) G are equal to zero

r € Ap . We shall show that the matrix function

matrix function

GQ

G d i f f e r s from the

satisfying inequality ( 6 . 6 ) only by a constant 2

multiplier

p. > 0 . For t h i s purpose, using the equation ||a|| = ||AA*||

and choosing the point

t e r

a r b i t r a r i l y , we write

||l-nG(t)||a =

= ||(I-uG(t))(l-RG(tJ»|| = ||I+|i a G(t)G(t)* - 2H Re G(t)|| . Inside the noim sign we have obtained a Hermitian non-negative matrix N the norm of which, as well-known, i s the maximum of the quantity taken over a l l vectors

x

of the unit sphere. Since

(Nx,x

( l x , x ) = ||x||l = 1

(H*G(t)G(t)*x,x) = n*||G(t)*x||* < u'||G(t)*P = H2l|G(t)||*

and 233

(-2H Re G ( t ) x , x ) s -2n(He G ( t ) x , x ) < -2ud(G(t)) , we deduce Hl-UGCtJH* < 1 + |i«||G(-b)||a - 2|id(G(t)) . Therefore, the inequality ||I-jiG(t)||* < 1 + (x*f|G||* - 2(i ess inf d ( G ( t ) ) holds a.e. on r . ~ t€r Choosing u = ||G||~2 ess iaf d ( G ( t ) ) , we obtain that t€r ||I-|iG(t)||a < 1 - ||G|f2 ess i a f d ( g ( t ) ) a.e. on T. Thus, due to condi~ ter tion (6.11), ||I-HG|| < 6 p j a ( r ) . From this and assertion 3) of Lemma 6.2 we conclude the $-factorability in

L r , r € Ap , of the matrix function

(iG . Moreover, we see that i t s partial r-indices are equal to zero. Clearly, the i n i t i a l matrix function

G i s , together with

jiG , also

^-factorable and the tuples of partial indices of these matrix functions coincide.

=

Acting as in the proof of Theorem 6.10, from Lemma 6.4- one can get the following result. THtSORBM 6.11. every point line

Let the matrix function

G € L„ 00 be such that,* f o r t € T , there exist an arc Y-fc ( * t ) aQ d a straight

1+. whose distance from zero i s greater than «

and which has the property« f o r almost a l l HCGi'O)

T €

||G|| l / l - ^ TQc F ) P> , the set

separated from zero by the straight line

matrix function

G i s ^-factorable in

Lp

for

r € Ap

. Then the and a l l i t s

partial indices coincide. Supposing, in addition, that the contour

r

satisfies the Lyapunov

condition, we are able to strengthen Lemma 6.4. To this end, we need the following result related to factorization of positive functions. LEMMA 6.5.

be a function given on the Lyapunov contour r +1 which takes only non-negative values and f o r which of" 6 L . Then +1 +T there exists a representation a = a + a_ with a~ € L^ , a_ 6 L~ .

Proof. a = a ot

Let

We set

a

a + = exp(P In a ) , a_ = exp(Q In a ) . Evidently,

and the function

degenerate in «

a

(a

, respectively) i s analytic and non-

~+1

(J> ) . A i s o , |a~ ( z ) | = exp(+ Re(P In a ) ( z ) ) .

Furthermore, i f we prove that Re(P In a ) i s a function bounded in Z>+, ±1 + then the inclusion a + € Lro w i l l be proved.

234

We have (P l a « ) ( « ) » s i r

/ la «(t) g -

,

r so that |Re(P l a a ) ( z ) | = &

/ l o « ( t ) 1» g - |
Re(Ax,x) > d(A)||x||z, on t h e other hand, which implies ||Ax|| > d(A)||x||. Hence A i s < d(A)~1.

^ - f a c t o r i z a t i o n i s a l r e a d y w e l l u n d e r s t o o d . This allows f o r making more p r e c i s e t h e r e s u l t s on t h e v a l u e s of p a r t i a l F i r s t of a l l , suppose t h a t

G € L* + C

indices.

(L~ + C) . According t o Theorem

5 . 2 , from t h e S - f a c t o r a b i l i t y of such a m a t r i x f u n c t i o n i n some we deduce t h e $ - f a c t o r a b i l i t y i n a l l

Lp

L p , 1 < p < » , moreover, w i t h

t h e same t u p l e of p a r t i a l i n d i c e s . Hence, i n t h e f o r m u l a t i o n s of Theorems 6.6 - 6.11' and Lemmas 6.4-, 6.4-' we may r e p l a c e maximal value

62

a

( r ) » which w i l l be denoted by

p , n•(r)

by i t s

6 Q ( r ) , and claim

t h e e x i s t e n c e of a ^ - f a c t o r i z a t i o n w i t h n o n - n e g a t i v e

(non-positive,

c o i n c i d i n g , e t c . ) p a r t i a l i n d i c e s i r r e s p e c t i v e of the p a r a m e t e r v a l u e p . In a d d i t i o n , Theorems 6 . 6 - 6 . 9 can be s i m p l i f i e d as f o l l o w s : i f G € L* + 0 , then i n a s s e r t i o n 2 ) of Theorems 6 . 6 and 6 . 9 t h e r e q u i r e —1 + ment that

X+ € M^ 1

X7 €

can be o m i t t e d , and i n Theorems 6 . 7 and 6 . 8 - t h e demand . If

G € L~ + 0 , t h e n i n a s s e r t i o n 1 ) of Theorems 6 . 6

and 6 . 9 we may cancel t h e requirement and 6 . 8 - t h e demand

X ^ e M~ , but i n Theorems 6 . 7

X ^ e M~ . 1

00

Now we a r e going t o s t a t e t h e e x a c t f o r m u l a t i o n and t o conduct t h e proof of t h e r e s t a t e d a s s e r t i o n 1 )' of Theorem 6 . 6 i n t h e c a s e

G 6 L~0 0 + 0 .

The o t h e r v a r i a n t s can be considered a n a l o g o u s l y , which i s l e f t t o t h e interested reader. THEOREM 6 . 1 2 .

Suppose t h a t

G € L~ + C ,

X+ €

t h e c o n d i t i o n s : 1 ) ||I-X + GXj| < 6 a ( r ) , 2 ) Then ( i )

1

partial Proof.

(1 < p < oo)

G admits a i - f a c t o r i z a - »

w i t h a n o n - n e g a t i v e t u p l e of

indices. X_ 6 L~

The r e l a t i o n

operator

implies t h a t

PX_Q = 0 . Moreover, t h e

PGQ i s compact by Lemma 5 . 1 . Consequently, P + X+GX_Q = (P+X+Q)(P+GQ)(P+X_G) + T ,

where

T

satisfy

L+ .

X~ e M~ , ( i i ) t h e m a t r i x f u n c t i o n

t i o n i n a l l spaces

, X_ 6 L~

(6.12)

i s compact.

From c o n d i t i o n 1 ) of t h e theorem we conclude t h e i n v e r t i b i l i t y of t h e operator

P+X+GX_Q

i n t h e space

L ° , and from c o n d i t i o n 2 ) and Theorem

5 . 2 we g e t t h e Fredholmness of t h e o p e r a t o r

P+X+Q

(in all

spaces 237

Therefore, equation (6.12) and Theorem 1.9 imply the Fredholmness of the product

(P+GQ)(P+X_Q) in the space L? . In turn, from the Fred-

holmaess of this product and the same Theorem 1.9 there follows that P+GQ

is a $_-operator.

Due to Corollary 5.1, if P+GQ the operator P+GQ of the product

is a $_-operator and

G € L~ + C , then

is Fredholm. In this case the second factor P+X_Q

(P+GQ)(P+X_Q) will also be Fredholm. Since

then, in virtue of Theorem 3»15', we may conclude that

X_ € L~ ,

iT^C M~ . In

this way, assertion (i) has been proved. As soon as this condition is fulfilled, we can apply assertion 1) of Theorem 6.6, according to which the partial 2-indices of the matrix function

G

are non-negative

(the $-factorability of this matrix function in L a FredholmnesB of the operator P+GQ

just proved). The result obtained

can be transferred to all parameter values beginning of the section.

results from the

p , as was mentioned at the

s

If the considered matrix function

G

is continuous, then we can apply

the strengthenings of Theorems 6.6 - 6.11 for the case well as for the case

G € L* + C

as

G € L~ + 0 mentioned at the beginning of the

section. In the case of a continuous matrix function

G , a particularly

clear form can be given to Theorem 6.11. DEFINITION 6.3. By

h(A) we denote the distance from the numerical

domain of matrix A

to zero. The matrix A will be called strictly

non-degenerate, if

h(A) > 0 , i.e. if

H(A) does not contain zero.

