Operator functions and localization of spectra (Lecture Notes in Mathematics 1830)

Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1830 3 Berlin Hei...

Author: Mikhail Iosifovich Gilʹ

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Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1830

3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Michael I. Gil’

Operator Functions and Localization of Spectra

13

Author Michael I. Gil’ Department of Mathematics Ben Gurion University of Negev P.O. Box 653 Beer-Sheva 84105 Israel e-mail: [email protected]

Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

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

41/3142/du - 543210 - Printed on acid-free paper

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 operators, quasi-Hermitian operators (i.e., linear operators with completely continuous imaginary components), quasiunitary operators (i.e., operators represented as a sum of a unitary operator and a compact one), etc. Note that singular integral operators and integro-diﬀerential ones are examples of quasi-Hermitian operators. On the other hand, Carleman, in the 1930s, obtained an estimate for the norm of the resolvent of ﬁnite dimensional operators and of operators belonging to the Neumann-Schatten ideal. In the early 1980s sharp estimates for norms of the resolvent of nonselfadjoint operators of various types were established, that supplement and extend Carleman’s estimates. In this book, we present the mentioned estimates and, as it was pointed out, systematically apply them to spectral problems. 2. The book consists of 19 chapters. In Chapter 1, we present some wellknown results for use in the next chapters. Chapters 2-5 of the book are devoted to ﬁnite dimensional operators and functions of such operators. In Chapter 2 we derive estimates for the norms of operator-valued functions in a Euclidean space. In addition, we prove relations for eigenvalues of ﬁnite matrices, which improve Schur’s and Brown’s inequalities.

VI

Preface

Although excellent computer softwares are now available for eigenvalue computation, new results on invertibility and spectrum inclusion regions for ﬁnite matrices are still important, since computers are not very useful, in particular, for analysis of matrices dependent on parameters. But such matrices play an essential role in various applications, for example, in the stability and boundedness of coupled systems of partial diﬀerential equations. In addition, the bounds for eigenvalues of ﬁnite matrices allow us to derive the bounds for spectra of inﬁnite matrices. Because of this, the problem of ﬁnding invertibility conditions and spectrum inclusion regions for ﬁnite matrices continues to attract the attention of many specialists. Chapter 3 deals with various invertibility conditions. In particular, we improve the classical LevyDesplanques theorem and other well-known invertibility results for matrices that are close to triangular ones. Chapter 4 is concerned with perturbations of ﬁnite matrices and bounds for their eigenvalues. In particular, we derive upper and lower estimates for the spectral radius. Under some restrictions, these estimates improve the Frobenius inequalities. Moreover, we present new conditions for the stability of matrices, which supplement the Rohrbach theorem. Chapter 5 is devoted to block matrices. In this chapter, we derive the invertibility conditions, which supplement the generalized Hadamard criterion and some other well-known results for block matrices. Chapters 6-9 form the crux of the book. Chapter 6 contains the estimates for the norms of the resolvents and analytic functions of compact operators in a Hilbert space. In particular, we consider Hilbert-Schmidt operators and operators belonging to the von Neumann-Schatten ideals. Chapter 7 is concerned with the estimates for the norms of resolvents and analytic functions of non-compact operators in a Hilbert space. In particular, we consider so-called P -triangular operators. Roughly speaking, a P triangular operator is a sum of a normal operator and a compact quasinilpotent one, having a suﬃciently rich set of invariant subspaces. Operators having compact Hermitian components are examples of P -triangular operators. In Chapters 8 and 9 we derive the bounds for the spectra of quasiHermitian operators. In Chapter 10 we introduce the notion of the multiplicative operator integral. By virtue of the multiplicative operator integral, we derive spectral representations for the resolvents of various linear operators. That representation is a generalization of the classical spectral representation for resolvents of normal operators. In the corresponding cases the multiplicative integral is an operator product. Chapters 11 and 12 are devoted to perturbations of the operators of the form A = D + W , where D is a normal boundedly invertible operator and D−1 W is compact. In particular, estimates for the resolvents and bounds for the spectra are established.

Preface

VII

Chapters 13 and 14 are concerned with applications of the main results from Chapters 7-12 to integral, integro-diﬀerential and diﬀerential operators, as well as to inﬁnite matrices. In particular, we suggest new estimates for the spectral radius of integral operators and inﬁnite matrices. Under some restrictions, they improve the classical results. Chapter 15 deals with operator matrices. The spectrum of operator matrices and related problems have been investigated in many works. Mainly, Gershgorin-type bounds for spectra of operator matrices with bounded operator entries are derived. But Gershgorin-type bounds give good results in the cases when the diagonal operators are dominant. In Chapter 15, under some restrictions, we improve these bounds for operator matrices. Moreover, we consider matrices with unbounded operator entries. The results of Chapter 15 allow us to derive bounds for the spectra of matrix diﬀerential operators. Chapters 16-18 are devoted to Hille-Tamarkin integral operators and matrices, as well as integral operators with bounded kernels. Chapter 19 is devoted to applications of our abstract results to the theory of ﬁnite order entire functions. In that chapter we consider the following problem: if the Taylor coeﬃcients of two entire functions are close, how close are their zeros? In addition, we establish bounds for sums of the absolute values of the zeros in the terms of the coeﬃcients of its Taylor series. They supplement the Hadamard theorem. 3. This is the ﬁrst book that presents a systematic exposition of bounds for the spectra of various classes of linear operators in a Hilbert space. It is directed not only to specialists in functional analysis and linear algebra, but to anyone interested in various applications who has had at least a ﬁrst year graduate level course in analysis. The functional analysis is developed as needed. I was very fortunate to have had fruitful discussions with the late Professors I.S. Iohvidov and M.A. Krasnosel’skii, to whom I am very grateful for their interest in my investigations.

Michael I. Gil’

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Table of Contents 1. Preliminaries 1.1 Vector and Matrix Norms . . . . 1.2 Classes of Matrices . . . . . . . . 1.3 Eigenvalues of Matrices . . . . . 1.4 Matrix-Valued Functions . . . . . 1.5 Contour Integrals . . . . . . . . . 1.6 Algebraic Equations . . . . . . . 1.7 The Triangular Representation of 1.8 Notes . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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1 1 2 3 4 5 6 7 8 8

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11 11 13 14 17 18 20

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21 22 24 27 28 30 30 32 33 33

3. Invertibility of Finite Matrices 3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . 3.2 lp -Norms of Powers of Nilpotent Matrices . . . . . . . . . . .

35 35 37

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2. Norms of Matrix-Valued Functions 2.1 Estimates for the Euclidean Norm of the Resolvent . . . . . 2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Relations for Eigenvalues . . . . . . . . . . . . . . . . . . . 2.4 An Auxiliary Inequality . . . . . . . . . . . . . . . . . . . . 2.5 Euclidean Norms of Powers of Nilpotent Matrices . . . . . . 2.6 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . 2.7 Estimates for the Norm of Analytic Matrix-Valued Functions . . . . . . . . . . . . . 2.8 Proof of Theorem 2.7.1 . . . . . . . . . . . . . . . . . . . . . 2.9 The First Multiplicative Representation of the Resolvent . . 2.10 The Second Multiplicative Representation of the Resolvent 2.11 The First Relation between Determinants and Resolvents . 2.12 The Second Relation between Determinants and Resolvents 2.13 Proof of Theorem 2.12.1 . . . . . . . . . . . . . . . . . . . 2.14 An Additional Estimate for Resolvents . . . . . . . . . . . . 2.15 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Invertibility in the Norm .p (1 < p < ∞) . 3.4 Invertibility in the Norm .∞ . . . . . . . 3.5 Proof of Theorem 3.4.1 . . . . . . . . . . . . 3.6 Positive Invertibility of Matrices . . . . . . 3.7 Positive Matrix-Valued Functions . . . . . . 3.8 Notes . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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65 65 67

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4. Localization of Eigenvalues of Finite Matrices 4.1 Deﬁnitions and Preliminaries . . . . . . . . . . . . . . 4.2 Perturbations of Multiplicities and Matching Distance 4.3 Perturbations of Eigenvectors and Eigenprojectors . . 4.4 Perturbations of Matrices in the Euclidean Norm . . . 4.5 Upper Bounds for Eigenvalues in Terms of the Euclidean Norm . . . . . . . . . . . . . . . . . 4.6 Lower Bounds for the Spectral Radius . . . . . . . . . 4.7 Additional Bounds for Eigenvalues . . . . . . . . . . . 4.8 Proof of Theorem 4.7.1 . . . . . . . . . . . . . . . . . . 4.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Block Matrices and π-Triangular Matrices 5.1 Invertibility of Block Matrices . . . . . . . . . . . . 5.2 π-Triangular Matrices . . . . . . . . . . . . . . . . 5.3 Multiplicative Representation of Resolvents of π-Triangular Operators . . . . . . . . . . . . . . 5.4 Invertibility with Respect to a Chain of Projectors 5.5 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . 5.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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69 70 72 74 74

6. Norm Estimates for Functions of Compact Operators in a Hilbert Space 6.1 Bounded Operators in a Hilbert Space . . . . . . . . . . 6.2 Compact Operators in a Hilbert Space . . . . . . . . . 6.3 Triangular Representations of Compact Operators . . . 6.4 Resolvents of Hilbert-Schmidt Operators . . . . . . . . . 6.5 Equalities for Eigenvalues of a Hilbert-Schmidt Operator 6.6 Operators Having Hilbert-Schmidt Powers . . . . . . . . 6.7 Resolvents of Neumann-Schatten Operators . . . . . . . 6.8 Proofs of Theorems 6.7.1 and 6.7.3 . . . . . . . . . . . . 6.9 Regular Functions of Hilbert-Schmidt Operators . . . . 6.10 A Relation between Determinants and Resolvents . . . . 6.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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75 75 77 79 83 84 86 88 88 91 93 95 95

Table of Contents

7. Functions of Non-compact Operators 7.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 P -Triangular Operators . . . . . . . . . . . . . . . . . . . 7.3 Some Properties of Volterra Operators . . . . . . . . . . . 7.4 Powers of Volterra Operators . . . . . . . . . . . . . . . . 7.5 Resolvents of P -Triangular Operators . . . . . . . . . . . 7.6 Triangular Representations of Quasi-Hermitian Operators 7.7 Resolvents of Operators with Hilbert-Schmidt Hermitian Components . . . . . . . 7.8 Operators with the Property Ap − (A∗ )p ∈ C2 . . . . . . . 7.9 Resolvents of Operators with Neumann - Schatten Hermitian Components . . . . . 7.10 Regular Functions of Bounded Quasi-Hermitian Operators 7.11 Proof of Theorem 7.10.1 . . . . . . . . . . . . . . . . . . . 7.12 Regular Functions of Unbounded Operators . . . . . . . . 7.13 Triangular Representations of Regular Functions . . . . . 7.14 Triangular Representations of Quasiunitary Operators . . 7.15 Resolvents and Analytic Functions of Quasiunitary Operators . . . . . . . . . . . . . . . . . . 7.16 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Bounded Perturbations of Nonselfadjoint Operators 8.1 Invertibility of Boundedly Perturbed P -Triangular Operators . . . . . . . . . . . . . . . . . 8.2 Resolvents of Boundedly Perturbed P -Triangular Operators . . . . . . . . . . . . . . . . . 8.3 Roots of Scalar Equations . . . . . . . . . . . . . . . . 8.4 Spectral Variations . . . . . . . . . . . . . . . . . . . . 8.5 Perturbations of Compact Operators . . . . . . . . . . 8.6 Perturbations of Operators with Compact Hermitian Components . . . . . . . . . 8.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XI

