C L A S S I C SI N M A T H E M AT I C S Tosio Kato Perturbation Theory for Linear Operators Springer Tosio Kato Perturbation Theory for Linear Operators Reprint of the 1980 Edition Springer Tosio Kato Department of Mathematics, University of California Berkeley, CA 94720-3840 USA Originally published as Vol. 132 of the Grundlehren der mathematischen Wissenschaften Mathematics Subject Classification (1991): 46BXX, 46CXX, 47AXX, 47BXX, 47D03, 47E05, 47F05, 81Q10, 81Q15, 81UXX ISBN 3-540-58661-X Springer-Verlag Berlin Heidelberg New York CIP data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustration, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provision 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. a Springer-Verlag Berlin Heidelberg 1995 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 specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. SPIN 10485260 41/3140 - 54 3 210 - Printed on acid-free paper Tosio Kato Perturbation Theory for Linear Operators Corrected Printing of the Second Edition Springer-Verlag Berlin Heidelberg New York 1980 Dr. Tosio Kato Professor of Mathematics, University of California, Berkeley AMS Subject Classification (1970): 46Bxx, 46Cxx, 47Axx, 47Bxx, 47D05, 47Exx, 47Fxx, 81A09, 81A10, 81A45 ISBN 3-540-07558-5 2. Auflage Springer-Verlag Berlin HeidelbergNew York ISBN 0-387-07558-5 2nd edition Springer-Verlag New York HeidelbergBerlin ISBN 3-540-03526-51. Auflage Berlin Heidelberg New York ISBN 0-387-03526-51st edition New York Heidelberg Berlin Library of Congress Cataloging in Publication Data. Kato, Tosio, 1917-Perturbation theory for linear operators. (Grundlehren der mathematischen Wissenschaften; 132). Bibliography: p. Includes indexes. 1. Linear operators. 2. Perturbation (Mathematics). I. Title. II. Series: Die Grundlehren der mathematischen Wissen- schaften in Einzeldarstellungen; Bd. 132. QA329.2.K37 1976. 515'.72.76-4553. This work is subject to copyright. All rights are reserved, whether thewhole or part of the material is concerned, specifically those of translation, reprinting, re-useof illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Lawwhere copies are made for other than private use, a fee is payable to thepublisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1966, 1976 Printed in Germany. Typesetting, printing and binding: Briihlsche Universit5tsdruckerei Giellen To the memory of my patents Preface to the Second Edition In view of recent development in perturbation theory, supplementary notes and a supplementary bibliography are added at the end of the new edition. Little change has been made in the text except that the para- graphs V-§ 4.5, VI-§ 4.3, and VIII-§ 1.4 have been completely rewritten, and a number of minor errors, mostly typographical, have been corrected. The author would like to thank many readers who brought the errors to his attention. Due to these changes, some theorems, lemmas, and formulas of the first edition are missing from the new edition while new ones are added. The new ones have numbers different from those attached to the old ones which they may have replaced. Despite considerable expansion, the bibliography is not intended to be complete. Berkeley, April 1976 Tosio KATO Preface to the First Edition This book is intended to give a systematic presentation of perturba- tion theory for linear operators. It is hoped that the book will be useful to students as well as to mature scientists, both in mathematics and in the physical sciences. Perturbation theory for linear operators is a collection of diversified results in the spectral theory of linear operators, unified more or less loosely by their common concern with the behavior of spectral properties when the operators undergo a small change. Since its creation by RAY- LEIGH and SCHRODINGER, the theory has occupied an important place in applied mathematics; during the last decades, it has grown into a mathematical discipline with its own interest. The book aims at a mathe- matical treatment of the subject, with due consideration of applications. The mathematical foundations of the theory belong to functional analysis. But since the book is partly intended for physical scientists, who might lack training in functional analysis, not even the elements of that subject are presupposed. The reader is assumed to have only a basic knowledge of linear algebra and real and complex analysis. The necessary tools in functional analysis, which are restricted to the most elementary part of the subject, are