This definition is justified by the fact that a strictly non-degenerate matrix is clearly non-degenerate. THEOREM 6.13. all

t e

If the matrix function

is continuous and, for

r, h(G(t)) >

Vl-6£(r) ||G(t)|| ,

then all the partial indices of Proof.

G

The continuity of

G

G

(6.13)

are equal to each other.

implies the continuity of the function

a(t) = ||G(t)|| . In accordance with inequality (6.13) this function does not vanish. 238

From relation (6.13) we deduce an analogous relation f o r the matrix function

F = a~1G . In addition, f o r a l l

Consequently, f o r a l l

t € T , we have

||F(t)|| = 1.

t € r ,

h ( F ( t ) ) > V l " 6 £ ( r ) ||F|| . The closed convex set which we denote by Let

H(F(t))

(6.14)

contains a unique point nearest to zero

z ( t ) . Due to inequality (6.14),

z(t) ^ 0 •

be the straight l i n e passing through the point

^ ( h ( F ( t ) ) + V l - 6 * ( r ) ||f|| ) z ( t )

and orthogonal to the radius vector of

this point. U t i l i z i n g inequality (6.14) again, we find that distance to zero greater than H(F(t)) where

V 1 - 6 ^ 1 " ) ||F|| and separates the set

from zero. Since the matrix functions

nuous, the straight l i n e T

F

and

G are contiH(F(T))

,

t .

Summarizing, f o r the matrix function indices of

F

separates from zero a l l sets

i s s u f f i c i e n t l y close to

6.11 are f u l f i l l e d ( f o r

has a

F , the assumptions of Theorem

p = 2 ) . In view of this theorem, the p a r t i a l

are equal to each other. Applying Lemma 6.3

we observe that the partial indices of which proves Theorem 6.13.

for

x = «

G are also equal to each other

=

For the total index of a continuous matrix function, Ivluskhelishvili's formula i s true. Therefore, i t i s not hard to evaluate the partial indices of a matrix function

G which meets the assumptions of Theorem

6.13« each of them i s equal to 2jj{arg det G ( t ) } r

de'b G ( ' b ) } r

("thus, ' b l i e value

automatically proves to be d i v i s i b l e by

n ) . Notioe

however, that in the suppositions of Theorem 6.13 the values of the p a r t i a l indices of

G are closely connected with the behaviour of i t s

numerical domain. To illuminate this connection, we f i r s t remark that the index of a function

i|r continuous on

r i s uniquely defined by the

condition \|/(t) € H ( G ( t ) ) . In f a c t , i f

i s any other function continuous on

this condition, then, f o r every = W("t) + ( 1 - X ) t 0 ( t )

(6.15) r

and s a t i s f y i n g

X ^[0,1] , the function tyj^(t) =

i s continuous and f u l f i l s condition (6.15),

239

therefore, i t does not vanish on

T . ^he mapping

X -»

0(r) . Conse-

interval [ 0 , 1 ] continuously into the functional space quently, the indices of a l l functions indices of

>|/Q and

maps the

coincide. Especially, the

\|/1 = ty are the same.

I t i s quite natural to c a l l the common value of the indices of a l l functions

\|/ continuous on

T

and satisfying condition (6.15) the winding

number Q± the numerical domain around zero. The partial indices of

G

coincide with precisely this number as soon as Theorem 6.13 applies. Indeed, the function ty may be chosen in such a way that the matrix function

ty~1(t)||G(t)||~^

s u f f i c e s to set, e . g . , F

s a t i s f i e s the hypothesis of Lemma 6.4 ( i t = z ( t ) ) . But then the partial indices of

are equal to zero, and, as follows from Lemma 6.3, the partial indi-

ces of

G coincide with the index of the function

, i . e . , with the

winding number of rotations of the numerical domain around zero. Olearly, the remark made at the end of the previous section can be applied to Theorem 6.13 in f u l l . Thus, not only the l e f t but also the right indices of an arbitrary matrix function

G satisfying the condi-

tions of this theorem coincide with the winding number of the numerical domain around zero. 6.6.

ON THE STABILITY OP THE FACTORIZATION FACTORS G+

Up to now, the question of s t a b i l i t y of the factors

G+

from the

^-factorization ( 2 . 1 ) of the matrix function «•

L P

has not been

G in

studied in f u l l d e t a i l . Hence, we r e s t r i c t ourselves to the explanation of results associated with the factorization problem in .a (C,R)-algebra ft • Following the agreement at the beginning of the chapter, instead of "factorization in ft" we shall simply write " f a c t o r i z a t i o n " , under "norm" we shall understand the norm in

A , etc.

Closeness of the factorization factors

G+

and

F.+

of the matrix function G

of the matrix function F(t) = F+(t) A,(t)F_(t)

(6.16)

i s characterized by the quantities l|G+-F+H • H G - - p J » H ^ - ^ l l 240

. ll€ 1 -iC 1 ll

(6.17)

THEOEM 6.14.

Let

G be a certain matrix function admitting a f a c -

t o r i z a t i o n ( 2 . 1 ) . Then there exists a

6 > 0

such that, i f ||G-F|| < 6,

(6.16) i s a f a c t o r i z a t i o n of the matrix function of the values ( 6 . 1 7 ) i s smaller than Proof. Since

F

and at l e a s t one

6 , then A i = A .

For the sake of definiteness, l e t

||G-F|| < 6

and ||G~1-F~1||< 6.

G+ A - F + A i = GG~1-FF21 = G(g3 1 -F~ 1 ) + (G-F)F2 1 , then l|G + A-F + Aill < l|G||l|G:1-F:1||+||G-F||||r1|| < 6(iigii + P : 1 ! ! ) < 6(I|G|I + IIG:1-iC1|l • l|Gl1||) < 6(iiqh + IIG:1II + &)

so that the d i f f e r e n c e of corresponding columns of the matrix functions G + A and

F+A1

can be made a r b i t r a r i l y small, i f

6

i s chosen in a

suitable manner. Consequently, i f we choose 6 small enough, then the norm of the vector K(1)_x. J can also be made a r b i t r a r i l y small, where function g . - f . t J g., u u XJ . Then

" * « ) T~1dT . Letting

6 - . 0 , the r i g h t -

hand side becomes a r b i t r a r i l y small and the l e f t one does not depend on

6 . Consequently,

function

G+

g ^ ( 0 ) = 0 , which i s impossible, as the matrix o i s non-degenerate in 31 * . The contradiction obtained

shows that, f o r s u f f i c i e n t l y small for

6

6 , x ^ 1 ) < x^ , j=1

small enough, the t o t a l indices of

clude that

G

and

F

• Since, agree, we con-

x^"1) = x.. , ¡j=1,...,n , which proves Theorem 6.14.

s

In t h i s way, f o r close matrix functions with d i f f e r e n t tuples of partia). indices, the f a c t o r i z a t i o n factors

G c a n n o t be too c l o s e . I t turns

out that the converse statement, the formulation of which must somewhat, be strengthened because of the lack of uniqueness of the f a c t o r s

G+ ,

i s also true. THBOBEM 6.15.

Let

we can s p e c i f y a

G be a factorable matrix function. For any e > 0, 6 > 0

such that, i f

||F-G|| < 6

and the p a r t i a l

241

indices of the matrix function of the matrix function

P

coincide with the partial indices

G , then, for every factorization of

there can be found a factorization of in (6.17) are smaller than

F

G ,

for which any of the values

e .

In order to prove this theorem, first of all remark that

||F+-G+|| =

= IIF(F:1-G:1) A~ 1 + (F-^)G:1 A- 1 U < (iifiiiif:1-G:iII+IIG:iIIIIF^II)II7\-iH . Thus, it can be guaranteed that the quantity —1 as

||F-G|| and

||F+-G+||

is small as soon

||P_ -G_ || are small. Since in every Banach algebra the

operation of taking the inverse element is continuous for fixed 1

1

||i'~ -G~ || will be small, if smallness of

G ,

||F+-G+|| is small. Analogously, —i —1 the

||F_-Gj| results form the smallness of

-G_ || .