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9. Spectrum Localization of Nonself-adjoint Operators 9.1 Invertibility Conditions . . . . . . . . . . . . . . . . . . . . 9.2 Proofs of Theorems 9.1.1 and 9.1.3 . . . . . . . . . . . . . . 9.3 Resolvents of Quasinormal Operators . . . . . . . . . . . . 9.4 Upper Bounds for Spectra . . . . . . . . . . . . . . . . . . . 9.5 Inner Bounds for Spectra . . . . . . . . . . . . . . . . . . . 9.6 Bounds for Spectra of Hilbert-Schmidt Operators . . . . . . 9.7 Von Neumann-Schatten Operators . . . . . . . . . . . . . . 9.8 Operators with Hilbert-Schmidt Hermitian Components . . 9.9 Operators with Neumann-Schatten Hermitian Components

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135 135 137 139 142 143 145 146 147 148

XII

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9.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 10. Multiplicative Representations of Resolvents 10.1 Operators with Finite Chains of Invariant Projectors 10.2 Complete Compact Operators . . . . . . . . . . . . . 10.3 The Second Representation for Resolvents of Complete Compact Operators . . . . . . . . . . . 10.4 Operators with Compact Inverse Ones . . . . . . . . 10.5 Multiplicative Integrals . . . . . . . . . . . . . . . . 10.6 Resolvents of Volterra Operators . . . . . . . . . . . 10.7 Resolvents of P -Triangular Operators . . . . . . . . 10.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Relatively P -Triangular Operators 11.1 Deﬁnitions and Preliminaries . . . . . . . . . . . . 11.2 Resolvents of Relatively P -Triangular Operators . 11.3 Invertibility of Perturbed RPTO . . . . . . . . . . 11.4 Resolvents of Perturbed RPTO . . . . . . . . . . . 11.5 Relative Spectral Variations . . . . . . . . . . . . . 11.6 Operators with von Neumann-Schatten Relatively Nilpotent Parts . . . . . . . . . . . . . . . . . . . . 11.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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151 . . . . . 151 . . . . . 154 . . . . . . .

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156 157 158 159 159 161 161

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163 163 165 166 167 167

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12. Relatively Compact Perturbations of Normal Operators 12.1 Invertibility Conditions . . . . . . . . . . . . . . . . . . . . 12.2 Estimates for Resolvents . . . . . . . . . . . . . . . . . . . . 12.3 Bounds for the Spectrum . . . . . . . . . . . . . . . . . . . 12.4 Operators with Relatively von Neumann - Schatten Oﬀ-diagonal Parts . . . . . . . . . . . . . . . . . . . . . . . 12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Inﬁnite Matrices in Hilbert Spaces and Diﬀerential Operators 13.1 Matrices with Compact oﬀ Diagonals . . . . . . . 13.2 Matrices with Relatively Compact Oﬀ-diagonals 13.3 A Nonselfadjoint Diﬀerential Operator . . . . . . 13.4 Integro-diﬀerential Operators . . . . . . . . . . . 13.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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181 181 184 185 186 187 188

Table of Contents

14. Integral Operators in Space L2 14.1 Scalar Integral Operators . . . . . . . . . . . . . . . . . 14.2 Matrix Integral Operators with Relatively Small Kernels 14.3 Perturbations of Matrix Convolutions . . . . . . . . . . 14.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIII

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189 189 191 193 196 197

15. Operator Matrices 15.1 Invertibility Conditions . . . . . . . . . . . . . . . . . . . . . 15.2 Bounds for the Spectrum . . . . . . . . . . . . . . . . . . . . 15.3 Operator Matrices with Normal Entries . . . . . . . . . . . . 15.4 Operator Matrices with Bounded oﬀ Diagonal Entries . . . . 15.5 Operator Matrices with Hilbert-Schmidt Diagonal Operators 15.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 202 204 205 207 209 212 212

16. Hille - Tamarkin Integral Operators 16.1 Invertibility Conditions . . . . . . . . . . . . . 16.2 Preliminaries . . . . . . . . . . . . . . . . . . . 16.3 Powers of Volterra Operators . . . . . . . . . . 16.4 Spectral Radius of a Hille - Tamarkin Operator 16.5 Nonnegative Invertibility . . . . . . . . . . . . . 16.6 Applications . . . . . . . . . . . . . . . . . . . . 16.7 Notes . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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17. Integral Operators in Space L∞ 17.1 Invertibility Conditions . . . . . 17.2 Proof of Theorem 17.1.1 . . . . . 17.3 The Spectral Radius . . . . . . . 17.4 Nonnegative Invertibility . . . . . 17.5 Applications . . . . . . . . . . . . 17.6 Notes . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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227 227 228 230 231 232 234 234

18. Hille - Tamarkin Matrices 18.1 Invertibility Conditions . . . 18.2 Proof of Theorem 18.1.1 . . . 18.3 Localization of the Spectrum 18.4 Notes . . . . . . . . . . . . . References . . . . . . . . . . . . . .

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19. Zeros of Entire Functions 19.1 Perturbations of Zeros . . . . . 19.2 Proof of Theorem 19.1.2 . . . . 19.3 Bounds for Sums of Zeros . . . 19.4 Applications of Theorem 19.3.1 19.5 Notes . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

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243 243 246 248 250 252 252

List of Main Symbols

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Index

255

1. Preliminaries

In this chapter we present some well-known results for use in the next chapters.

1.1

Vector and Matrix Norms

Let Cn be an n-dimensional complex Euclidean space. A function ν : Cn → [0, ∞) is said to be a norm on Cn (or a vector norm), if ν satisﬁes the following conditions: ν(x) = 0 iﬀ x = 0, ν(αx) = |α|ν(x), ν(x + y) ≤ ν(x) + ν(y)

(1.1)

for all x, y ∈ Cn , α ∈ C. Usually, a norm is denoted by the symbol .. That is, ν(x) = x. The following important properties follow immediately from the deﬁnition: x − y ≥ x − y and x = − x. There are an inﬁnite number of norms on Cn . However, the following norms are most commonly used in practice: xp = [

n

|xk |p ]1/p (1 ≤ p < ∞) and x∞ = max |xk |

k=1

k=1,...,n

for an x = (xk ) ∈ Cn . The norm x2 is called the Euclidean norm. Throughout this chapter x means an arbitrary norm of a vector x. We will use the following matrix norms: the operator norm and the Frobenius M.I. Gil’: LNM 1830, pp. 1–9, 2003. c Springer-Verlag Berlin Heidelberg 2003

2

1. Preliminaries

(Hilbert-Schmidt) norm. The operator norm of a matrix (a linear operator in Cn ) A is Ax . A = sup n x x∈C The relations A > 0 (A = 0), λA = |λ|A (λ ∈ C), AB ≤ AB, and A + B ≤ A + B are valid for all matrices A and B. The Frobenius norm of A is n N (A) = |ajk |2 . j,k=1

Here ajk are the entries of matrix A in some orthogonal normal basis. The Frobenius norm does not depend on the choice of an orthogonal normal basis. The relations N (A) > 0 (A = 0); N (λA) = |λ|N (A) (λ ∈ C), N (AB) ≤ N (A)N (B) and N (A + B) ≤ N (A) + N (B) are true for all matrices A and B.

1.2

Classes of Matrices

For an n × n-matrix A, A∗ denotes the conjugate matrix. That is, if ajk are entries of A, then akj (j, k = 1, ..., n) are entries of A∗ . In other words (Ax, y) = (x, A∗ y) (x, y ∈ C). The symbol (., .) = (., .)C n means the scalar product in Cn . We use I to denote the unit matrix in Cn . Deﬁnition 1.2.1 A matrix A = (ajk )nj,k=1 is 1. symmetric (Hermitian) if A∗ = A; 2. positive deﬁnite (negative deﬁnite ) if it is Hermitian and (Ah, h) ≥ (≤) 0 (h ∈ Cn ); 3. unitary if A∗ A = AA∗ = I; 4. normal if AA∗ = A∗ A; 5. nilpotent if An = 0. Let A be an arbitrary matrix. Then the matrices AI = (A − A∗ )/2i and AR = (A + A∗ )/2 are the imaginary Hermitian component and the real Hermitian one of A, respectively. A matrix A is dissipative if its real Hermitian component is negative deﬁnite. By A−1 the matrix inverse to A is denoted: AA−1 = A−1 A = I.

1.3. Eigenvalues of Matrices

1.3

3

Eigenvalues of Matrices

Let A be an arbitrary matrix. Then if for some λ ∈ C, the equation Ah = λh has a nontrivial solution, λ is an eigenvalue of A and h is its eigenvector. An eigenvalue λ has the (algebraic) multiplicity r if dim(∪nk=1 ker(A − λI)k ) = r. Here ker B denotes the kernel of a mapping B. Let λk (A) (k = 1, . . . , n) be the eigenvalues of A, including with their multiplicities. Then the set σ(A) = {λk (A)}nk=1 is the spectrum of A. All the eigenvalues of a Hermitian matrix A are real. If, in addition, A is positive (negative) deﬁnite, then all its eigenvalues are non-negative (nonpositive). Furthermore, rs (A) = max |λk (A)| k=1,...,n

is the spectral radius of A. Denote α(A) = max Reλk (A), β(A) = k=1,...,n

min Reλk (A).

k=1,...,n

A matrix A is said to be a Hurwitz matrix if all its eigenvalues lie in the open left half-plane, i.e., α(A) < 0. A complex number λ is a regular point of A if it does not belong to the spectrum of A, i.e., if λ = λk (A) for any k = 1, . . . , n. The trace of A is sometimes denoted by T r (A): T race (A) = T r (A) =

n

λk (A).

k=1

So the Frobenius norm can be deﬁned as N 2 (A) = T race (A∗ A) = T race (AA∗ ). Recall that T r(AB) = T r(BA) and T r(A + B) = T r(A) + T r(B) for all matrices A and B. In addition, det(A) means the determinant of A: det(A) =

n

λk (A).

k=1

The polynomial p(λ) = det(λI − A) =

n

(λ − λk (A))

k=1

is said to be the characteristic polynomial of A. All the eigenvalues of A are the roots of its characteristic polynomial. The algebraic multiplicity of an eigenvalue of A coincides with the multiplicity of the corresponding root of the characteristic polynomial. A polynomial is said to be a Hurwitz one if all its roots lie in the open left half-plane. Thus, the characteristic polynomial of a Hurwitz matrix is a Hurwitz polynomial.

