Factorization of Matrix Functions and Singular Integral Operators PDF
Preview Factorization of Matrix Functions and Singular Integral Operators
OT3: 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, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. © 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
The list of books you might like