In summary, it suffices to prove a simplified form of Theorem 6.15 in mm'X —1

which it is only stated that

||G_

columns of the matrix function

G_

|| < e . We shall show that the may be interpreted as vectors from

_]R

the kernel of an operator of the form

P+Qt

G . The Fredholmness of

such operators results from the following proposition. LEMMA 6.6.

The kernels of the operators

P+GQ

and

T^(G)

have

equal dimensions and the images are closed only simultaneously and have equal codimensions. Lemma 6.6 is close to assertion 1) of Lemma 3.5 with respect to its formulation and way of proving. Thus, its proof is omitted here. We only wish to explain that if and only if

+

x

LBMMA 6.7. Let ces of which are 1)

(= Px) = 0 G

x~€ ker TQ(G) .

be a factorable matrix function the partial indi-

x, > ... > x Q . Then

if

X

(Pt~1 + Qt"*iG)x = 0 |

—/1

G_

is a solution of the (6.18)

is a matrix function which is non-degenerate at least at

one point of

r

(6.18), then

X m a y

factorization of 242

and

the j th column of the matrix function equation

2)

ker(P+GQ) = 0 + ker TQ(G) , i.e., x € ker(P+GQ)

and the 0 th column of which solves the equation

G .

play the role of a right factor in some

Proof. and

Relation (6.18) is equivalent to both equations P t ^ x s 0

Qt""wlGx = 0 • The first of the latter two equations means that

t ^ x €(fl~)a

or, what is the same, x €(ft~)Q ; the second amounts to

saying that

t'^Gx €(ft+)n . If we take

j th column of the matrix function

-1

G_

x = x., , where

x^

, then obviously

is the -

x. €(ft~) •

Moreover,

t~M*Gx. = t^^G. A G x. is the j th column of the matrix u *** *" J

function

G + . Thus, assertion 1) has been proved. Now we proceed to

the proof of 2). If the vector function x. u is the solution of equation (6.18), then, as mentioned above, x. €(fl~) and t- H lG_AG x. €(fl+)a , therefore, we deduce that

tíG

the kernels of the operators

with them, also of the operators

A

F+Qt 1 ~ w íF

and

and

(and, togethe

B ) axe equal. Due to a

theorem from GOHBERG, KRUPHIK [4], p. 170, for any vector ker A

there exists a vector

where the constant

c

y € ker B

from

||x-y|| < c ||A-B|| ,

such that

is determined only by the operator

||A—B|| < ||y||*||t1"Kí||.||G-P|| , for any

x

A • Since

e.. > 0 , one can find a

> 0

such that, for

||G-F|| < 6 . , there exists a vector y. € ker B with u u . Now we form the matrix function Y = (y1 ... y u ) .

||y.-x.|| < o j o

Since the matrix function sufficiently small

(x1 ... x Q ) (=G~ )

e• u , the matrix

least at one point of

Y

is non-singular, for

will also be non-singular at

T . Owing to assertion 2 ) of Lemma 6.7, there

exists a factorization of the matrix function Clearly, for given the inequalities ii j?:1 c

1

p
0 , one can choose ||y-¡-x.|| < e. J

J

J

(3=1,..,, n)

F

such that

e- , 3=1,..,,n imply

F_ = "í"^ such that

||X-Y|| < e , i.e.,

6.

In order to finish the proof of the simplified form of Theorem 6.15» 6 = min{6.i 3=1,...,n} . At the same time, we u have proved Theorem 6.15 in full, = it remains only to set

COROLLARY 6.5»

Assume the matrix function

G

to be factorable and

the tuple of its partial indices to be stable. Then, for every e > 0 there exists a tion

F

6 > 0

such that, for

||F-G|| < 6 , the matrix func-

admits a factorization in which each of the quantities

(6.17) is less then

s .

In fact, if the tuple of partial indices of the matrix function stable, then, for

6 > 0

small enough, the condition

is

and

G

||F-G|| < 6

implies the coincidence of the tuples of partial indices of Thus, Theorem 6.15 can be applied.

G

F

=

In brief, the latter result can be formulated in the following way: If the middle factorization factor A

is stable, then the factors

are also stable. Now vie intend to formulate Theorem 6.15 in terms of sequences.

244

G+

THEOREM 6,15' . Let

iG^}^

be a sequence of matrix functions

admitting a factorizat;ion with, one and the same tuple of partial indices and converging to the matrix function

G

which is also as-

sumed to be factorable and to have the same tuple of partial indices. Then one may choose factorizations manner that

= G^^A G~1

-» G + ,

in such a

(in the norm (5.8), where

G + are factorization factors of G . COROLLARY 6.5*»

Let

be a sequence of matrix functions

which converge to the factorable matrix function

G

with a stable

tuple of partial indices. Then, beginning with some element of the sequence, the conclusion of Corollary 6.5 is valid. For a factorable matrix function all partial indices of which axe equal to each other, Corollary 6.5 can be made more precise. This refinement amounts for stating that the closeness of the factorization factors of close matrix functions

F

and

G may be ensured by the requirement

G_(oo) = F_(co) . Up to now, in the present chapter we have been concerned with the following problems knowing the properties of a given matrix function, determine the properties of matrix functions sufficiently close to it. Now we turn to the "inverse" problem consisting in the determination of the factorization properties for the limit of a sequence of matrix functions, if we know the factorization properties of the terms of this sequence. Theorem 6.5 shows that any factorable matrix function tuple of partial indices a sequence

G with the

may be represented as the limit of

of factorable matrix functions the tuples of

partial indices

C*^^)!? = of which obey the condition J tj I

(*.) ^ ¿J

J

but, in other respect, they are not interconnected or joined with the tuple

(*,) . Vice versa, if the sequence of factorable matrix func-

tions

{G^ J } ^

converges to the factorable matrix function

according to Theorem 6.4, for sufficiently large 0 J— I

y

J

J— *

G , then,

k , the condition

is fulfilled. Finally, the limit of a sequence of 245

factorable matrix functions may not permit any f a c t o r i z a t i o n at a l l » However, we are able to indicate conditions under which the passage to the l i m i t does not go beyond the class of factorable matrix functions and does not influence on the values of p a r t i a l indices. THEOREM 6.16.

Let

{G^}^.«,

be a sequence of factorable matrix

functions converging to the matrix function

G . Then the following

assertions are equivalent: 1)

the f a c t o r i z a t i o n f a c t o r s of the matrix functions

chosen in such a way that || does not depend on 2)

G^^

may be

< M , where the constant

M < oo

k ; G can be factored and a l l matrix functions

the matrix function

G^^ , beginning with some number

k , have one and the same tuple

of p a r t i a l indices which agrees with that of the matrix function Proof.

G .

Suppose that 2 ) i s f u l f i l l e d . Due to Theorem 6.15 1 , the f a c -

t o r i z a t i o n s of {(G|k^)"1}

and

G^^

can be chosen in such a fashion that the sequences

{(G^)-1}

converge ( t o

G~1

and

G~1, r e s p e c t i v e l y ) .

In this case these sequences w i l l be bounded e i t h e r . Thus, 1 ) f o l l o w s from 2 ) . Conversely, l e t 1 ) be v a l i d . As i t was mentioned in the proof of Theorem 5.6, the operator

^ ( G ^ )

i s a regularizer f o r

- 1

considered in

(«")n

. Hence, there e x i s t s an equibounded

sequence of regularizers of the operators

. In accordance

with Theorem 1.11 this means that the l i m i t of the sequence i . e . , the operator operator

Tp(G')

0« (G< k >) , si T^(G) , i s also iredholm. The Fredholmness of the

can be shown in a similar way. With regard to Theorem

5.6, the matrix function

G can be factored. I t s tuple of p a r t i a l

indices w i l l be denoted by certain neighbourhood of

(*.)!?_,. . In virtue of Theorem 6.4, in a o J" ' G we encounter only such matrix functions

(x 1 .) f u l f i l the condition ( k . ) >- ( * ' • ) . J J J Since there e x i s t s only a f i n i t e number of such tuples, one of the

whose tuples of p a r t i a l indices

following two situations always applies: a ) there i s a neighbourhood of

246

G

such that a l l matrix functions

G^^

contained in i t have one

and the same tuple of partial indices ( * J } b) it is possible to (k ) " choose a sequence {G s all terms of which have one and the same tuple of partial indices

(x.,) ± ( x p . (k ) (k ) A (k ) Assume that situation b) applies. As the sequence G = G+ ^ G_ converges to

G

and

(x.) , where

|| A -(&+ k £ J V 1 -A (G^8^)""1!! =

= II(G^ ks) r 1 (G (ks) -G)(G^ ks) )" 1 || < M'llG^-Gll , then the sequence EV = (G.