4

1. Preliminaries

1.4

Matrix-Valued Functions

Let A be a matrix and let f (λ) be a scalar-valued function which is analytical on a neighborhood M of σ(A). We deﬁne the function f (A) of A by the generalized integral formula of Cauchy 1 f (A) = − f (λ)Rλ (A)dλ, (4.1) 2πi Γ where Γ ⊂ M is a closed smooth contour surrounding σ(A), and Rλ (A) = (A − λI)−1 is the resolvent of A. If an analytic function f (λ) is represented in some domain by the Taylor series ∞

f (λ) =

ck λk ,

k=0

and the series

∞

ck Ak

k=0 n

converges in the norm of space C , then f (A) =

∞

ck Ak .

k=0

In particular, for any matrix A, eA =

∞ Ak k=0

k!

.

Example 1.4.1 Let A be a diagonal matrix: 

a1  0 A=  . 0 Then

 0 ... 0 a2 . . . 0  . ... . .  . . . 0 an



 f (a1 ) 0 ... 0  0 0  f (a2 ) . . . . f (A) =    . ... . . 0 ... 0 f (an )

Example 1.4.2 If a matrix J is an n × n-Jordan block:

1.5. Contour Integrals

     J =     then

      f (J) =     

1.5

λ0 0 . . . 0 0

f (λ0 ) 0 . . . 0 0

1 λ0 . . . 0 0

0 ... 0 1 ... 0 . ... . . ... . . ... . . . . λ0 1 . . . 0 λ0

f (λ0 ) 1!

f (λ0 ) . . . ... ...

... ... ... ... ... f (λ0 ) 0

5

     ,    

f (n−1) (λ0 ) (n−1)!

. . .

f (λ0 ) 1! f (λ0 )

      .    

Contour Integrals

Lemma 1.5.1 Let M0 be the closed convex hull of points x0 , x1 , ..., xn ∈ C and let a scalar-valued function f be regular on a neighborhood D1 of M0 . In addition, let Γ ⊂ D1 be a Jordan closed contour surrounding the points x0 , x1 , ..., xn . Then 1 f (λ)dλ 1 |≤ sup |f (n) (λ)|. | 2πi Γ (λ − x0 )...(λ − xn ) n! λ∈M0 Proof: First, let all the points be distinct: xj = xk for j = k (j, k = 0, ..., n), and let Df (x0 , x1 , ..., xn ) be a divided diﬀerence of function f at points x0 , x1 , ..., xn . The divided diﬀerence admits the representation f (λ)dλ 1 (5.1) Df (x0 , x1 , ..., xn ) = 2πi Γ (λ − x0 )...(λ − xn ) (see (Gel’fond, 1967, formula (54)). But, on the other hand, the following estimate is well-known: | Df (x0 , x1 , ..., xn ) | ≤

1 sup |f (n) (λ)| n! λ∈M0

(Gel’fond, 1967, formula (49)). Combining that inequality with relation (5.1), we arrive at the required result. If xj = xk for some j = k, then the claimed inequality can be obtained by small perturbations and the previous reasonings. 2

6

1. Preliminaries

Lemma 1.5.2 Let x0 ≤ x1 ≤ ... ≤ xn be real points and let a function f be regular on a neighborhood D1 of the segment [x0 , xn ]. In addition, let Γ ⊂ D1 be a Jordan closed contour surrounding [x0 , xn ]. Then there is a point η ∈ [x0 , xn ], such that the equality 1 f (λ)dλ 1 (n) = f (η) 2πi Γ (λ − x0 )...(λ − xn ) n! is true. Proof: First suppose that all the points are distinct: x0 < x1 < ... < xn . Then the divided diﬀerence Df (x0 , x1 , ..., xn ) of f in the points x0 , x1 , ..., xn admits the representation Df (x0 , x1 , ..., xn ) =

1 (n) f (η) n!

with some point η ∈ [x0 , xn ] (Gel’fond, 1967, formula (43)), (Ostrowski, 1973, page 5 ). Combining that equality with representation (5.1), we arrive at the required result. If xj = xk for some j = k, then the claimed inequality can be obtained by small perturbations and the previous reasonings. 2

1.6

Algebraic Equations

Let us consider the algebraic equation z n = P (z) (n > 1), where P (z) =

n−1

cj z n−j−1

(6.1)

j=0

with non-negative coeﬃcients cj (j = 0, ..., n − 1). Lemma 1.6.1 The extreme right-hand root z0 of equation (6.1) is nonnegative and the following estimates are valid: z0 ≤ [P (1)]1/n if P (1) ≤ 1,

(6.2)

1 ≤ z0 ≤ P (1) if P (1) ≥ 1.

(6.3)

and Proof: Since all the coeﬃcients of P (z) are non-negative, it does not decrease as z > 0 increases. From this it follows that if P (1) ≤ 1, then z0 ≤ 1. So z0n ≤ P (1), as claimed. Now let P (1) ≥ 1, then due to (6.1) z0 ≥ 1, because P (z) does not decrease. It is clear that P (z0 ) ≤ z0n−1 P (1) in this case. Substituting this inequality in (6.1), we get (6.3). 2

1.7. The Triangular Representation

7

Setting z = ax with a positive constant a in (6.1), we obtain xn =

n−1

cj a−j−1 xn−j−1 .

(6.4)

j=0

Let a≡2 Then

n−1

cj a−j−1 ≤

j=0

max

j=0,...,n−1

n−1

√

j+1

cj .

2−j−1 = 1 − 2−n+1 < 1.

j=0

Let x0 be the extreme right-hand root of equation (6.4), then by (6.2) we have x0 ≤ 1. Since z0 = ax0 , we have derived Corollary 1.6.2 The extreme right-hand root z0 of equation (6.1) is nonnegative. Moreover, √ j+1 c . z0 ≤ 2 max j j=0,...,n−1

1.7

The Triangular Representation of Matrices

Let B(Cn ) be the set of all linear operators (matrices) in Cn . A subspace M ⊂ Cn is an invariant subspace of an A ∈ B(Cn ), if the relation h ∈ M implies Ah ∈ M . If P is a projector onto an invariant subspace of A, then P AP = AP.

(7.1)

By Schur’s theorem (Marcus and Minc, 1964, Section I.4.10.2 ), for a linear operator A ∈ B(Cn ), there is an orthogonal normal basis {ek }, such that A is a triangular matrix. That is, Aek =

k

ajk ej with ajk = (Aek , ej ) (j = 1, ..., n),

(7.2)

j=1

where (., .) is the scalar product. This basis is called Schur’s basis of the operator A. In addition, ajj = λj (A), where λj (A) are the eigenvalues of A. According to (7.2), A=D+V with a normal (diagonal) operator D deﬁned by Dej = λj (A)ej (j = 1, ..., n)

(7.3)

8

1. Preliminaries

and a nilpotent (upper-triangular) operator V deﬁned by V ek =

k−1

ajk ej (k = 2, ..., n).

j=1

We will call equality (7.3) the triangular representation of matrix A. In addition, D and V will be called the diagonal part and the nilpotent part of A, respectively. Put j Pj = (., ek )ek (j = 1, ..., n), P0 = 0. k=1

Then

0 ⊂ P1 Cn ⊂ ... ⊂ Pn Cn = Cn .

Moreover, APk = Pk APk ; V Pk−1 = Pk V Pk ; DPk = DPk (k = 1, ..., n).

(7.4)

So A, V and D have the same chain of invariant subspaces. Lemma 1.7.1 Let Q, V ∈ B(Cn ) and let V be a nilpotent operator. Suppose that all the invariant subspaces of V and of Q are the same. Then V Q and QV are nilpotent operators. Proof: Since all the invariant subspaces of V and Q are the same, these operators have the same basis of the triangular representation. Taking into account that the diagonal entries of V are equal to zero, we easily determine that the diagonal entries of QV and V Q are equal to zero. This proves the required result. 2

1.8

Notes

This book presupposes a knowledge of basic matrix theory, for which there are good introductory texts. The books (Bellman, 1970), (Gantmaher, 1967), ( Marcus and Minc, 1964) are classical. For more details about the notions presented in Sections 1.1-1.4 also see (Collatz, 1966) and (Stewart and Sun, 1990). Estimates for roots of algebraic equations similar to Corollary 1.6.2 can be found in (Ostrowski, 1973, page 277).

References [1] Bellman, R.E. (1970). Introduction to Matrix Analysis. McGraw-Hill, New York.

1.8. Notes

9

[2] Collatz, L. (1966). Functional Analysis and Numerical Mathematics. Academic Press, New York-London. [3] Gantmaher, F. R. (1967). Theory of Matrices. Nauka, Moscow (In Russian). [4] Gelfond, A. O. (1967). Calculations of Finite Diﬀerences. Nauka, Moscow (In Russian ). [5] Marcus, M. and Minc, H. (1964). A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston . [6] Ostrowski, A. M. (1973). Solution of Equations in Euclidean and Banach spaces. Academic Press, New York - London. [7] Stewart, G. W. and Sun Ji-guang (1990). Matrix Perturbation Theory. Academic Press, New York.

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2. Norms of Matrix-Valued Functions

In the present chapter we derive estimates for the norms of operator-valued functions in a Euclidean space. In addition, we prove relations for eigenvalues of ﬁnite matrices, which improve Schur’s and Brown’s inequalities.

2.1

Estimates for the Euclidean Norm of the Resolvent

Throughout the present chapter . means the Euclidean norm. That is, . = .2 (see Section 1.1). Let A = (ajk ) be an n × n-matrix (n > 1). The following quantity plays a key role in the sequel: g(A) = (N 2 (A) −

n

|λk (A)|2 )1/2 .

(1.1)

k=1

Recall that I is the unit matrix, N (A) is the Frobenius (Hilbert-Schmidt) norm of A, and λk (A) (k = 1, ..., n) are the eigenvalues taken with their multiplicities. Since n |λk (A)|2 ≥ |T race A2 |, k=1

we get

g 2 (A) ≤ N 2 (A) − |T race A2 |.

(1.2)

In Section 2.2 we will prove the relations g 2 (A) ≤ M.I. Gil’: LNM 1830, pp. 11–34, 2003. c Springer-Verlag Berlin Heidelberg 2003

1 2 ∗ N (A − A) 2

(1.3)

12

2. Norms of Matrix Functions

and

g(eiτ A + zI) = g(A)

(1.4)

for all τ ∈ R and z ∈ C. To formulate the result, for a natural n > 1 introduce the numbers

k Cn−1 γn,k = (k = 1, ..., n − 1) and γn,0 = 1. (n − 1)k Here k Cn−1 =

(n − 1)! (n − k − 1)!k!

are binomial coeﬃcients. Evidently, for all n > 2, 2 = γn,k

1 (n − 1)(n − 2) . . . (n − k) (k = 1, 2, ..., n − 1). ≤ (n − 1)k k! k!

(1.5)

Theorem 2.1.1 Let A be a linear operator in Cn . Then its resolvent Rλ (A) = (A − λI)−1 satisﬁes the inequality Rλ (A) ≤

n−1 k=0

g k (A)γn,k for any regular point λ of A, ρk+1 (A, λ)

where ρ(A, λ) = mink=1,...,n |λ − λk (A)|. The proof of this theorem is divided into a series of lemmas which are presented in Sections 2.3-2.6. Theorem 2.1.1 is exact: if A is a normal matrix, then g(A) = 0 and Rλ (A) =

1 for all regular points λ of A. ρ(A, λ)

Let A be an invertible n × n-matrix. Then by Theorem 2.1.1, A−1 ≤

n−1

g k (A)

k=0

γn,k , k+1 ρ0 (A)

where ρ0 (A) = ρ(A, 0) is the smallest modulus of the eigenvalues of A: ρ0 (A) =

inf

k=1,...,n

|λk (A)|.