developed in the text as the need for them arises (Chapters I, III and parts of Chapters V, VI). An introduction, containing a brief historical account of the theory, precedes the main exposition. There are ten chapters, each prefaced by a VIII Preface summary. Chapters are divided into sections, and sections into para- graphs. I-§ 2.3, for example, means paragraph three of section two of chapter one; it is simply written § 2.3 when referred to within the same chapter and par. 3 when referred to within the same section. Theorems, Corollaries, Lemmas, Remarks, Problems, and Examples are numbered in one list within each section: Theorem 2.1, Corollary 2.2, Lemma 2.3, etc. Lemma 1-2.3 means Lemma 2.3 of chapter one, and it is referred to simply as Lemma 2.3 within the same chapter. Formulas are numbered consecutively within each section; I-(2.3) means the third formula of section two of chapter one, and it is referred to as (2.3) within the same chapter. Some of the problems are disguised theorems, and are quoted in later parts of the book. Numbers in [ ] refer to the first part of the bibliography containing articles, and those in QJI to the second part containing books and mono- graphs. There are a subject index, an author index and a notation index at the end of the book. The book was begun when I was at the University of Tokyo and completed at the University of California. The preparation of the book has been facilitated by various financial aids which enabled me to pursue research at home and other institutions. For these aids I am grateful to the following agencies: the Ministry of Education, Japan; Com- missariat General du Plan, France; National Science Foundation, Atomic Energy Commission, Army Office of Ordnance Research, Office of Naval Research, and Air Force Office of Scientific Research, U.S.A. I am indebted to a great many friends for their suggestions during the long period of writing the book. In particular I express my hearty thanks to Professors C. CLARK, K. O. FRIEDRICHS, H. FUJITA, S. GOLD- BERG, E. HILLE, T. IKEBE, S. KAKUTANI, S. T. KURODA, G. NEUBAUER, R. S. PHILLIPS, J. and O. TODD, F. WOLF, and K. YOSIDA. I am especially obliged to Professor R. C. RIDDELL, who took the pains of going through the whole manuscript and correcting innumerable errors, mathematical as well as linguistic. I am indebted to Dr. J. HOWLAND, Dr. F. McGRATH, Dr. A. MCINTOSH, and Mr. S.-C. LIN for helping me in proofreading various parts of the book. I wish to thank Professor F. K. SCHMIDT who suggested that I write the book and whose constant encouragement brought about the completion of the book. Last but not least my gratitudes go to my wife, MIZUE, for the tedious work of typewriting the manuscript. Berkeley Tosio KATO August, 1966 Contents page Introduction . XVII Chapter One Operator theory in finite-dimensional vector spaces § 1.Vector spaces and normed vector spaces . I 1. Basic notions. 1 2. Bases .. 2 3. Linear manifolds . 3 4. Convergence and norms . 4 5. Topological notions in a normed space . 6 6. Infinite series of vectors . 7 7. Vector-valued functions . 8 § 2. Linear forms and the adjoint space . 10 1. Linear forms . 10 2. The adjoint space . 11 3. The adjoint basis . 12 4. The adjoint space of a normed space . 13 5. The convexity of balls . 14 6. The second adjoint space . 15 § 3. Linear operators. 16 1. Definitions. Matrix representations . 16 2. Linear operations on operators . 18 3. The algebra of linear operators . 19 4. Projections. Nilpotents . 20 5. Invariance. Decomposition . 22 6. The adjoint operator . 23 § 4. Analysis with operators . 25 1. Convergence and norms for operators . 25 2. The norm of T" . 27 3. Examples of norms . 28 4. Infinite series of operators . 29 5. Operator-valued functions . 31 6. Pairs of projections . 32 § 5. The eigenvalue problem . 34 1. Definitions . 34 2. The resolvent . 36 3. Singularities of the resolvent . 38 4. The canonical form of an operator . 40 5. The adjoint problem . 43 6. Functions of an operator . 44 7. Similarity transformations . 46 X Contents § 6. Operators in unitary spaces . 47 1. Unitary spaces . 47 2. The adjoint space . 48 3. Orthonormal families . 49 4. Linear operators . 51 5. Symmetric forms and symmetric operators . 52 6. Unitary, isometric and normal operators . 54 7. Projections. 55 8. Pairs of projections . 56 9. The eigenvalue problem . 58 10. The minimax principle. 60 Chapter Two Perturbation theory in a finite-dimensional space 62 § 1. Analytic perturbation of eigenvalues .