)

A(G

) "

converges to A if

. Consequently, in any neigh-

bourhood of the matrix function A there exists a matrix function the tuple of partial indices of which is just the same as that of the matrix function G . According to Theorem 6.4, contradiction to the conditions V

(x.) A

a n d(x.)

, and A this is a ^ *

Summarizing, situation a) holds by all means, i.e., assertion 2) is true. The proof of Theorem 6.16 is complete. 6.7.

=

COMMENTS

The question of stability of partial indices was first considered in MANDZHAVIDZE [ 2 ] in connection with the problem of the approximate solution of the Riemann boundary value problem, theorems 6.1 and 6.2 as well as the criterion of stability of partial indices (Theorem 6.3)

were obtained by GOHBjSRG and KBE& [2,3] and BOJABSKI [1], The proof of Theorems 6.1 and 6.2 conducted here is close to the one suggested in MANDZHAVIDZE [ 5 ] . The same applies to the proof of the sufficiency in Theorem 6.3» The proof of the necessity in Theorem 6.3 given above generalizes correspondent arguments from BOJARSKI

[ 1 ] , GOHBERG, KBBiN [3] to the case of matrix functions with unbounded factorization factors. In Section 10.2 of the book OLANCEY, GOHBERG [3] another way of proving the necessity is explained, which is based on the following general result of WIDOM [3] related to perturbations of Fredholm operators: Let

T

be a Fredholm operator in the Banach space

be a subspace of there exists an and a number

[X] A € S

r > 0

such that, for every non-zero such that

f(Ax) 4=

such that, for any

0

X , and let S

x € X,

f € X* ,

• Then there exist

B € S

e 6(0,r) , at least one of

247

the defect numbers of the operator

T + eB

T = P+t~sGQ , S = {AP+B^: A,B 6

application of this result to the caae € L^} , where

s

is equal to zero. The

is an integer strictly included between the smallest

and largest partial index of the matrix function

G , proves the

necessity of condition (6.2) for the stability of partial indices. Theorems 6.4 and 6.5 and Lemma 6.1 are due to GOHBiSRG and KEfiiN [3]. The topological interpretation of Theorems 6.3 - 6.5 is given by the authors under the influence of BOJABSKl's paper [3]. Remark, incidentally, that in this paper there was proved the connectedness of the intersection of the sets functions continuous on r

with the class of matrix ^i"''*1 ^ ^ class of

and of the sets

matrix functions continuous on T

and factorable in

nectedness of the intersection of

C(r) . The con-

and the class of piecewise con-

tinuous matrix function was studied in GLEBOV, DEUNDYAK [1]. Furthermore, the question of connectedness of the sets of the density in Lra

of the set $

$M

themselves as also

of all ^-factorable matrix func-

tions remains to be solved. Theorems 6.6 - 6.13 are obtained by one of the authors (SPITKOVSKl¥ [2,3,5|10]). The idea of transition from the matrix function

G, to

its Cayley transform used in the proof of Lemma 6.2 is borrowed from KBUPNIK, NYAGA [1], where this device was applied to the study of unitary matrix functiona. V

In connection with Lemma 6.3, note that in the article of SPITKOVSKII [9] one can find the description of the class

S

consisting of all

those functions the multiplication by which does not violate the factorability in L^ . It appeared that this class does not depend on p and, being multiplied by a function

x

arbitrary ^-factorable matrix function

6

® » the partial indices of an G

behave precisely as in case

of a continuous function % , i.e., they are shifted by its index. Lemma 6.4 is due to DANILYUK [4,5], its precising for the case of a Lyapunov contour (Lemma 6.41) was obtained by one of the authors (SPITKOVSKII [2]). Lemma 6.5, with the help of which the latter lemma

248

was derived, is well-known for a long time and was already used in SIMOIffiNKO [4], The proof of Lemma 6.5 is borrowed from the book GOKBERG, KRUPNIK [4], In SHELEPOV [1] it was mentioned that, with regard to RADON's [1] results, Lemma 6.5 may be transferred to the case when

T

is a Radon

curve without cusps. Therefore, Theorem 6.11* can also be transferred to this case. Concerning the considerations after Theorem 6.13 related to the values of the partial indices, let us still mention the following. Generally speaking, starting from a matrix function it is not always possible to construct a tuple of j=1,...,n

continuous on

T

G n

and coinciding, for all

continuous on r, functions

A-i(t), ¿J

t e r , with the

tuple of its eigenvalues counted with regard to their algebraic multiplicity (as a matter of fact, in moving along the components of the contour

r , the continuous branches of the solutions of the equation

det(G(t)-\l) = 0

move into another, not necessarily coinciding with

the original ones). But if we do succeed in doing so, then it makes sense to speak about indices of eigenvalues of the matrix funktion t

G.

In GAKHOV's review article [3] among others there was posed the problem of describing all those matrix functions the partial indices of which coincide with the indices of eigenvalues. The complete solution of this problem has not been found yet

The relations

u

€ H(G(t)) ,

however, imply that, under the conditions of Theorem 6.13» the indices of eigenvalues coincide with the winding number of the numerical domain of the matrix function dices of

G

around zero and, thus, with the partial in-

G . In this way, matrix functions satisfying the conditions

of Theorem 6.13 form a certain subclass of the desired set. The application of Theorems 6.6 - 6.13 is impossible without the knowledge

of lower estimates for the values

n

( r ) . Assertion 1 ) of

Le&ma 6.2 allows for replacing the problem of finding such estimates by the problem of evaluating (or obtaining upper estimates for) the norms The problem of the description of all triangular matrix functions - whose partial indices coincide with the indices of the diagonal elements solved in Chapter 4 is a special case of this problem.

249

of operators Q

aad S . la case when the coatour I* is a circle, the

norm of the operator S has been calculated in all spaces

,

1 < p < oo (VERBITSKlí, KRUPNIK [1,2], KRUPNIK [4]). This fact will be used in the next chapter. With our definition of the noun in L®

(see

Section 1.2) the norms of the operators S and Q , generally speaking» depend on n . However, for p = 2 , such a dependence fails to •xiat. In SPITKOVSKlí [3] it was shown that the noma of the operators S

and Q in L»

are connected via the relation

||Q|| = ^(||S||+||8||~'1).

Therefore, the estimates for 6 2 n(r) mentioned in assertion 1) of Lemma 6.2 coincide. Unfortunately, we do not know whether these estimates are exact or not.

It is also unknown, which of these estimates is

the most exact one for p 4= 2 and a contour r

different from the

circle. Theorem 6.15 (on the continuity of the factorization factors for fixed partial indices) is due to SHUBIN [1]. He has also proved there that, if the matrix function G is a k times continuously differentiable (analytic) function of some parameter u

and the partial indices do

not depend on a> , then the factorization factors G +

are also k

times continuously differentiable (analytic) functions of (•) . The factorization of matrix functions depending on a parameter was also examined by NIKOLENKO [1,2]. In NIKOLENKO [1] he derived the stability criterion for partial indices in case when the parameter ranges over even-dimensional spheres or triangulations. In this paper it was furthefr shown that the triple fibre bundle, where

(T x T , p_

H X

T+

) forms a locally trivial

is the class of nxn matrix functions non-

singular and satisfying a Holder condition in 1 " U r and analytic in + 3H~ , is the class of Holder continuous matrix functions with the

h

tuple of partial indices from T x T

into X. _

X

_ k (=(x 1 ,...,h

)) , and p_ is the mapping x acting according to formula p_(P.,P ) = +J M ? +

X

-

t

In NTKQLBNKO [2] it is shown that this fibre bundle is trivial if and only if x1 = ... = x Q . Theorem 6.16 was first formulated in LITVINCHUK, NIKQLAICHUK, SPITK0V8K$I

[1]. 250

A series of papers is devoted to the approximate solution of singular integral equations (and, especially, to the vector-valued Riemann boundary value problem). A systematic explanation of the state of the arts (up to 1978) related to methods of their approximate solution the reader can find in the last chapter of the monograph

PRftSSDORF [1],

Concerning later results, see BABBSHKO [3], DIDENKO [1-3], DIDEHKO, TIKHONENKO [1,2], ZCEiOTAREVSKlf [1], PR&SSDOHF, SCHMIDT [1], SILHERMANN [1]. In MBUNARGIYA [1,2] several cases are examined in which the solution of the vector-valued Riemann boundary value problem with matrix G

+ XG 1

may be found as a series

k=o

, where

J?.[k) are suc-

cessively determined as solutions of Riemann problems with matrix and a free term depending on

GQ

J?^) (j=0,... ,k-1).