Moreover, Theorem 2.1.1 and inequalities (1.5) imply Corollary 2.1.2 Let A be a linear operator in Cn . Then Rλ (A) ≤

n−1 k=0

√

g k (A) for any regular point λ of A. k!ρk+1 (A, λ)

An independent proof of this corollary is presented in Section 2.6.

2.2. Examples

2.2

13

Examples

In this section we present some examples of calculations of g(A). Example 2.2.1 Consider the matrix A=

a11 a21

a12 a22

where ajk (j, k = 1, 2) are real numbers. First, consider the case of nonreal eigenvalues: λ2 (A) = λ1 (A). It can be written det(A) = λ1 (A)λ1 (A) = |λ1 (A)|2 and |λ1 (A)|2 + |λ2 (A)|2 = 2|λ1 (A)|2 = 2det(A) = 2[a11 a22 − a21 a12 ]. Thus,

g 2 (A) = N 2 (A) − |λ1 (A)|2 − |λ2 (A)|2 = a211 + a212 + a221 + a222 − 2[a11 a22 − a21 a12 ].

Hence, g(A) =

(a11 − a22 )2 + (a21 + a12 )2 .

(2.1)

Let n = 2 and a matrix A have real entries again, but now the eigenvalues of A are real. Then |λ1 (A)|2 + |λ2 (A)|2 = T race A2 . Obviously, 2

A =

a211 + a12 a21 a21 a11 + a21 a22

a11 a12 + a12 a22 a222 + a21 a12

.

We thus get the relation |λ1 (A)|2 + |λ2 (A)|2 = a211 + 2a12 a21 + a222 . Consequently, g 2 (A) = N 2 (A) − |λ1 (A)|2 − |λ2 (A)|2 = a211 + a212 + a221 + a222 − (a211 + 2a12 a21 + a222 ). Hence, g(A) = |a12 − a21 |. Example 2.2.2 Let A be an upper-triangular matrix:

(2.2)

14

2. Norms of Matrix Functions



a11  0 A=  . 0

a12 a21 ... ...

 . . . a1n . . . a2n  . . .  0 ann

n k−1 g(A) = |ajk |2 ,

Then

(2.3)

k=1 j=1

since the eigenvalues of a triangular matrix are its diagonal elements. Example 2.2.3 Consider the matrix 

−a1  1 A=  . 0

. . . −an−1 ... 0 ... . ... 1

 −an 0   .  0

with complex numbers ak . Such matrices play a key role in the theory of scalar ordinary diﬀerential equations. Take into account that   2 a1 − a2 . . . a1 an−1 − a1 a1 an  −a1 ... −an−1 −an    1 . . . 0 0  A2 =   .  . ... . .  0 ... 0 0 Thus, we obtain, T race A2 = a21 − 2a2 . Therefore g 2 (A) ≤ N 2 (A) − |T race A2 | = n − 1 − |a21 − 2a2 | +

n

|ak |2 .

(2.4)

k=1

Now let ak be real. Then (2.4) gives us the inequality g 2 (A) ≤ n − 1 + 2a2 +

n

a2k .

(2.5)

k=2

2.3

Relations for Eigenvalues

Theorem 2.3.1 For any linear operator A in Cn , 2

2

g (A) := N (A) −

n k=1

2

2

|λk (A)| = 2N (AI ) − 2

n

|Im λk (A)|2 ,

(3.1)

k=1

where λk (A) are the eigenvalues of A with their multiplicities and AI = (A − A∗ )/2i.

2.3. Relations for Eigenvalues

15

To prove this theorem we need the following two lemmas. Lemma 2.3.2 For any linear operator A in Cn , N 2 (V ) = g 2 (A) ≡ N 2 (A) −

n

|λk (A)|2 ,

k=1

where V is the nilpotent part of A (see Section 1.7). Proof: Let D be the diagonal part of A. Then, due to Lemma 1.7.1 both matrices V ∗ D and D∗ V are nilpotent. Therefore, T race (D∗ V ) = 0 and T race (V ∗ D) = 0.

(3.2)

It is easy to see that T race (D∗ D) =

n

|λk (A)|2 .

(3.3)

k=1

Since A = D + V,

(3.4)

due to (3.2) and (3.3) N 2 (A) = T race (D + V )∗ (V + D) = T race (V ∗ V + D∗ D) = N 2 (V ) +

n

|λk (A)|2 ,

k=1

and the required equality is proved. 2 Lemma 2.3.3 For any linear operator A in Cn , N 2 (V ) = 2N 2 (AI ) − 2

n

|Im λk (A)|2 ,

k=1

where V is the nilpotent part of A. Proof:

Clearly, −4(AI )2 = (A − A∗ )2 = AA − AA∗ − A∗ A + A∗ A∗ .

But due to (3.2) and (3.4) T race (A − A∗ )2 = T race (V + D − V ∗ − D∗ )2 = T race [(V − V ∗ )2 + (V − V ∗ )(D − D∗ )+ (D − D∗ )(V − V ∗ ) + (D − D∗ )2 ] =

(3.5)

16

2. Norms of Matrix Functions

T race (V − V ∗ )2 + T race (D − D∗ )2 . Hence, N 2 (AI ) = N 2 (VI ) + N 2 (DI ),

(3.6)

where VI = (V − V ∗ )/2i and DI = (D − D∗ )/2i. It is not hard to see that N 2 (VI ) =

n m−1 1 1 |akm |2 = N 2 (V ), 2 m=1 2 k=1

where ajk are the entries of V in the Schur basis. Thus, N 2 (V ) = 2N 2 (AI ) − 2N 2 (DI ). But N 2 (DI ) =

n

|Im λk (A)|2 .

k=1

Thus, we arrive at the required equality. 2 The assertion of Theorem 2.3.1 follows from Lemmas 2.3.2 and 2.3.3. The inequality (1.3) follows from Theorem 2.3.1. Furthermore, take into account that the nilpotent parts of the matrices Aeiτ and Aeiτ + zI with a real number τ and a complex one z, coincide. Hence, due to Lemma 2.3.2 we obtain the following Corollary 2.3.4 For any linear operator A in Cn , a real number τ , and a complex one z, relation (1.4) holds. Corollary 2.3.5 For arbitrary commuting linear operators A, B in Cn , g(A + B) ≤ g(A) + g(B). In fact, A and B have the same Schur basis. This clearly forces VA+B = VA + VB , where VA+B , VA and VB are the nilpotent parts of A + B, A and B, respectively. Due to Lemma 2.3.2 the relations g(A) = N (VA ), g(B) = N (VB ), g(A + B) = N (VA+B ) are true. Now the property of the norm implies the result.

2.4. An Auxiliary Inequality

17

Corollary 2.3.6 For any n × n matrix A and real numbers t, τ the equality N 2 (Aeit − A∗ e−it ) −

n

|eit λk (A) − e−it λk (A)|2 =

k=1

N 2 (Aeiτ − A∗ e−iτ ) −

n

|eiτ λk (A) − e−iτ λk (A)|2

k=1

is true. The proof consists in replacing A by Aeit and Aeiτ and using Theorem 2.3.1. In particular, take t = 0 and τ = π/2. Due to Corollary 2.3.6, N 2 (AI ) −

n

|Im λk (A)|2 = N 2 (AR ) −

k=1

n

|Re λk (A)|2

k=1

with AR = (A + A∗ )/2 .

2.4

An Auxiliary Inequality

Lemma 2.4.1 For arbitrary positive numbers a1 , ..., an and m = 1, ..., n, we have n ak1 . . . akm ≤ n−m Cnm [ ak ]m . (4.1) k=1

1≤k1
Proof:

Consider the following function of n positive variables y1 , ..., yn : yk1 yk2 . . . ykm . Rm (y1 , ..., yn ) ≡ 1≤k1
Let us prove that under the condition n

yk = n b

(4.2)

k=1

where b is a given positive number, function Rm has a unique conditional maximum. To this end denote Fj (y1 , ..., yn ) ≡

∂Rm (y1 , ..., yn ) . ∂yj

Obviously, Fj (y1 , ..., yn ) does not depend on yj , symmetrically depends on other variables, and monotonically increases with respect to each of its variables. The conditional extremums of Rm under (4.2) are the roots of the equations n ∂ Fj (y1 , ..., yn ) − λ yk = 0 (j = 1, ..., n), ∂yj k=1

18

2. Norms of Matrix Functions

where λ is the Lagrange factor. Therefore, Fj (y1 , ..., yn ) = λ (j = 1, ..., n). Since Fj (y1 , ..., yn ) does not depend on yj , and Fk (y1 , ..., yn ) does not depend on yk , equality Fj (y1 , ..., yn ) = Fk (y1 , ..., yn ) = λ for all k = j is possible if and only if yj = yk . Thus Rm has under (4.2) a unique extremum when y1 = y2 = ... = yn = b. But

Rm (b, ..., b) = bm

(4.3) 1 = bm Cnm .

(4.4)

1≤k1
Let us check that (4.3) gives us the maximum. Letting y1 → nb and yk → 0 (k = 2, ..., n), we get Rm (y1 , ..., yn ) → 0. Since the extremum (4.3) is unique, it is the maximum. Thus, under (4.2) Rm (y1 , ..., yn ) ≤ bm Cnm (yk ≥ 0, k = 1, ..., n). Take yj = aj and b=

a1 + ... + an . n

Then Rm (a1 , ..., an ) ≤ Cnm n−m [

n

ak ]m .

k=1

We thus get the required result. 2

2.5

Euclidean Norms of Powers of Nilpotent Matrices

Theorem 2.5.1 For any nilpotent operator V in Cn , the inequalities V k ≤ γn,k N k (V ) (k = 1, . . . , n − 1) are valid.

2.5. Powers of Nilpotent Matrices

19

Proof: Since V is nilpotent, due to the Shur theorem we can represent it by an upper-triangular matrix with the zero diagonal: V = (ajk )nj,k=1 with ajk = 0 (j ≥ k). Denote xm = (

n

|xk |2 )1/2 for m < n,

k=m

where xk are coordinates of a vector x. We can write V x2m =

n−1

n

|

ajk xk |2 for all m ≤ n − 1.

j=m k=j+1

Now we have (by Schwarz’s inequality) the relation V

x2m

≤

n−1

hj x2j+1 ,

(5.1)

j=m

where

n

hj =

|ajk |2 (j < n).

k=j+1

Further, by Schwarz’s inequality V 2 x2m =

n−1

|

n

ajk (V x)k |2 ≤

j=m k=j+1

n−1

hj V x2j+1 .

j=m

Here (V x)k are coordinates of V x. Taking into account (5.1), we obtain V 2 x2m ≤

n−1 j=m

hj

n−1

hk x2k+1 =

k=j+1

Hence, V 2 2 ≤

hj hk x2k+1 .

m≤j

hj hk .

