Operator Functions and Localization of Spectra
"Operator Functions and Localization of Spectra" is the first book that presents a systematic exposition of bounds for the spectra of various linear nonself-adjoint operators in a Hilbert space, having discrete and continuous spectra. In particular bounds
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Michael I. Gil’
Operator Functions and Localization of Spectra
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Author Michael I. Gil’ Department of Mathematics Ben Gurion University of Negev P.O. Box 653 Beer-Sheva 84105 Israel e-mail: [email protected]
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Mathematics Subject Classification (2000): 47A10, 47A55, 47A56, 47A75, 47E05, 47G10, 47G20, 30C15, 45P05, 15A09, 15A18, 15A42 ISSN 0075-8434 ISBN 3-540-2246-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10964781
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Preface 1. A lot of books and papers are concerned with the spectrum of linear operators but deal mainly with the asymptotic distributions of the eigenvalues. However, in many applications, for example, in numerical mathematics and stability analysis, bounds for eigenvalues are very important, but they are investigated considerably less than asymptotic distributions. The present book is devoted to the spectrum localization of linear operators in a Hilbert space. Our main tool is the estimates for norms of operator-valued functions. One of the first estimates for the norm of a regular matrix-valued function was established by I. M. Gel’fand and G. E. Shilov in connection with their investigations of partial differential equations, but this estimate is not sharp; it is not attained for any matrix. The problem of obtaining a precise estimate for the norm of a matrix-valued function has been repeatedly discussed in the literature. In the late 1970s, I obtained a precise estimate for a regular matrixvalued function. It is attained in the case of normal matrices. Later, this estimate was extended to various classes of nonselfadjoint operators, such as Hilbert-Schmidt operator
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