Now we focus our attention on another results associated with the stability problem and estimates of partial indices. KRAVCHENKO and v NIKGLAICHUK [1J have obtained formulae for partial indices of secondorder matrix functions such that at least one element of them does not degenerate on

r . These formulae involve the dimension of the kernel

of an integral operator which can be effectively constructed by a given matrix function. CHORIEV [1-3] has studied the relationships between stability of partial indices of sufficiently smooth matrix functions

G

and the validity

of Liouville's theorem for the equation 3U = BU , where

1

z eJ>+ , B(z) = 0

B(z) = £(z)"" a&(z) for

is an extension of the matrix function differentiable in Here

L^ (p > 2)

D(t) = diag[t k+1

GD ''

for

z

, and

into the domain J5+

in the generalized sense.

t k+1 ,t k ,...,t k ]

nulling the total index of

-

is a matrix function an-

G .

In LAX [1] it was established that the property of the partial indices of a matrix function

G

given on the circle to be equal to zero is

equivalent to the solvability of the Dirichlet problem for the differ-

251

eatial equation

= G^G

Gz .

In the author's opinion, the problem of finding estimates for the partial indices and of obtaining effectively verifiable conditions of their stability or non-negativity is still far from being completely solved and any new result towards its solution iH of considerable interest.

252

CHAPTER 7.

FACTORIZATION ON THE CIRCLE

The results of this chapter may be divided into two groups, which are associated with each other in a natural way. The first of them is formed by problems of factorization and ^-factorization on the circle

T

of

Hermitian (in particular, of definite) and unitary matrix functions» The properties in which these matrix functions are peculiar enable for obtaining more complete and exact theorems on factorization. Some of these theorems are useful, especially, for the study of relationships between necessary and exact conditions of factorability, a problem, which was already stated in Chapters 2 and 3» The second group represents essential strengthenings for the space L a ( T ) of a series of important results from the preceding chapter related to ^-factorization with non-negative, non-positive, stable, in particular, with equal partial indices of arbitrary matrix functions + from the classes

Lro , L ^ + C

and

0 . This group of results is based

on the utilization of exact estimates for the norms of the operators S, P,

PGCi, QGP, which hold in

L 2 ( T ) , but aLso on the application

of results of the first group concerning ^-factorization of matrix functions with the algebraic properties mentioned above. In Section 7.1 a definition of factorization specific for the circle T will be introduced, from which we immediately shall obtain the connection between factorizations of the matrix function the matrix function

G*

in

) ,

G

in

Lp(T)

and

q = p/(p-l) , which is of some

significance to what follows. In Section 7.2 we shall be concerned with factorization and 9-factorization of Hermitian matrix functions, where the structure of the factorability domain will be revealed, relations for partial indices will be derived and a necessary condition for the domain of factorability to be non-empty will be established* A representation for a Hemiitian matrix function factorable in

L2(T )

the total index of which is

equal to zero will be described. This representation enables us to

253

estimate the cumber of zero partial indices of a given matrix function G

with the aid of the absolute value of the signature of G .

Section 7»3 is dedicated to the factorization problem in L 2

of those

Hermitian matrix function for which the corresponding quadratic form preserves the sign* (The main result of this section is a theorem which asserts that, for+a positively definite matrix function G , the necessary condition G € L., of its factorability is also sufficient. The proof of this theorem rests upon the claim on the existence of a representation of a positively definite matrix function of a erertain outer matrix function

A

G

and its adjoint

as the product A* • This pro-

position is of independent interest and significance for applications. The aim of Section 7«4 consists in making more precise various results of 6.3 and 6.4. The most general result of this section is the theorem on the existence of a ^-factorization in L^

with coinciding partial

indices for those matrix functions the numerical domain of which may be locally (i.e., for any sufficiently small arc of the contour T ) included into a sector whose apex angle is less than

2it/max(p,q.) •

A special case of this theorem is provided by the assertion that every uniformly positively definite matrix function admits a ^-factorization with zero partial indices. It appears that the factorization factors of such matrix functions are necessarily bounded. This assertion, which is of independent interest (cf. Theorem 5.2 from Section 5.1), allows for reducing in Section 7*5 the problem of $-factorability of an arbitrary matrix function

G € LM

to the same problem for a certain unitary

matrix function. Moreover, this assertion permits us with the help of this reduction to obtain criteria of $-factorability in L a

with non-

negative, non-positive and stable (in particular, zero) partial indices. Hiffls criteria are the conversions (and that even in a stronger form) of corresponding sufficient conditions from 6.3. Thus, for the problem of ^-factorization in L S (T ) they are also necessary. Note that for unitary matriac functions the criterion of existence of a ^-factorization in L a

254

with definite partial indices will be derived from the sophis-

ticated result OQ the calculation of the norms of the Haakel operators PGQ

and

(¿GP , which has important applications beyond the factori-

zation theory for matrix functions either. Furthermore, we intend to specify the results of 7.4 and 7.5 for the case of a matrix function

G

which satisfies certain analytic condi-

tions, mainly for the case of a continuous matrix function

G . This

topic is pursued in Section 7«6. Here, in particular, it is established that the partial indices of a continuous strictly non-singular (i.e., satisfying the condition

0 ^ H(G(t)) ) matrix function

G

are equal

to each other. This result is a strengthening but also a simplification of Theorem 6.13 for the case when the contour

r

is a circle.

Section 7«7 contains information on the history of the subject and comments related to current literature. 7.1.

DEFINITION OF FACTORIZATION ON THE CIRCLE

In the present chapter we deal with the factorization problem in case when the contour T suppose that T

is a circle. Without loss of generality, we may

is the unit circle T = {£ s

= 1} •

In this case complex conjugation realizes a one-to-one correspondence between the classes L~

and L* (= H^) . Indeed, the function

defined in the unit disk belongs to the Hardy class the function ^

H^

f

if and only if

defined outside the unit disk by the equation

¥(£) = f(g"*1) lies in the class E^ , where the boundaxy values of the functions

f

and "i? are connected via the relation £(t) = f(t) .

Redenoting the matrix function and the matrix function

G_

by

we may claim that, in the case

G+

from the representation (2.1) by A

B* , in view of the remark made above, T = T , Definition 2.1 is equivalent

to the following one. DEFINITION 7.1.

A representation G = AAB*

will be called a factorization in

(7.1) of the matrix function

defined on the unit circle, if the matrix functions

A

and

G

B 255

(A-

and

B)

belong to the class

H^ (H^ , q = p/-(p—1))

and .A

i s a diagonal matrix function of the form ( 2 . 2 ) . From Theorems 1.17 and 1.18 i t i s clear that the requirements A-1

€ H^

A € L

are f u l f i l l e d i f and only i f

and

function

A

€ L

i s an outer matrix function 1 5 ,

. A similar assertion i s true f o r the matrix

B .

THEOREM 7.1. in

A

A € H^ ,

Lp

The matrix function

defined on T can be factored

G

i f and only i f the matrix function

L^ . In this case the tuple the tuple

(>1-;) u

)

G*

can be factored in

of p a r t i a l (¡.-indices of

of p a r t i a l p-indices of

G* and

G are connected v i a the

relations Hi = - * n _o+1 ,

j = 1,2,...,n .