1≤j
Repeating these arguments, we arrive at the inequality hk1 . . . hkp . V p 2 ≤ 1≤k1
Therefore due to Lemma 2.4.1, n−1

2 V k 2 ≤ γn,k (

j=1

hj )k .

(5.2)

20

2. Norms of Matrix Functions

But

n−1

hj ≤ N 2 (V ).

j=1

We thus have derived the required result. 2 Theorem 2.5.1 and (1.5) imply Corollary 2.5.2 For any nilpotent operator V in Cn , the inequalities V k ≤

N k (V ) √ (k = 1, . . . , n − 1) k!

are valid. An independent proof of this corollary can be found in (Gil’, 1995, p. 50, Lemma 2.3.1).

2.6

Proof of Theorem 2.1.1

Let D and V be the diagonal and nilpotent parts of A, respectively. According to Lemma 1.7.1, Rλ (D)V is a nilpotent operator. So by virtue of Theorem 2.5.1, (6.1) (Rλ (D)V )k ≤ N k (Rλ (D)V )γn,k (k = 1, ..., n − 1). Since D is a normal operator, we can write down Rλ (D) = ρ−1 (D, λ). It is clear that N (Rλ (D)V ) ≤ N (V )Rλ (D) = N (V )ρ−1 (D, λ).

(6.2)

According to (3.4), A − λI = D + V − λI = (D − λI)(I + Rλ (D)V ). We thus have Rλ (A) = (I + Rλ (D)V )−1 Rλ (D) =

n−1

(Rλ (D)V )k Rλ (D).

(6.3)

k=0

Now (6.1) and (6.2) yield the inequality Rλ (A) ≤

n−1

N k (V )γn,k ρ−k−1 (D, λ).

k=0

This relation proves the stated result, since A and D have the same eigenvalues and N (V ) = g(A), due to Lemma 2.3.2. 2 An additional proof of Corollary 2.1.2: Corollary 2.5.2 implies N k (Rλ (D)V ) √ (k = 1, ..., n − 1). k! Now the required result follows from (6.2) and (6.3), since N (V ) = g(A) due to Lemma 2.3.2. 2 (Rλ (D)V )k ≤

2.7. Analytic Matrix Functions

2.7

21

Estimates for the Norm of Analytic Matrix-Valued Functions

Recall that g(A) and γn,k are deﬁned in Section 2.1, and B(Cn ) is the set of all linear operators in Cn . Theorem 2.7.1 Let A ∈ B(Cn ) and let f be a function regular on a neighborhood of the closed convex hull co(A) of the eigenvalues of A. Then f (A) ≤

n−1

sup |f (k) (λ)|g k (A)

k=0 λ∈co(A)

γn,k . k!

(7.1)

The proof of this theorem is divided into a series of lemmas, which are presented in the next section. Theorem 2.7.1 is exact: if A is a normal matrix and sup |f (λ)| = sup |f (λ)|,

λ∈co(A)

λ∈σ(A)

then we have the equality f (A) = supλ∈σ(A) |f (λ)|. Theorem 2.7.1 and inequalities (1.5) yield Corollary 2.7.2 Let A ∈ B(Cn ) and let f be a function regular on a neighborhood of the closed convex hull co(A) of the eigenvalues of A. Then f (A) ≤

n−1

sup |f (k) (λ)|

k=0 λ∈co(A)

g k (A) . (k!)3/2

(7.2)

An additional proof of this corollary is presented in the next section. Example 2.7.3 For a linear operator A in Cn , Theorem 2.7.1 and Corollary 2.7.2 give us the estimates exp(At) ≤ eα(A)t

n−1 k=0

g k (A)tk

n−1

g k (A)tk γn,k ≤ eα(A)t k! (k!)3/2 k=0

(t ≥ 0)

where α(A) = maxk=1,...,n Re λk (A). In addition, Am ≤

n−1 k=0

n−1 γn,k m!g k (A)rsm−k (A) m!g k (A)rsm−k (A) ≤ (m = 1, 2, ...) (m − k)!k! (m − k)!(k!)3/2 k=0

where rs (A) is the spectral radius. Recall that 1/(m − k)! = 0 if m < k.

22

2. Norms of Matrix Functions

2.8

Proof of Theorem 2.7.1

Lemma 2.8.1 Let {dk } be an orthogonal normal basis in Cn , A1 , . . . , Aj n × n-matrices and φ(k1 , ..., kj+1 ) a scalar-valued function of arguments k1 , ..., kj+1 = 1, 2, ..., n; j < n. Deﬁne projectors Q(k) by Q(k)h = (h, dk )dk (h ∈ Cn , k = 1, ..., n), and set

T =

φ(k1 , ..., kj+1 )Q(k1 )A1 Q(k2 ) . . . Aj Q(kj+1 ).

1≤k1 ,...,kj+1 ≤n

Then T ≤ a(φ)|A1 ||A2 | . . . |Aj |, where a(φ) =

max

1≤k1 ,...,kj+1 ≤n

|φ(k1 , ..., kj+1 )|

and |Ak | (k = 1, ..., j) are the matrices, whose entries in {dk } are the absolute values of the entries of Ak in {dk }. Proof: have

For any entry Tsm = (T ds , dm ) (s, m = 1, ..., n) of operator T we Tsm =

(1)

1≤k2 ,...,kj ≤n

(j)

φ(s, k2 , ..., kj , m)ask2 . . . akj ,m ,

(i)

where ajk = (Ai dk , dj ) are the entries of Ai . Hence,

|Tsm | ≤ a(φ)

1≤k2 ,...,kj ≤n

(1)

(j)

|ask2 . . . akj m |.

This relation and the equality 2

T x =

n

|(T x)j |2 (x ∈ Cn ),

j=1

where (.)j means the j-th coordinate, imply the required result. 2 Furthermore, let |V | be the operator whose matrix elements in the orthonormed basis of the triangular representation (the Schur basis) are the absolute values of the matrix elements of the nilpotent part V of A with respect to this basis. That is, |V | =

n k−1

|ajk |(., ek )ej ,

k=1 j=1

where {ek } is the Schur basis and ajk = (Aek , ej ).

2.8. Proof of Theorem 2.7.1

23

Lemma 2.8.2 Under the hypothesis of Theorem 2.7.1, the estimate f (A) ≤

n−1

sup |f (k) (λ)|

k=0 λ∈co(A)

|V |k k!

is true, where V is the nilpotent part of A. Proof:

It is not hard to see that the representation (3.4) implies the equality Rλ (A) ≡ (A − Iλ)−1 = (D + V − λI)−1 = (I + Rλ (D)V )−1 Rλ (D)

for all regular λ. According to Lemma 1.7.1 Rλ (D)V is a nilpotent operator because V and Rλ (D) have common invariant subspaces. Hence, (Rλ (D)V )n = 0. Therefore, Rλ (A) =

n−1

(Rλ (D)V )k (−1)k Rλ (D).

(8.1)

k=0

Due to the representation for functions of matrices 1 f (A) = − 2πi where Ck = (−1)k+1

1 2πi

f (λ)Rλ (A)dλ =

Γ

n−1

Ck ,

(8.2)

k=0

f (λ)(Rλ (D)V )k Rλ (D)dλ.

Γ

Here Γ is a closed contour surrounding σ(A). Since D is a diagonal matrix with respect to the Schur basis {ek } and its diagonal entries are the eigenvalues of A, then Rλ (D) =

n j=1

Qj , λj (A) − λ

where Qk = (., ek )ek . We have Ck =

n

Qj1 V

j1 =1

Here Ij1 ...jk+1 =

n

Qj2 V . . . V

j2 =1

(−1)k+1 2πi

n

Qjk Ij1 j2 ...jk+1 .

jk =1

Γ

f (λ)dλ . (λj1 − λ) . . . (λjk+1 − λ)

24

2. Norms of Matrix Functions

Lemma 2.8.1 gives us the estimate Ck ≤

max

1≤j1 ≤...≤jk+1 ≤n

|Ij1 ...jk+1 | |V |k .

Due to Lemma 1.5.1 |Ij1 ...jk+1 | ≤

|f (k) (λ)| . k! λ∈co(A) sup

This inequality and (8.2) imply the result. 2 Proof of Theorem 2.7.1: Theorem 2.5.1 implies |V |k ≤ γn,k N k (|V |) (k = 1, ..., n − 1). But N (|V |) = N (V ). Moreover, thanks to Lemma 2.3.2, N (V ) = g(A). Thus |V |k ≤ γn,k g k (A) (k = 1, ..., n − 1). Now the previous lemma yields the required result. 2 An additional proof of Corollary 2.7.2: Corollary 2.5.2 implies |V |k ≤

N k (V ) √ (k = 1, ..., n − 1) k!

Now the required result follows from Lemma 2.8.2, since N (V ) = g(A) due to Lemma 2.3.2. 2

2.9

The First Multiplicative Representation of the Resolvent

Recall that B(Cn ) is the set of linear operators in Cn , I is the unit operator. Let Pk (k = 1, . . . , n) be the maximal chain of the invariant projectors of an A ∈ B(Cn ). That is, Pk are orthogonal projectors, APk = Pk APk (k = 1, . . . , n) and

0 = P0 Cn ⊂ P1 Cn ⊂ ... ⊂ Pn Cn = Cn .

So dim ∆Pk = 1. Here ∆Pk = Pk − Pk−1 (k = 1, ..., n). We use the triangular representation A = D + V.

(9.1)

2.9. The First Multiplicative Representation

25

(see Section 1.7). Here V is the nilpotent part of A and n

D=

λk (A)∆Pk

k=1

is the diagonal part. For X1 , X2 , ..., Xn ∈ B(Cn ) denote →

Xk ≡ X1 X2 ...Xn .

1≤k≤n

That is, the arrow over the symbol of the product means that the indexes of the co-factors increase from left to right. Theorem 2.9.1 For any A ∈ B(Cn ), →

λRλ (A) = −

(I +

1≤k≤n

A∆Pk ) (λ ∈ σ(A)), λ − λk (A)

where Pk , k = 1, ..., n is the maximal chain of the invariant projectors of A. Proof:

Denote Ek = I − Pk . Since A = (Ek + Pk )A(Ek + Pk ) for any k = 1, ..., n

and E1 AP1 = 0, we get the relation A = P1 AE1 + P1 AP1 + E1 AE1 . Take into account that ∆P1 = P1 and P1 AP1 = λ1 (A)∆P1 . Then A = λ1 (A)∆P1 + P1 AE1 + E1 AE1 = λ1 (A)∆P1 + AE1 .

(9.2)

Rλ (A) = Ψ(λ),

(9.3)

Now, we check the equality

where Ψ(λ) ≡

∆P1 ∆P1 − AE1 Rλ (A)E1 + E1 Rλ (A)E1 . λ1 (A) − λ λ1 (A) − λ

In fact, multiplying this equality from the left by A − Iλ and taking into account equality (9.2), we obtain the relation (A − Iλ)Ψ(λ) = ∆P1 − ∆P1 AE1 Rλ (A)E1 + (A − Iλ)E1 Rλ (A)E1 .

