Factorization of Measurable Matrix Functions PDF
Preview Factorization of Measurable Matrix Functions
D OT25: Operator Theory: Advances and Applications Vol. 25 Editor: I. Gohberg Tel Aviv University Ramat-Aviv, Israel Editorial Office School of Mathematical Sciences Tel Aviv University Ramat-Aviv, Israel Editorial Board A. Atzmon (Tel-Aviv) M.A. Kaashoek (Amsterdam) J. A. Ball (Blacksburg) T. Kailath (Stanford) L. de Branges (West Lafayette) H. G. Kaper (Argonne) K Clancey (Athens, USA) S. T. Kuroda (Tokyo) L.A. Coburn (Buffalo) P. Lancaster (Calgary) R. G. Douglas (Stony Brook) L. E. Lerer (Haifa) H. Dym (Rehovot) M. S. Livsic (Beer Sheva) A. Dynin (Columbus) E. Meister (Darmstadt) P. A. Fillmore (Halifax) B. Mityagin (Columbus) C. Foias (Bloomington) J.D. Pincus (Stony Brook) P. A. Fuhrmann (Beer Sheva) M. Rosenblum (Charlottesville) S. Goldberg (College Park) J. Rovnyak (Charlottesville) B. Gramsch (Mainz) D. E. Sarason (Berkeley) J. A. Helton (La Jolla) H. Widom (Santa Cruz) D. Herrero (Tempe) D. Xia (Nashville) Honorary and Advisory Editorial Board P. R. Halmos (Santa Clara) R. Phillips (Stanford) T. Kato (Berkeley) B. Sz.-Nagy (Szeged) S. G. Mikhlin (Leningrad) Springer Basel AG Georgii S. Litvinchuk Ilia M. Spitkovskii Factorization of Measurable Matrix Functions Edited by Georg Heinig Translated from the Russian by Bernd Luderer With a foreword by Bernd Silbermann 1987 Springer Basel AG Authors Address: Prof. Dr. Georgii S. Litvindtuk Dr. Ilia M. Spitkovskii Institute of Hydromedlanica Academy of Science USSR-170100 Odessa, Ukraine Ubrary of Congress Cataloging In PubUeation Data LltYiDdtuk, G. S. (Geoqli Semenovkh), 1931- Factorization of measurable matrix functions. (Operator theory, advances and applications ; v. 25) Bibliography: p. Includes index. 1. Matrices. 2. Factorization (Mathematics) I. Spitkovskii, Ilia M. 1953- II. Heinig, Georg. m. Title. IV. Series. QA188.LS8 1987 512.9'434 87-6632 CJP-Kantltelallfnahme der Deutsdaen Bibliothek Utrillcbuk, GecqU S.: Factorization of measurable matrix functions I Georgii S. Litvinchuk ; Ilia M. Spitkovskii. Ed. by Georg Heinig. Transl. from the Russian by Bernd Luderer. With a foreword by Bernd Silbermann. - Basel ; Boston ; Stuttgart : Birkhauser,1987. (Operator theory ; Vol. 25) (Basel ...) NE: Spitkovskii, Dia M. All rights reserved. No part of this publication may be reproduced stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. © Springer Basel AG 1987 Originally published by Akademie Verlag Berlin in 1987 Softcover reprint of the hardcover 1st edition 1987 ISBN 978-3-0348-6268-4 ISBN 978-3-0348-6266-0 (eBook) DOI 10.1007/978-3-0348-6266-0 Foreword In treating mathematical problems, it is often necessary to factorize matrix-valued (or even operator-valued) functions in a certain manner. Factorizations closely connected with the notion of Wiener-Hopf factorization play a prominent role. The development of factorization theory was, among other things, stimulated by needs arising in the theory or singular integral operators and its manifold modifications. Richness and profundity of the results obtai.ned as well as their significance for the theory of singular integral operators and -other branches of mathematics required explanation of this theory in .mono graphs. After the book ''Factorization of Matrix Functions and Si~gular Integral Operators" (Birkhiiuser Verlag, Basel 1981 ), excellently written by K. CLANCEY and I. GORBERG, the present monograph of Go S. LITVINCHUK and I. M. SPITKOVSKI~ •Factorization of Measurable Matrix Functions• ·is now the second work available on problems of Wiener-Hopf factorization. In addition to problems already discussed in the book of CLANCEY and GORBERG, it contains a series of more recent results partly due to the authors and of considerable interest. It should be emphasized that the monograph reflects not only the state of the art in Wiener-Hopf facto rization of matrix functions, but it also focusses the reader1s attention on many problems which remain to be solved. The rich biblio g~aphy containing both theoretical papers and work on practical appli cations is a further advantage of the book, which will greatly benefit all those interested in factorization 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 prob lems and singular integral equations in 19?3-7?, 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. MUSKHELISHVlLI and N. P. VEKUA, 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 reader1s judgment will show in how far this attempt has been success ful. The authors ? CONTENTS Introduction 11 Chapter 1. Background information 1? 