In particular, the total q-index of p-index of Proof.

G*

(7.2)

i s opposite to the t o t a l

G .

We introduce the constant matrix

(

0 ...

1 The matrix

T

1

• ' ?..

| •

(7.3)

0

i s both Hermitian and unitary, i . e .

T = T * = T .

Taking these properties into consideration, the expression

G* = B/V*A*,

which i s obtained from ( 7 . 1 ) , may be rewritten in the equivalent form G* = B T ( T A » T ) ( A T ) * . Since

(7.4)

T A * T = U " 1 T = d i a g [ t ~ * n , . . . , t " " H l ] , the representation ( 7 . 1 )

i s a f a c t o r i z a t i o n of the matrix function

G in

equation ( 7 . 4 ) provides a f a c t o r i z a t i o n of

G*

i f and only i f in

L^

( t h e r e f o r e , the

f a c t o r i z a t i o n s e x i s t only simultaneously) and the corresponding p a r t i a l indices of

G

and

G*

are connected by the relations ( 7 . 2 ) , which

proves the theorem. THEOREM 7.1 .

A matrix function

G of class

L^,

defined on T

is

^-factorable in L p i f and only i f the matrix function G* i s fl-fadA nxn matrix function A i s called outer, i f i t s entries belong to - the class D and det A i s an outer function*

256

torable in

L^ • Moreover, the assertions concerning the values of

t o t a l and p a r t i a l indices formulated in Theorem 7.1 continue to be valid. Proof.

I f the matrix function

G i s not f a c t o r a b l e in

owing to the previous theorem, the matrix function able in

G*

i s not f a c t o r -

L^ , which i s consistent with the assertion to be proved. I t

remains to consider the case of a matrix function tored in

, then,

G which can be f a c -

Lp • According to Corollary 3*3 and the definition of

f a c t o r i z a t i o n , the representation ( 7 . 1 ) i s a ^ - f a c t o r i z a t i o n of Lp

i f and only i f the operator

K& = ( B * )

-1

A^QA

-1

K ^ = [(Al)*]"" 1 (TA*T)~ 1 Q(BT)" 1

in

i s bounded in

Lp , and the representation (7.4-) i s a ^ - f a c t o r i z a t i o n of i f f the operator

G

G* in

L^

i s bounded in

.

But the operator K^, = (A*)"" 1 A T^TB -1 =(A*)~ 1 AQB - 1 may be viewed as adjoint ( i n the sense of Definition 1 . 6 ) to the operator (B*)~ Q A A , i f we i d e n t i f y

with the dual of the space

L^

with the aid of the

b i l i n e a r form (f,g)

J g(t)*f(t)|dt| T In virtue oif assertions 2 ) and 3 ) of Lemma 3«3 the operators (B')-1QA'1A-1

=

Kg

and

Eire densely defined and closed. Therefore, a s s e r t i o n 4 )

of the same lemma may be applied, owing to which these operators are bounded only simultaneously. Thus, the operators

Kg

and

Kg*

considered in

L^

and L° , respec-

t i v e l y , are bounded only simultaneously. From here and Theorem 3*8 we deduce the assertion to be proved. COROLLARY 7 . 1 . provided that

=

The matrix functions

G and

G 6 L w , also $ - f a c t o r a b l e ) in

G* L2

are f a c t o r a b l e (and, only simultaneously.

In addition, t h e i r t o t a l 2-indices are opposite and the p a r t i a l indices are connected by r e l a t i o n ( 7 . 2 ) . Corollary 3 . 6 taken together with Theorem 7 . 1 ' leads to the following conclusion. O This can be established j u s t as the corresponding part from the - proof of Lemma 3»4-»

257

COROLLARY 7.2.

The matrix function

G € Lra

(G*)""1

if and only if the matrix function in

is left «-factorable in

L ^ , where the corresponding tuples of partial indices coincide.

Due to Corollary 7.2, a unitary matrix function for which

G(t)* = G(t)~

1

a.e.

on T

)

G

(i.e., a function

is left «-factorable in

if and only if it is right «-factorable in 7.2.

is right «-factorable

Lp

L^ .

FACTORIZATION OF HERMITIAN MATRIX FUNCTIONS

From Theorems 7.1 and 7.11 it is clear that Hermitian matrix functions (i.e. such that

G(t) = G(t)* a.e. on T

) play a special role in the

factorization problem with respect to T . THEOREM 7.2.

Under the mapping

ability (and, if

p -* p/(p-l), the domain of factor-

G € L ^ , then also the domain of $-fac to rability)

of a Hermitian matrix function

G

is transformed into itself. In

addition, for its partial p-indices, the relations H

d

*d

+

H

n-j+1

+

M

n-j+1

+

w

n-j+1

< 0

if

p > 2 ,

= 0

if

P = 2 ,

> 0

if

p < 2

(7.5)

are valid. The assertion of Theorem 7«2 follows from Corollary 7.1 and Theorem 2.1 (on the monotony of partial p-indices). COROLLARY 7.3.

The components of the factorability domain of a

Hermitian matrix function which correspond to positive

(negative)

values of the total index are situated strictly to the left (right) of the point

p = 2 . The component

T (G) , provided that it is not

empty, necessarily contains the point

p = 2 .

Consider the Hermitian matrix function

G

with non-empty factorability

domain. In accordance with Theorem 7.2

G

can be factored in

some since

p > 2 . If (7.1) provides a factorization of G

in

Lp

for

, then,

is Hermitian, equation (7.4-) yields a factorization of this

matrix function in

258

G

L^

L

. According to formulae (2.5)» which express the

connection between factorization factors from different factorizations of one and the same matrix function, we have A = BTH , where

H

(7.6)

is a polynomial matrix function the structure of which is

described in Chapter 2 immediately after formulae (2.5)» In particular, there was also shown that

det

H

is a polynomial whose degree is

equal to the difference of the total indices of G

in the considered

factorizations. In the case under study, thanks to Theorem 7.1 and Corollary 7.3» we may assert that this degree is equal to * (< 0) is the total p-index of

-2x , where

G .

Utilizing equations (7.1) and (7.6), we may represent the matrix function

G

in the form G = BTHAB* .

(7.7)

Consequently, for almost all

t 6 T , the quadratic forms associated

with the matrices

TH(t)A(t)

G(t) and

are congruent. Hence, the

signatures (i.e. the differences of the numbers of positive and negative squaxes of the corresponding quadratic forms) of these matrices also coincide a.e. on T . Since the matrix function

THA

is continuous

on T , its signature is a piecewise constant function of

t

the dis-

continuities of which are contained in the set of zeros of the function det (THA ) = (-1)QtKdet H , i.e. in the set of zeros of the polynomial det H . However, the latter vanishes at no more than

2|x|

points of

the circle T . In this way, the following statement has been proved. THEOREM 7.3. 7X(G) t. e T ,

Let

G

be a Hermitian matrix function. For the set

to be non-empty, it is necessary that there exist points j = 1,...,N

which partition T

into

N < 2|x|

arcs such that

u the signature of the matrix G(t) is constant on a each of them a.e.» COROLLARY 7.4-. For T 0 (G) to be non-empty, it is necessary that the signature of the matrix COROLLARY 7.5. For

G(t) is constant a.e..

T(G)

to be non-empty, it is necessary that

there exists a partition of TP into a finite number of axes such that»' on each of them the signature of the matrix

G(t) is constant a.e..

25S

With the help of Corollary 7*5 one can e a s i l y construct an example of a scalar function taking the values

1

a f a c t o r i z a t i o n in any of the spaces f(t) = 1

if

[V|t-l|]

and

-1

which does not admit

L^ • In particular, one may set

i s even, and

f ( t ) = -1

otherwise.

The conditions found in Theorem 7«3 are not s u f f i c i e n t f o r the domain of f a c t o r a b i l i t y of the matrix function G to be non-empty, even in +1 case we require G € L^ and N = 0 . As an example we consider the matrix function

G(t) = d i a g [ f ( t ) , - f ( t ) ]

, where the function ±1

real-valued, changes i t s sign i n f i n i t e l y many times and all

t € T , the matrix

G(t)

f

f

is

€ L^ . For

i s Hermitian and i t s signature i s equal

to zero. At the same time, the matrix function in question d i f f e r s from the Hermitian matrix function

c :)•

d i a g [ f , f ] , the signature of which

changes i n f i n i t e l y many times, by the constant multiplier Consequently, the f a c t o r a b i l i t y domain of

G i s empty.