26

2. Norms of Matrix Functions

But E1 AE1 = E1 A and thus E1 Rλ (A)E1 = E1 Rλ (A). I.e. we can write (A − Iλ)Ψ(λ) = ∆P1 + (−∆P1 A + A − Iλ)E1 Rλ (A) = ∆P1 + E1 (A − Iλ)Rλ (A) = ∆P1 + E1 = I. Similarly, we multiply (9.3) by A − Iλ from the right and take into account (9.2). This gives I. Therefore, (9.3) is correct. Due to (9.3) I − ARλ (A) = (I − (λ1 (A) − λ)−1 A∆P1 )(I − AE1 Rλ (A)E1 ).

(9.4)

Now we apply the above arguments to operator AE1 . We obtain the following expression which is similar to (9.4): I − AE1 Rλ (A)E1 = (I − (λ2 (A) − λ)−1 A∆P2 )(I − AE2 Rλ (A)E2 ). For any k < n, it similarly follows that I − AEk Rλ (A)Ek = (I −

A∆Pk+1 )(I − AEk+1 Rλ (A)Ek+1 ). λk+1 (A) − λ

Substitute this in (9.4), as long as k = 1, 2, ..., n − 1. We have I − ARλ (A) = → 1≤k≤n−1

(I +

A∆Pk )(I − AEn−1 Rλ (A)En−1 ). λ − λk (A)

(9.5)

It is clear that En−1 = ∆Pn . I.e., I − AEn−1 Rλ (A)En−1 = I +

A∆Pn . λ − λn (A)

Now the identity I − ARλ (A) = −λRλ (A) and (9.5) imply the result. 2 Let A be a normal matrix. Then A=

n

λk (A)∆Pk .

k=1

Hence, A∆Pk = λk (A)∆Pk . Since ∆Pk ∆Pj = 0 for j = k, Theorem 2.9.1 gives us the equality λRλ (A) = −

n k=1

(I + (λ − λk (A))−1 λk (A)∆Pk ).

2.10. The Second Multiplicative Representation

But I=

n

27

∆Pk .

k=1

The result is λRλ (A) = −

n

[1 + (λ − λk (A))−1 λk (A)]∆Pk ) =

k=1

−

n

λ

k=1

Or

∆Pk . λ − λk (A)

n

Rλ (A) =

k=1

∆Pk . λk (A) − λ

We have obtained the well-known spectral representation for the resolvent of a normal matrix. Thus, Theorem 2.9.1 generalizes the spectral representation for the resolvent of a normal matrix.

2.10

The Second Multiplicative Representation of the Resolvent

Lemma 2.10.1 Let V ∈ B(Cn ) be a nilpotent operator and Pk , k = 1, ..., n, be the maximal chain of its invariant projectors. Then →

(I − V )−1 =

(I + V ∆Pk ).

(10.1)

2≤k≤n

Proof: In fact, all the eigenvalues of V are equal to zero, and V ∆P1 = 0. Now Theorem 2.9.1 gives us relation (10.1). 2 Relation (10.1) allows us to prove the second multiplicative representation of the resolvent of A. Theorem 2.10.2 Let D and V be the diagonal and nilpotent parts of A ∈ B(Cn ), respectively. Then Rλ (A) = Rλ (D)

→

[I +

2≤k≤n

V ∆Pk ] (λ ∈ σ(A)), λ − λk (A)

(10.2)

where Pk , k = 1, ..., n, is the maximal chain of invariant projectors of A.

28

2. Norms of Matrix Functions

Proof:

Due to (9.1)

Rλ (A) = (A − λI)−1 = (D + V − λI)−1 = Rλ (D)(I + V Rλ (D))−1 . But V Rλ (D) is a nilpotent operator. Take into account that Rλ (D)∆Pk = (λk (A) − λ)−1 ∆Pk . Now (10.1) ensures the relation (10.2). 2

2.11

The First Relation between Determinants and Resolvents

In this section, a relation between the determinant and resolvent of a matrix is derived. It improves the Carleman inequality. We recall the Carleman inequality in Section 2.15. Theorem 2.11.1 Let A ∈ B(Cn ) (n > 1) and I − A be invertible. Then (I − A)−1 det (I − A) ≤ [1 +

1 (N 2 (A) − 2Re T race (A) + 1)](n−1)/2 . n−1

(11.1)

The proof of this theorem is presented in this section below. Corollary 2.11.2 Let A ∈ B(Cn ). Then (Iλ − A)−1 det (λI − A) ≤

[

N 2 (A) − 2Re (λ T race (A)) + n|λ|2 (n−1)/2 ] (λ ∈ σ(A)). n−1 (11.2)

In particular, let V be a nilpotent matrix. Then (Iλ − V )−1 ≤

1 N 2 (V ) (n−1)/2 1 [1 + (1 + )] (λ = 0). |λ| n−1 |λ|2

(11.3)

Indeed, inequality (11.2) is due to Theorem 11.1 with λ−1 A instead of A, and the equality |λ|2 λ−1 = λ taken into account. If V is nilpotent, then |det (λI − V )|2 = det (Iλ − V ) det (Iλ − V ∗ ) = |λ|2n . Moreover, T race V = 0. So (11.2) implies (11.3). To prove Theorem 2.11.1, we need the following

2.11. The First Relation for Determinants

29

Lemma 2.11.3 Let A ∈ B(Cn ) be a positive deﬁnite Hermitian matrix: A = A∗ > 0. Then T race A n−1 ] A−1 det A ≤ [ . n−1 Proof:

Without loss of generality assume that λn (A) =

min λk (A).

k=1,...,n

Then A−1 = λ−1 n (A) and A−1 det A =

n−1

λk (A).

k=1

Hence, due to the inequality between the arithmetic and geometric mean values we get A−1 det A ≤ [(n − 1)−1

n−1

λk ]n−1 ≤ [(n − 1)−1 T race A]n−1 ,

k=1

since A is positive deﬁnite. As claimed. 2 Proof of Theorem 2.11.1: For any A ∈ B(Cn ), the operator B ≡ (I − A)(I − A∗ ) is positive deﬁnite and det B = det (I − A)(I − A∗ ) = det (I − A)det (I − A∗ ) = |det (I − A)|2 . Moreover, T race [(I − A)(I − A∗ )] = T race I − T race (A + A∗ ) + T race (AA∗ ) = n − 2Re T race A + N 2 (A). But

B −1 = (I − A)−1 (I − A∗ )−1 = (I − A)−1 2 .

Now Lemma 2.11.2 yields B −1 det B = (I − A)−1 det (I − A)2 ≤ 1 (n + N 2 (A) − 2Re T race (A))]n−1 = . n−1 1 [1 + (N 2 (A) − 2Re T race (A) + 1)]n−1 , n−1 [

as claimed. 2

30

2.12

2. Norms of Matrix Functions

The Second Relation between Determinants and Resolvents

Without any loss of generality assume that for a regular λ the relation |λ − λ1 (A)| = mink=1,...,n |λ − λk (A)|

(12.1)

is valid. Recall that g(A) and γn,k are deﬁned in Section 2.1. Theorem 2.12.1 Let A ∈ B(Cn ) and (12.1) hold. Then Rλ (A)det(A − Iλ) ≤

n

max {1, |λj (A) − λ|}G(A),

(12.2)

j=2

where G(A) =

n−1

g k (A)γn,k .

k=0

The proof of this theorem is given in the next section. Theorem 2.12.1 is exact. In fact, for instance, let A be a unitary operator and λ = 0. Since any unitary operator is normal and |λk (A)| = 1, then G(A) = 1 and due to (12.3) |detA| = 1 ≤ A. But the norm of a unitary operator equals 1. Thus, we arrive at the equality |detA| = A = 1. Let rs (A) be the spectral radius of a matrix A. Let A be nonsingular. Put A0 = r−1 (A)A. Due to Theorem 2.12.1, A−1 0 det(A0 ) ≤ G(A0 ). But g(A0 ) = rs−1 (A) g(A). Therefore, the following result holds Corollary 2.12.2 Let A ∈ B(Cn ) be nonsingular. Then A−1 det(A) ≤

n−1

rsn−k−1 (A)g k (A)γn,k .

k=0

2.13

Proof of Theorem 2.12.1

For brevity, put λj (A) = λj (j = 1, ..., n). Without any loss of generality assume that |λ1 | =

min |λk |

k=1,...,n

(13.1)

2.13. Proof of Theorem 2.12.1

31

Lemma 2.13.1 Let A be a nonsingular operator in Cn and condition (13.1) hold. Then with the notations χk = max{1, |λk |} and ω(A) =

n

χj ,

j=2

the inequality

det (A)A−1 ≤ G(A).

holds. Proof:

We have

A−1 = D−1 (I + V D−1 )−1 ≤ (I + V D−1 )−1 D−1 .

(13.2)

Clearly, D−1 = |λ1 |−1 . To estimate (I + V D−1 )−1 , observe that V D−1 is a nilpotent matrix and, due to Lemma 2.10.1, →

(I + V D−1 )−1 =

(I − λ−1 k V ∆Pk ),

2≤k≤n

since

D−1 ∆Pk = λ−1 k ∆Pk .

This yields (I + V D−1 )−1 =

n k=2

where K=

−1 λ−1 K, k K = λ1 [det(A)]

→

(13.3)

(Iλk − V ∆Pk ).

2≤k≤n

It not hard to check that |Iλk − V ∆Pk | ≤ (I + |V |∆Pk ) χk . Here |A| means the matrix ( |aij | ), if A = (aij ). The inequality B ≤ C for non-negative matrices B, C is understood in the natural sense. So we have |K| ≤

→ 2≤k≤n

→

χk (I + |V |∆Pk ) = ω(A)

2≤k≤n

ω(A) (I − |V |)−1 .

