Table Of ContentOT3:
Operator Theory: Advances and Applications
Vol.3
Edited by
I. Gohberg
Editorial Board
K.Clancey B.Gramsch M. Livsic
L. Coburn W.Helton E. Meister
R.Douglas D.Herrero B. Mityagin
H.Dym M.Kaashoek J. Pincus
A.Dynin T. Kailath M. Rosenblum
P. Fillmore H.Kaper J.Rovnjak
C.Foias S.Kuroda D. Sarason
P.Fuhrman P. Lancaster H.Widom
S.Goldberg L. Lerer D.Xia
Honorary and Advisory
Editorial Board
P.Halmos R. Phillips
T.Kato B. Sz.-Nagy
S. Mikhlin
Editorial Office
School ofMathematical Sciences
Tel-Aviv University
Ramat-Aviv (Israel)
Springer Basel AG
K Clancey and I. Gohberg
Factorization of Matrix
Functions and Singular
Integral Operators
1981
Springer Basel AG
Author's addresses
Prof. Kevin F. Clancey
The U niversity of Georgia
Department of Mathematics
Athens, GA 30602 (USA)
Prof. Israel Gohberg
Tel-Aviv University
School of Mathematical Sciences
Ramat-Aviv (Israel)
Library of Congress Cataloging in Publication Data
Clancey, Kevin, 1944-
F actorization of Matrix Functions and Singular
Integral Operators.
(Operator Theory, Advances and Applications v. 3)
Bibliography: p.
Includes index.
1. Matrices. 2. IntegralOperators. 3. Factoriza
tion (Mathematics) im Error I. Gokhberg, I. ts. (Izrail
Tsudikovich), 1928- II. Title. III. Series.
QR 188.C58 512.9'434 81-18023
AACR2
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Oancey, Kevin:
Factorization ofmatrix functions and singular
integral operators / K. Clancey ; I. Gohberg. -
Basel ; Boston ; Stuttgart : Birkhäuser, 1981. -
(Operator theory ; Vol.3)
ISBN 978-3-0348-5494-8 ISBN 978-3-0348-5492-4 (eBook)
DOI 10.1007/978-3-0348-5492-4
NE: Gohberg, Izrail' C.:; GT
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,
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© 1981 Springer Basel AG
Originally published by Birkhäuser Verlag Basel in 1981.
Softcover reprint of the hardcover I st edition 1981
Dedicated to the memory of
N.I. Muskhelishvili
for outstanding contributions
to the theory of singular
integral equations
PREFACE
A few years aga the authors started a project of a book on
the theory of systems of one-dimensional singular integral equa
tions which was planned as a continuation of the monograph by
one of the authors and N.Ya. Krupnik ~~ concerning scalar equa
tions. This set of notes was initiated as a chapter dealing with
problems of factorization of matrix functions vis-a-vis appli
cations to systems of singular integral equations. Working
systematically onthischapter and adding along the way new points
of view, new proofs and results, we finally saw that the material
connected with factorizations is of independent interest and we
decided to publish this chapter as aseparate volume. In fact,
because of recent activity, the amount of material was quite
large and we quickly learned that we cannot cover all of the
results in complete detail. We have tried to include a represen
tative variety of all kinds of methods, techniques,results and
applications.
Apart of the current work exposes results from the Russian
literature which have never appeared in English translation. We
have also decided to reflect some of the recent results which
make interesting connections between factorization of matrix
functions and systems theory.
The field remains very active and many results and connec
tions are still not weIl understood. These notes should be viewed
as a stepping stone to further development. The authors hope
that sometime they will return to complete their original plan.
CindyWilliams typed a major portion of the present monograp~
We are extremely grateful for her patient excellent work. We
would also like to thank Cathy Craft for typing the finishing
touches of the manuscript.
Athens, Ga.-Tel Aviv, May 1981. The authors
TABLE OF CONTENTS
Introduction 1
I. The factorization of rational matrix functions 6
1. Preliminaries on factorizations of non
singular continuous matrix functions 6
2. Factorization of non-singular rational
matrix functions 13
3. Factorization and singular integral operators 20
4. The partial indices of rational matrix func-
tions 25
Notes 31
11. Decomposing algebras of matrix functions 34
1. Canonical factorization in decomposing
Banach algebras 34
2. Factorization in decomposing algebras of
scalar functions 39
3. Factorization in decomposing R-algebras of
matrix functions 44
4. The general theorem on factorization in
decomposing algebras 47
5. Complements to the theorem on factoriza
tion in decomposing algebras 56
6. Examples 58
7. Systems of homogeneous barrier problems 63
Notes 71
111. Canonical factorizations of continuous matrix
functions 73
1. Sufficient conditions for canonical fac
torization 73
2. Canonical factorization of self-adjoint
matrix functions on the unit circle 78
Table of Contents VIII
3. The fai1ure of canonica1 factorization
of dissipative matrix functions on non
circu1ar contours 81
4. Canonica1 factorization of near1y ana-
1ytic dissipative matrix functions 86
Notes 89
IV. Factorization of triangular matrix functions 91
1. Factorization of tri angular matrix
functions 91
2. The partial indices of triangular matrix
functions of the second order 98
3. Matrix functions of mixed tri angular
type 101
Notes 112
V. Factorization of continuous se1f-adjoint
matrix functions on the unit circ1e 113
1. General resu1ts on the partial indices
of se1f-adjoint matrix functions 113
2. The factorization of se1f-adjoint matrix
functions relative to the unit circ1e 115
Notes 121
VI. Misce11aneous resu1ts on factorization
relative to a contour 122
1. Multiple extensions of matrix functions 122
2. A method of reduction to canonica1 fac-
torization 128
3. Factorization of operator functions 133
Notes 140
VII. Genera1ized factorization 141
1. Definition and basic properties of
genera1ized factorization 141
Table of Contents IX
2. Generalized factorization and singu
lar integral operators on L 146
p
3. Factorization and Fredholm operators 155
Notes 164
VIII. Further results concerning generalized
factorization 165
1. Generalized factorization of continu
ous matrix functions 165
2. Generalized factorization of piecewise
continuous matrix functions 168
3. Factorization and shift bases 173
4. Extensions of results on dissipative
and self-adjoint matrix functions on
the unit circle 177
5. Generalized factorization relative
182
188
IX. Local principles in the theory of
factorization 189
1. A local principle for factorization
relative to a contour 189
2. A local principle for generalized
factorization 194
3. Further remarks on the local principle 202
Notes 203
X. Perturbations and stability 204
1. Stability of canonical factorization 204
2. The stability of partial indices 207
3. The partial indices of matrix func
tions depending analytically on a
parameter 214
4. Rational matrix functions of two variables 216
Notes 223