Now we intend to consider in more detail an Hermitian matrix function G f o r which the component

T Q (G)

in this case the inclusion indices of

G

k^ > . . . > k., > 0

set

T(G)

2 € T Q (G)

necessarily holds and the p a r t i a l

are connected by relations

p a r t i a l indices of

THEOREM 7»4.

i s not empty. According to CoicLlary 7.3,

(7.5)»

Thus, the tuple of

G i s uniquely determined by the numbers (m > 0 )

of d i f f e r e n t positive indices and t h e i r

Assume that f o r the Hermitian matrix function i s non-empty. Then

G the

G admits a representation in the

form G

where

±1 A0

=

A

OAOAS

(7.8)

.

€ H, , hkn

\

t k l £, A0(t)= \ t"1^' X Il„

260

/

(7.9)

\| ,

J = I

sum i s equal to

and

1. = n-2 0

signature

a

1+

of

and

m E 1-t d=1 3

1

are integers such that t h e i r

and t h e i r d i f f e r e n c e equals the

G .

Before proving the theorem, l e t us remark that by

a

one has to under-

stand that value of the signature which i s taken by the matrix funcG a . e . on T • Because of Corollary 7.4-> such a value e x i s t s .

tion Proof«

Starting with an arbitrary f a c t o r i z a t i o n ( 7 . 1 ) of the matrix

function

G in

L2

and arguing as in the proof of Theorem 7.3» we +1

obtain equation ( 7 . 7 ) in which from one f a c t o r i z a t i o n of structure of

G in

B € Ha L2

and

H

i s a transition matrix

to another. Consequently, the

H i s described by assertion 2 ) of Theorem 2.2. With r e -

gard to relations ( 7 . 5 ) » we deduce the following description of the structure of the matrix function L e t the matrix function as the matrix function

W = TH .

W be partitioned into blocks in the same way A0

defined by equation (7.9)> and l e t these

blocks be numbered in l i n e s and columns by the indices from

-m

to m

from bottom to top and from the l e f t to the r i g h t , r e s p e c t i v e l y . Then: 1)

if

2)

W_ _

3)

if

r < s , then

W,, r , s„ = 0 , i s a constant non-singular matrix,

r > s , then

W_ I ,s

i s a polynomial matrix function such that

the degrees of i t s elements do not exceed formity, s e t

k g = -k —a f o r

s < 0 , and

( f o r the sake of unikQ = 0 ) .

Now we want to show that there exists a polynomial matrix function

¥

with a constant non-zero determinant such that the equation W(t)A(t) = ¥(t)A0(t)v(t)*

(7.10)

is valid. P a r t i t i o n i n g the matrix function as

¥

into blocks in the same manner

W and imposing on i t the additional condition ^•n I® os = 0

f

°r

r+s < 0 ,

(7.11)

261

we get a system which is equivalent to (7.10); t

= V^ »r.-vW V s . v ^ * *

V

•

where the summation on the right-hand side is carried out for v between -s and r , J = I for v 4= 0 Since the matrix function

lying

and J Q = J .

WA is Hermitian, the system (7.12) is trans-

formed into itself after conjugation and changing r

and s with each

other. Consequently, those equations with r < s may be omitted. The remaining equations of the system (7.12) are partitioned into 2m+1 groups, assigning to the j th group those of the equations for which r+s = d (o=0,...,2m) . The zero group consists of the equations w

r r ^ = „ = ^0,0„ 0,0

-r^-r r^* »

.

(7.13)

0,0„ .

For the solvability of the last equation, it is necessary and sufficient to equate the integers 1 +

and 1_ to the number of positive and

negative squares, respectively, of the matrix

j0

. The remaining

equations are always solvable. The solutions of all the equations (7.13) may be chosen in the class of constant matrices. Now we take advantage of the fact that the equations of the j th group involve only those unknown matrices Suppose that, for

_ for which 0 — < r+s — < J• r,s ¡j < 3 0 (< 2m+1) , the equations of the j th group

are solved, where the matrix functions Hr,s already obtained are polynomial with degree of its elements not exceeding k._+k_ r s . Bach equation of the (oq+1) th group may be rewritten in the form k +k » r . s ^ V S . - s *rf-r r*B,r( >* * ' = t+k w (t) S r," J » r ^ W v V v W * • +

J

t

t

The summation on the right-hand side is carried out for v lying between -s+1

and r-1 .

In view of the assumption stipulated the right-hand side of (7.14-) is a, polynomial matrix function the degree of the entries of which are not

262

g r e a t e r than 6®t

k_+k„ . S e t t i n g ¥ _ = 0 f o r s i r , from ( 7 . 1 4 ) we 1 9 S i n the form o f a polynomial m a t r i x f u n c t i o n the degrees o f

r , s„

the e n t r i e s o f which are a l s o l e a s than or equal t o For

k^+kg .

s = r , equation ( 7 . 1 4 ) reads as f o l l o w s ! 2 Re(t t

r

*\

(t)»*

_) »

—k rW (t) Z * r,—r* ' |v| 0 )

In f a c t , i f

t*1 \

\

0

and

A(t)*

,

/

- k , (= k 2 )

are i t s p a r t i a l 2 - i n d i c e s .

x., > 0 , then the mentioned representation exists by

Theorem 7.4-. I f representation

264

1 0

x 1 = 0 , then Theorem 7.4- ensures the existence of the G = AQ ^ ^

® j A* . I t remains only to remark that

7.3.

FACTORIZATION OF DEFINITE MATRIX FUNCTIONS

An important subclass of Hermitian matrix functions are positively (negatively) definite matrix functions, i.e. such that the signature of the matrix

G(t) is equal to

n

(-n) a.e. on T

. Since negatively

definite matrix functions become positively definite after multiplication by -1 , it is sufficient to study only positively definite matrix functions. If a positively definite matrix function is factorable in Lj , then, by Corollary 7.6, all its partial 2-indices are equal to zero. Moreover, 1+ = n

and

1_ = 0 , since only in this case the equation

1+ -

= n

may be guaranteed. Thus, for positively definite matrix functions, the factor A 0

is the unit matrix. Consequently, if a positively definite

matrix function can be factored in L 2 , then it admits also a representation G = AA* with

(7.15)

A-'U H 2 .

The necessary condition for T 0 (G) to be non-empty established in Corollary 7.4- (i.e. of factorability in L,) is fulfilled for a positively definite matrix function. If we complete this condition by the requirements

G

€ L, , which, axe obviously necessary for factorability«

then the obtained collection of conditions proves already to be sufficient. In other words, the following result holds» THEOREM 7.5.

A positively definite matrix function ±1 can be factored in h 2 if and only if G 6 L 1 .

G

defined on T

Theorem 7.5 may be derived as a consequence of a more general result on the representation of a positively definite matrix function

G

in the

form of the product (7.15) of an outer matrix function and its adjoint1). Let us begin with the scalar case. D The reader who is only interested in the problem of ^-factorization of matrix functions from the class L^ may omit all the material up to the end of the present section. The question of S—factorability of positively definite matrix functions of the mentioned class will be solved in 7.4- irrespective of Theorem 7.5.

265

LBMMA 7.1»

Let

representation

g

be a non-negative function defined on T

g(t) = |f(t)|* , where

tion, is possible if and only if tion

f

f

. The

is a certain outer func-

log g ( L 1 . If the desired func-

exists, then it is defined up to a constant multiplier with

absolute value 1. Proof.

Prom Definition 1.12 of an outer function and Theorem 1.15. it g = |f|a

is clear that the relation

holds if and only if

• Since the role of the function

a.e. on T

k

g(t) = k(t)2

in (1.7) may be played

by any non-negative function with summable logarithm, it follows that the condition

log g € L,

is necessary and sufficient for the existence

of the desired representation. If this condition is fulfilled, then the corresponding function defined up to the constant 7.1.

v

f

is

from relation (1.7), which proves Lemma

s

In particular, Lemma 7.1 allows for solving^ the question of factorability

in

L2

of a real-valued function to the very end.