(I + |V |∆Pk ) =

32

2. Norms of Matrix Functions

Taking into account that N (V ) = N (|V |) and N (V ) = g(A), by virtue of Theorem 2.1.1 we get the inequality (I − |V |)−1 ≤

n

γk,n N k (V ) = G(A).

k=0

That is, K ≤ ω(A)G(A). Due to (13.2) and (13.3) A−1 ≤ |det(A)|−1 ω(A)G(A). As claimed. 2 The assertion of Theorem 2.12.1 follows from the latter lemma with A−λI instead of A. 2

2.14

An Additional Estimate for Resolvents

Theorem 2.14.1 Let A ∈ B(Cn ), n > 1. Then (Iλ − A)−1 ≤

1 g 2 (A) 1 [1 + (1 + 2 ) ](n−1)/2 ρ(A, λ) n−1 ρ (A, λ)

for any regular λ of A. Proof:

Due to the triangular representation (see Section 1.7),

(Iλ − A)−1 = (Iλ − D − V )−1 = (Iλ − D)−1 (I + Bλ )−1 ,

(14.1)

where Bλ := −V (Iλ−D)−1 . But operator Bλ is a nilpotent one. So Theorem 2.11.1 implies (I + Bλ )−1 ≤ [1 +

1 + N 2 (Bλ ) (n−1)/2 . ] n−1

(14.2)

Take into account that N (Bλ ) = N (V (Iλ − D)−1 ) ≤ (Iλ − D)−1 N (V ) = ρ−1 (D, λ)N (V ). Moreover, σ(D) and σ(A) coincide and due to Lemma 2.3.2, N (V ) = g(A). Thus, N (Bλ ) ≤ ρ−1 (D, λ)g(A) = ρ−1 (A, λ)g(A). Now (14.1) and (14.2) imply the required result. 2

2.15. Notes

2.15

33

Notes

The quantity g(A) was introduced both by P. Henrici (1962) and independently by M.I. Gil’ (1979b). Theorem 2.1.1 was derived in the paper (Gil’, 1979a) in a more general situation and was extended in (Gil’, 1995) (see also (Gil’, 1993b)). Recall that Carleman has derived the inequality Rλ (A)

n

(1 − λ−1 λk (A))exp[λ−1 λk (A)] ≤

k=1

|λ|exp[1 + N 2 (Aλ−1 )/2], cf. (Dunford, N and Schwartz, 1963, p. 1023). Theorem 2.3.1 was published in (Gil’, 1993a). It improves Schur’s inequality n |λk (A)|2 ≤ N 2 (A) k=1

and Brown’s inequality n

|Im λk (A)|2 ≤ N 2 (AI )

k=1

(see (Marcus and Minc, 1964)). A very interesting inequality for eigenvalues of matrices was derived in (Kress et. al, 1974). Gel’fand and G.E. Shilov (1958) have established the estimate f (A) ≤

n−1

sup |f (k) (λ)|(2A)k .

k=0 λ∈co(A)

About other estimations for the matrix exponent, see (Coppel, 1978). Theorem 2.7.1 was derived in the paper (Gil’, 1979b) in the case of the Hilbert-Schmidt operators (see also (Gil’, 1993b)). Theorems 2.9.1 and 2.10.1 were published in the more general situation in (Gil’, 1973). Theorems 2.11.1 and 2.12.1 are probably new.

References [1] Coppel, W.A. (1978). Dichotomies in Stability Theory. Lecture Notes in Mathematics, No. 629, Springer-Verlag, New York. [2] Dunford, N and Schwartz, J. T. (1963). Linear Operators, part II. Spectral Theory. Interscience publishers, New York, London.

34

2. Norms of Matrix Functions

[3] Gel’fand, I.M. and Shilov, G.E. (1958). Some Questions of Theory of Diﬀerential Equations. Nauka, Moscow (in Russian). [4] Gil’, M. I. (1973). On the representation of the resolvent of a nonselfadjoint operator by the integral with respect to a spectral function, Soviet Math. Dokl., 14 , 1214-1217. [5] Gil’, M. I. (1979a). An estimate for norms of resolvent of completely continuous operators, Mathematical Notes, 26 , 849-851. [6] Gil’, M. I. (1979b). Estimating norms of functions of a Hilbert-Schmidt operator (in Russian), Izvestiya VUZ, Matematika, 23, 14-19. English translation in Soviet Math., 23 , 13-19. [7] Gil’, M. I. (1983). One estimate for resolvents of nonselfadjoint operators which are ”near” to selfadjoint and to unitary ones, Mathematical Notes, 33, 81-84. [8] Gil’, M. I. (1992). On an estimate for the norm of a function of a quasihermitian operator, Studia Mathematica, 103 (1), 17-24. [9] Gil’, M. I. (1993a). On inequalities for eigenvalues of matrices, Linear Algebra and its Applications, 184, 201-206. [10] Gil’, M. I. (1993b). Estimates for norm of matrix-valued functions , Linear and Multilinear Algebra, 35, 65-73. [11] Gil’, M. I. (1995). Norm Estimations for Operator-valued Functions and Applications. Marcel Dekker, New York. [12] Henrici, P. (1962). Bounds for iterates, inverses, spectral variation and ﬁeld of values of nonnormal matrices. Numerische Mathematik, 4, 24-39. [13] Kress, R., De Vries, H. L. and Wegmann, R. (1974). On non-normal matrices. Linear Algebra Appl., 8, 109-120. [14] Marcus, M. and Minc, H. (1964). A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston.

3. Invertibility of Finite Matrices

The present chapter deals with various types of invertibility conditions for ﬁnite matrices. In particular, we improve the classical Levy-Desplanques theorem and other well-known invertibility results for matrices that are close to triangular ones.

3.1

Preliminary Results

For a matrix put



0 a12  0 0 V+ =   . ... 0 0

A = (ajk )nj,k=1 (n > 1),   0 . . . a1n  a21 . . . a2n   , V− =   . . .  ... 0 an1

... 0 ... 0 ... . . . . an,n−1

 0 0    0

and D = diag (a11 , a22 , ..., ann ). So A = D + V+ + V− . In this chapter it is assumed that all the diagonal entries are nonzero. So d0 :=

min |ajj | > 0.

k=1,...,n

In the present section . is an arbitrary norm in Cn . Recall that I is the unit operator. Set W± = D−1 V± . M.I. Gil’: LNM 1830, pp. 35–48, 2003. c Springer-Verlag Berlin Heidelberg 2003

36

3. Invertibility of Finite Matrices

Theorem 3.1.1 With the notation J(W± ) := (I − W± )−1 , let

1 1 − W− , − W+ } > 0. J(W+ ) J(W− ) Then A is invertible and the inverse matrix satisﬁes the inequality ν(A) ≡ max{

A−1 ≤

D−1 . ν(A)

(1.1)

(1.2)

To prove this theorem we need the following two simple lemmas. Lemma 3.1.2 Let

ψ0 ≡ (I + W− )−1 W+ < 1.

Then the operator B ≡ I − W− + W+ is boundedly invertible. Moreover, B −1 ≤

J(W− ) . 1 − ψ0

Proof: Clearly, operators W± are nilpotent. So operators I + W± are invertible. We have B = (I + W− )(I + (I + W− )−1 W+ ). Moreover, (I + (I + W− )−1 W+ )−1 ≤

∞

((I + W− )−1 W+ )k ≤

k=0 ∞

ψ0k = (1 − ψ0 )−1 .

k=0

Thus,

B −1 ≤ (I + (I + W− )−1 W+ )−1 (I + W− )−1 ≤ (1 − ψ0 )−1 (I + W− )−1 ,

as claimed. 2 Lemma 3.1.3 Let at least one of the following inequalities: W+ J(W− ) < 1

(1.3)

W− J(W+ ) < 1

(1.4)

or hold. Then relation (1.1) is valid. Moreover, the operator B ≡ I + W− + W+ is invertible and 1 B −1 ≤ . ν(A)

3.2. lp Norms of Powers of Nilpotent Matrices

Proof:

37

If condition (1.3) holds, then Lemma 3.1.2 yields the inequality B −1 ≤

1 J(W− ) = −1 . 1 − W+ J(W− ) J (W− ) − W+

(1.5)

Interchanging W− and W+ , under condition (1.4), we get B −1 ≤

J −1 (W

1 . + ) − W−

This relation and (1.5) yield the required result. 2 Proof of Theorem 3.1.1: Clearly, condition (1.1) implies that at least one of inequalities (1.3) or (1.4) holds. But A = D + V+ + V− = D(I + W+ + W− ) = DB. Now the required result follows from Lemma 3.1.3.

lp -Norms of Powers of Nilpotent Matrices

3.2

Recall that hp = [

n

|hk |p ]1/p (h = (hk ) ∈ Cn ; 1 < p < ∞).

k=1

In the present and next sections Ap is an operator norm of a matrix A with respect to the vector norm .p . Put m (n − 1)−m ]1/p (m = 1, ..., n − 1) and γn,0,p = 1, γn,m,p = [Cn−1

where Cnm = n!/(n − m)!m! are the binomial coeﬃcients. Note that p = γn,m,p

(n −

(n − 1)...(n − m) (n − 1)! ≤ . − 1 − m)!m! (n − 1)m m!

1)m (n

Hence,

1 (m = 1, ...n − 1). m! Lemma 3.2.1 For any upper triangular nilpotent matrix p γn,m,p ≤

V+ = (ajk )nj,k=1 with ajk = 0 (1 ≤ k ≤ j ≤ n) the inequality V+m p ≤ γn,m,p Mpm (V+ ) (m = 1, . . . , n − 1) is valid with the notation n−1

Mp (V+ ) = (

[

n

j=1 k=j+1

|ajk |q ]p/q )1/p (p−1 + q −1 = 1).

(2.1)

38

3. Invertibility of Finite Matrices

Proof:

For a natural s = 1, ..., n − 1, denote n |xk |p )1/p , xp,s = ( k=s

where xk are the coordinates of a vector x ∈ Cn . We can write out n−1

V+ xpp,s =

n

|

ajk xk |p .

j=s k=j+1

By H¨older’s inequality, V+ xpp,s ≤

n−1 j=s

where hj = [

n

hj xpp,j+1 ,

(2.2)

|ajk |q ]p/q .

k=j+1

Similarly, V+2 xpp,s =

n−1

n

|

ajk (V+ x)k |p ≤

j=s k=j+1 n−1 j=s

hj V+ xpp,j+1 .

Here (V+ x)k are the coordinates of V+ x. Taking into account (2.2), we obtain V+2 xpp,s ≤

n−1

hj

j=s

s≤j
Therefore,

n−1 k=j+1

hk xpp,k+1 =

hj hk xpp,k+1 .

V+2 pp = V+2 pp,1 ≤

hj hk .

1≤j
Repeating these arguments, we arrive at the inequality V+m pp ≤ hk1 . . . hkm . 1≤k1
Since,

n j=1

hj =

n n [ |ajk |q ]p/q = Mpp (V+ ), j=1 k=j+1

(2.3)

3.3. Invertibility in the Norm .p

39

due to Lemma 2.4.1 and (2.3) we have m (n − 1)−m , V+m pp ≤ Mpmp (V+ )Cn−1

as claimed. 2 Similarly we can prove Lemma 3.2.2 For any lower triangular nilpotent matrix V− = (ajk )nj,k=1 with ajk = 0 (1 ≤ j ≤ k ≤ n). the inequality V−m p ≤ γn,m,p Mpm (V− ) (m = 1, . . . , n − 1) is valid, where

j−1 n |ajk |q ]p/q )1/p . Mp (V− ) = ( [ j=2 k=1

Consider the case p = 2. The Euclidean norm .2 is invariant with respect to an orthogonal basis. Moreover, M22 (V+ ) = T race V+∗ V+ =

n−1

n

|ajk |2 = N22 (V+ ),

j=1 k=j+1

where N2 (.) = N (.) is the Hilbert-Schmidt norm. Due to Lemma 3.2.1 we have Corollary 3.2.3 Any n × n-nilpotent matrix V satisﬁes the inequalities V m 2 ≤ γn,m,2 N2m (V ) (m = 1, ..., n − 1). Thus, Lemma 3.2.1 gives us the new proof of Lemma 2.5.1, since γn,m,2 = γn,m .

