Table Of ContentD
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