THEOREM 7.6. Let 2 e J(G) 1) 2)

Proof.

be a real-valued scalar function. The inclusion

holds if and only if

€ L, G

G

and

takes values of one and the same sign a.e. The necessity of condition 1) is obvious} the necessity of

condition 2 ) results from Corollary 7.4-. If condition 1) and 2) are satisfied, then ±1 such that Lemma 7.1,

g

G = eg , where

€ L,

and

g = |f|* , where

its inverse to the class factorization of Now let

s

G € L1

G

is a non-negative (a.e.) function

is a constant equal to f

1

or

-1 . Due to

is an outer function belonging with

H 2 . Setting

A = f ,

B = ef , we get a

in L 2 . Thus Theorem 7.6 is proved.

=

be a positively definite matrix function of n th order.

We introduce the Hilbert space ^ (rows) f

g

defined on T

of n-dimensional vector functions

for which the value ^

/ f(e ie )G(e ie )f(e ie )»d© 1 >

All the integrals in the present section are taken over the interval [-m,it] . 266

makes sense and is finite. The scalar product in formula

is defined by the

/ f(eie)G(el6)g(eie)»de . We denote by £

(f,g) =

sure (with respect to the norm in ^ ) of the space L

the clo-

consisting of

all vector functions of the form f(e ie ) = where Let X Q space L Q

? e i k 9 f( k ) , k=o

(7.16)

(k=0,...,Nj N=0,1,...) are constant row vectors. be the closure (again with respect to the nonm in ^ ) of the of those vector functions

Clearly, HoX, Q

and

IiBMMA 7.2. Let

f €L

for which

dim Z |J£0 < n .

G € L1

be a positively definite matrix function of

n th order such that log det G € L., . Then

function. Since

implies that G = |G+|a , where G € L 1 , then ie

i0

From this it is clear that, for all ie

G+

is an outer

G + 6 H a . For an arbitrary function

ie

instead of J f(e )G(e )f(e )de (f,f) =

dim X. | |G+(0)f(0)|1 . Setting

g 6 L Q , we thus obtain the distance from is not less than

1

f,

i6

f = 1-g with

1 2

||1-g||^ = (l-g.l-g) ^ > |G+(0)|. Consequently,

to L Q

(and, therefore, also to the subspace X,

|G+(0)| . Thus, 3tQ

and

Returning to the general case, we denote by

dim( X J A

X.|(e

0)

= 1 •

) > ... > XQ(e "i A

the collection of all eigenvalues of the matrix G(e

J

A

)

) . Since

log X a (e i e ) < 1 | log X 3 (e i e ) = Ja log and

16 n XnCe ) = aJ log det G(e i6 ) 3 0=1

log det G(e ie ) =

E log \ A e i Q ) < log Xn(ei6)+(n-1 )log \,(e ie ) 0 d=i

< log \ n (e i e Hn-1)Xi(e l e ) < log \.(ei6)+(n-1) E \di (e i 6 ) 0=1 = log X a (e i e ) + (n-1)tr G(e ie ) , the conditions

log det G € L i

and

log ^

€ L,

are equivalent for a 267

)

positively definite matrix function JA

G- € L 1 • By '

iA tr G(e ) we denote

the trace * of the matrix G(eAW) . We shall make use of Its summability. Choosing a vector function

f €L

and denoting its j th component by

f • , we have u

/ f(eie)G(eie)f(eie)*d6

(f,f) = ^

* i(ei6)*Q(ei0)f(eie)* d6 = ^ Jf / XQ(eie)|f.,(eie)|de .

> h

The inclusion 1

X Q = |X+|

log X Q € L,

, where 1

inequality

established above permits us to assert that

is an outer function. In addition, due to the ft 1 ft

0
|*+(0)f¿(0)1» .

Hence (f,f)> letting

E U + (o)f 1 (o)|» = |x+(o)|*!|f(o)||a . J

0=1

f = a - g , where

a is a fixed constant vector and

over L q , we find that the distance from |X+(0)|||a|| . Since L Q tor does not belong to

a to g

g

ranges

is not less than

is dense in £ Q , every constant non-zero vec. Consequently,

dim( X|5C0) = n, which

proves Lemma 7.2. = THEOREM 7.7. The matrix function

G

defined on V admits a repre-

sentation G = AA* with an outer matrix function 1) G

A

if and only if

is positively definite,

2) log det S C L ) ( 3) 1 og||G|| € L, . If the indicated representation exists, then the matrix function

A

is defined up to a constant right unitary multiplier. Proof. Assume that G = XX* , where

A = (ajjk)jtk=1

i s 8X1

outer ma-

trix function* Then, for an arbitrary n-dimensional vector x , (G(t)x,x) = (A(t)A(t)*x,x) a HA(t)*xj]» > 0 1

holds a.e. on T , i.e.,

> tr G(e ie ) is the sum of the diagonal elements of G(ei6), where G € L 1 a

266

the matrix function G

is positively definite and condition 1) is ful-

filled. Moreover, from Lemma 7.15 and the relation

det G = |det A|1

we deduce that condition 2) is also satisfied. Purthermore, we denote by values of the matrix

Xi(t) > «.* > ^("O

G(t) at those points

is positively definite. Then

tlie

collection of eigen-

t € T where this matrix

||G(t)|= (Afc^CoKt^Ct)*))1/2 = ^ ^ ( t ) )

= Xi(t) . We have, on the one hand, Xi(t) > (Xi(t)...Xn(t))1^n = (det G(t))1/Q . On the other hand, \.,(t) < X1(t)+...+ X a (t) = tr G(t) =

J i 8 3d ( t ) = ;},Li | a ^ ( t ) | 2 - a* ^ l a d k ( t ) l • T h u a ^ log det G(t) < log||G(t)|| < 2 log n + max log|ajk(t)| 0

a.e. on T

.

With regard to the definition of an outer function each of the functions a ^

belongs to the class D . Therefore, it is either identi-

cally equal to zero or, taken absolutely, coincides with a certain jA outer function. In any case, / logla^j^e

)|d6 < + co and, conse-

i

quently, J max log|a;.j£(e ®)|de < + » . Prom the inequalities obtained for log||G(t^|j and the summability of the function log det G , we thus conclude that log||G|j € L 1 .In this way, condition 3) is also necessary for the existence of a representation (7.15). It is obvious that, replacing in (7.15) the matrix function A by AU, where U

is a constant unitary multiplier, we do not violate equation

(7.15) and do not go beyond the class of outer matrix functions.Vice versa, assume that, besides of the representation (7.15), the equation G s BB* with an outer matrix function B 1

is valid* Then

1

A~ (t)B(t) = (B~ (t)A(t))a.e. on

T .

—"1

The latter equation means that the matrix function

U = A B

is uni-

tary and, in particular, belongs to the class L^ . Moreover, this matrix function is outer along with

A

and

B. Therefore, on the basis

of Theorem 1.18,9 from the condition U € LCO we can conclude that U € H w . To verify that the matrix function U* (= B~1A) lies in the class

^

can be done in a similar way. But then

U € ^

H L~ , so

that, by Theorem 1*28, the matrix function U is constant. Thus B s All , where U is a constant unitary matrix*

269

It remains to prove the existence of a representation (7.15) in case the conditions 1) - 3 ) a-r-e fulfilled. ^Replacing the matrix function by

f

G , where

G

f(t) = ||G(t)|| , we get a summable positively definite

matrix function with summable logarithm of the determinant. If the existence of a representation tion

B

f

G = BB*

will be proved, then setting

function with the property

|f + |

4

with an outer matrix func-

A = Bf + (where

= f

f+

is an outer

existing in view of Lemma 7.1),

we shall obtain a representation (7.15) of the original matrix function

G • Hence, without loss of generality, we may assume the matrix

function

G

to be summable.

If we stipulate the mentioned assumption and use the notations introduced before Lemma 7.2, we observe that, due to this lemma, the orthogonal complement of X

with respect to X

choose an orthonoimal basis bitrary

f eJf and

{» d )J a 1

is n-dimensional. Now we

in

*

0

siQCe)

k = 1,2,... , e l k f € t Q , then for all

/ f(e ie )G(e ie )q. ;j (e ie )»e ikti de = 0 ,

la particular, taking

f = ^

for

^

f € Jt : (7.17)

, we have

/