1.1. Some facts on the geometry of Banach spaces and 1? operator theory in these spaces 1.2. Some information on boundary properties of func- 29 tions analytic and meromorphic in finitely connected domains 1.3 On the operator of singular integration in spaces 45 of summable functions Chapter 2. General properties of faotorization 55 2.1. The definition of factorization 56 2.2. Properties of factorization factors 58 2.3. The domain of factorability 63 2.4. Factorization of meromorphic matrix functions 6? and their products 2.5. Comments 80 Obapter 3. The criterion of factorability. ~-factorization 82 and its basic properties 3.1. On solvability of the Riemann boundary value 84 problem with factorable matrix coefficient 3.2. A criterion of factorability 93 3.3. On the normal solvability of the vector-valued 99 Riemann boundary value problem and the problem associate to it 3.4. Left and right w-factorization. A criterion of 105 simultaneous Fredholmness of the Riemann boundary value problem and the associate problem to it 3.5. ~-factorization of unbounded matrix functions 109 on contoum of class S.. 3.6. ~-factorization of bounded measurable matrix 116 functions on contours of class ~ 3.7. Comments 126 Chapter 4. ~-factorization of triangular matrix functions 132 and reducible to them 4.1. Existence problems of a of 135 ~factorization triangular matrix functions 4.2. Effective construction of a \!?-factorization and 140 estimates for the partial indices of triangular matrix functions \vi-th @-factorable diagonal elements 4.3. Calculation of partial indices of second-order 147 triangular matrix functions 4.4. of functionally commutable matrix 154 ~-factorization functions 4.5. Comments 163 Chapter 5. Some classes of factorable matrix functions 169 5.1. <li-factorization of matrix functions from classes 172 L.:t + C 00 5.2. Factorization in Banach algebras of matrix func- 180 tions 5.3. ~factorization of piecewis~ continuous matrix 195 functions 5.4. Comments 202 9 Chapter 6. On the stability of factorization factors 211 6.1. Stability criterion for partial indices 214 6.2. On the behaviour of partial indices under small 218 perturbations. The structure of the family of ~factorable matrix functions Several estimates for the partial indices of meas- 223 urable bounded matrix functions Sufficient conditions for coincidence of partial 229 indices of matrix functions whose Hausdorff set is separated from zero On partial indices of matrix functions of classes 236 L± + C 00 On the stability of the factorization factors G_+ 240 Comments 24? 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 ?.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 ?.6. On partial indices of continuous matrix functions 285 7.7. Comments 288 Chapter 8. Conditions of ~-factorability in the space ~n· 296 Criterion of ~-factorability in L1 of boundeu measurable matrix functions 8.1. Auxiliary results 297 8.2. Sufficient conditions of 304 ~factorability 8.3. Criterion of ~factorability in L1 307 8.4. Comments 309 Chapter 9. The generalized Riemann boundary value problem 315 9.1. Criterion of Fredholmness of the generalized 316 Riemann boundary value problem in the space Lp 9.2. Sufficient conditions of Fredholmness and 325 stability. Estimates for the defect numbers of the generalized Riemann boundary value problem in the space L 9.3. The generalized 1 Riemann boundary value problem 333 with continuous coefficients. A stability cri- terion 9.4. Comments 336 References 340 Subject index 3?1 Notation index 372 10 INTRODUCTION The notion of factorization is widely used in different branches of mathematics. The term factorization itself simply means the representa tion of some mathematical object (number, fUnction, matrix, operator, etc.) as a product of objects of the same type, but having certain additional properties. Acco»ding 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 num bers 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 repre sentation 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 way: Let some operator A be represented in the form (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 equa tions 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 Hilbert space along a chain of projections as well (GOHBEBG, KREIN [5), BARKAR1, GOHBERG [1,2)). 11
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