3.3

Invertibility in the Norm .p (1 < p < ∞)

Recall that A, d0 and V± are deﬁned in Section 3.1; γn,m,p and Mp (V± ) are deﬁned in the previous section. In addition, W± = D−1 V± . So j−1 n |ajk |q p/q 1/p Mp (W− ) = ( [ ] ) |ajj |q j=2 k=1

and

n−1

Mp (W+ ) = (

[

n |ajk |q p/q 1/p −1 ] ) (p + q −1 = 1). |ajj |q

j=1 k=j+1

40

3. Invertibility of Finite Matrices

Theorem 3.3.1 With the notation Jp (W± ) :=

n−1

γn,k,p Mpk (W± ),

k=0

let

1 1 − W− p , − W+ p } > 0. Jp (W+ ) Jp (W− ) Then A is invertible and the inverse matrix satisﬁes the inequality νp (A) := max{

A−1 p ≤ Proof:

1 . d0 νp (A)

Clearly, (I − W± )−1 p ≤

n−1

W±k p .

k=0

Lemmas 3.2.1 and 3.2.2 imply (I − W± )−1 p ≤ Jp (W± ). Now Theorem 3.1.1 yields the required result. 2

3.4

Invertibility in the Norm .∞

For a matrix A = (ajk )nj=1 take the norm A∞ ≡ max

j=1,...,n

n

|ajk |.

k=1

Recall that d0 is deﬁned in Section 3.1. Under the condition d0 > 0, introduce the notation: v˜k := max |ajk | (k = 2, ..., n); j=1,...,k−1

w ˜k := mup (A) :=

n k=2

max

j=k+1,...,n

(1 +

|ajk | (k = 1, ..., n − 1),

n−1 v˜k w ˜k ) and mlow (A) := ). (1 + |akk | |akk | k=1

Theorem 3.4.1 Let the condition mup (A)mlow (A) < mup (A) + mlow (A)

(4.1)

be fulﬁlled. Then matrix A is invertible and the inverse matrix satisﬁes the inequality A−1 ∞ ≤

mup (A)mlow (A) . (mup (A) + mlow (A) − mup (A)mlow (A))d0

(4.2)

3.5. Proof of Theorem 3.4.1

41

The proof of this theorem is divided into a series of lemmas which are presented in the next section. Note that condition (4.1) is equivalent to the following one: θ(A) := (mup (A) − 1)(mlow (A) − 1) < 1.

(4.3)

Inequality (4.2) can be written as A−1 ∞ ≤

mup (A)mlow (A) . d0 (1 − θ(A))

(4.4)

If matrix A is triangular and has nonzero diagonal entries, then (4.1) obviously holds.

3.5

Proof of Theorem 3.4.1

Recall that V± and D are introduced in Section 3.1, and W± = V± D−1 . In this section A is the operator norm of A with respect to an arbitrary vector norm. Lemma 3.5.1 Let the condition θ˜0 ≡

n−1

(−1)k+j W−k W+j < 1

(5.1)

j,k=1

hold. Then A is invertible and the inverse matrix satisﬁes the inequality A−1 ≤ D−1 (I + W− )−1 (I + W+ )−1 (1 − θ˜0 )−1 . Proof:

(5.2)

Clearly, A = D + W− + W = (I + W− + W+ )D = [(I + W− )(I + W+ ) − W− W+ ]D.

But W+ and W− are nilpotent: W−n = W+n = 0.

(5.3)

So the operators, I + W− and I + W+ are invertible: (I + W− )−1 =

n−1

(−1)k W−k ,

k=0

(I + W+ )−1 =

n−1 k=0

(−1)k W+k .

(5.4)

42

3. Invertibility of Finite Matrices

Thus A = (I + W− )[I − (I + W− )−1 W− W+ (I + W+ )−1 ](I + W+ )D. Thanks to (5.4) we have (I + W− )−1 W− =

n−1

(−1)k−1 W−k , W+ (I + W+ )−1 =

k=1 n−1

(−1)k−1 W+k .

k=1

So

n−1

A = (I + W− )[I −

(−1)k+j W−k W+j ](I + W+ )D.

j,k=1

Therefore, if (5.1) holds then A is invertible. Moreover A−1 = D−1 (I + W+ )−1 [I −

n−1

(−1)k+j W−k W+j ]−1 (I + W− )−1 .

(5.5)

j,k=1

Condition (5.1) yields [I −

n−1

(−1)k+j W−k W+j ]−1 ≤ (1 − θ˜0 )−1 .

j,k=1

Now inequality (5.2) is due to (5.5). 2 Denote n n−1 m(V ˜ +) = (1 + v˜k ) and m(V ˜ −) = (1 + w ˜k ). k=2

k=1

Lemma 3.5.2 The inequalities

and

˜ +) (I − V+ )−1 ∞ ≤ m(V

(5.6)

(I − V− )−1 ∞ ≤ m(V ˜ −)

(5.7)

are valid. Proof:

Let Qk be the projectors onto the standard basis: Qk h = (h1 , h2 , ..., hk , 0, 0, ..., 0) (k = 1, ..., n), Q0 = 0

for an arbitrary vector h = (h1 , ..., hn ) ∈ Cn . Clearly, Qk project onto the invariant subspaces of V+ . So according to Lemma 2.10.1, (I − V+ )−1 =

→ 2≤k≤n

(I + V+ ∆Qk ), where ∆Qk = Qk − Qk−1 .

(5.8)

3.5. Proof of Theorem 3.4.1

43

It is not hard to check that V+ ∆Qk ∞ = v˜k . ˜ k by Now inequality (5.6) follows from (5.8). Further, deﬁne a projector Q ˜ k h = (0, 0, ..., hn−k+1 , hn−k+2 , ..., hn ) Q ˜ 0 = 0. (k = 1, ..., n), Q ˜ k project onto invariant subspaces of V− . So Simple calculation show that Q according Lemma 2.10.1, →

(I − V− )−1 =

˜ k ), (I + V− ∆Q

2≤k≤n

˜k = Q ˜k − Q ˜ k−1 ). (∆Q (5.9) ˜ k ∞ = w It is not hard to check that V− ∆Q ˜n−k+1 . Now inequality (5.7) follows from (5.9). 2 Lemma 3.5.3 The inequalities

n−1

(−1)k V+k ∞ ≤ m(V ˜ + ) − 1 and

k=1

n−1

(−1)k V−k ∞ ≤ m(V ˜ − ) − 1 (5.10)

k=1

are valid. Proof:

Let B = (bjk )nk=1 be a nonnegative matrix with the property Bh ≥ h

(5.11)

for any nonnegative h ∈ Cn . Then bjj ≥ 1 (j = 1, ..., n). Hence, B − I∞ = max [ j=1,...,n

n

bjk − δjk ] = B∞ − 1.

(5.12)

k=1

Here δjk is the Kronecker symbol. Furthermore, since V+ is nilpotent,

n−1

(−1)k V+k ∞ ≤

k=1

n−1

|V+ |k ∞ =

k=1

(I − |V+ |)−1 − I∞ where |V+ | is the matrix whose entries are the absolute values of the entries of V . Moreover, clearly, n−1 |V+ |k h ≥ h k=0

44

3. Invertibility of Finite Matrices

for any nonnegative h ∈ Cn . So according to (5.11) and (5.12),

n−1

(−1)k V+k ∞ ≤

k=1

n−1

|V+ |k ∞ =

k=1

(I − |V+ |)−1 − I∞ Since ˜ + |), m(V ˜ + ) = m(|V equality (5.6) with V+ = |V+ | yields the ﬁrst inequality (5.10). Similarly, the second inequality (5.10) can be proved. 2 Proof of Theorem 3.4.1: Due to Lemma 3.5.2, (I + W− )−1 ∞ ≤ mup (A), (I + W+ )−1 ∞ ≤ mlow (A). Lemma 3.5.3 yields

n−1

(−1)k W−k ∞ ≤ mup (A) − 1,

k=1

and

n−1

(−1)k W+k ∞ ≤ mlow (A) − 1.

k=1

Hence,

n−1

(−1)k+j W−k W+j ∞ ≤ θ(A).

j,k=1

Now the required result follows directly from Lemma 3.5.1. 2

3.6

Positive Invertibility of Matrices

For h = (hk ), w = (wk ) ∈ Rn , we write h ≥ (>) g, if hk ≥ (>) wk , (k = 1, ..., n). A matrix A is (non-negative ) positive: A > (≥) 0, if all its entries are (non-negative) positive. For matrices A, B we write A > (≥) B if A−B > (≥) 0. Again V+ , V− and D are the upper triangular, lower triangular, and diagonal parts of matrix A = (ajk )nj,k=1 , respectively (see Section 3.1). Matrix A = (ajk )nj=1,k=1 is called a Z-matrix, if the conditions ajk ≤ 0 for j = k, and akk > 0 (j, k = 1, ..., n) hold. That is, D > 0, V± ≤ 0.

(6.1)

3.7. Positive Matrix-Valued Functions

45

Lemma 3.6.1 Let A be a Z-matrix, and let condition (4.1) hold. Then A is positively invertible and the inverse operator satisﬁes the inequalities (4.2) and A−1 ≥ D−1 . (6.2) Proof: As it was proved in the previous section, condition (4.1) implies the invertibility of A. Moreover, relation (5.5) holds. But W± ≤ 0. So (−1)k+j W−k W+j ≥ 0 and thus [I −

n−1

(−1)k+j W−k W+j ]−1 ≥ 0,

j,k=1

since

n−1

(−1)k+j W−k W+j ≤ θ0 < 1.

j,k=1

In addition, −1

(I + W± )

=

n−1

(−1)k W±k ≥ 0.

k=1

Now (5.5) implies (6.2). Inequality (4.2) is due to Theorem 3.4.1. 2

3.7

Positive Matrix-Valued Functions

We will call A an anti-Hurwitz matrix if its spectrum σ(A) lies in the open right half-plane: β(A) := inf Re σ(A) > 0. For an anti-Hurwitzian matrix A, deﬁne the matrix-valued function F (A) by

∞

e−At f (t)dt,

(7.1)

e−β(A)t f (t) ∈ L1 [0, ∞).

(7.2)

F (A) = 0

where

Lemma 3.7.1 Let A be an anti-Hurwitzian Z-matrix. In addition, let relations (7.1), (7.2) hold, and f (t) ≥ 0 for almost all t ≥ 0. Then F (A) ≥ F (D) ≥ 0.

(7.3)

46

3. Invertibility of Finite Matrices

Proof:

Since D > 0 and V± ≤ 0, we have e−At = lim (I − An−1 )nt = lim (I − n−1 D − n−1 V )nt ≥ n→∞

n→∞

lim (I − n−1 D)nt = e−Dt ,

n→∞

where I is the unit matrix. Thus ∞ −At F (A) = e f (t)dt ≥ 0

∞

e−Dt f (t)dt = F (D).

0

As claimed. 2 In, particular, let A be an anti-Hurwitzian matrix, and f (t) ≡

∞ k=1

ak tsk Γ(sk + 1)

(sk = const > 0, ak = const ≥ 0, k = 1, 2, ...), where Γ(.) is the Euler Gamma function. Then F (A) =

∞

ak A−sk −1 ,

(7.4)

k=1

provided the series converges. The following functions are examples of the functions deﬁned by (7.4): A−ν , A−ν eb0 A

−1

(b0 = const > 0; 0 < ν ≤ 1).

Lemma 3.7.2 Let A be a Z-matrix. Then it is anti-Hurwitzian and satisﬁes the inequality (6.2) if and only if it is positively invertible. Proof:

Let A be anti-Hurwitzian. Then due to Lemma 3.7.1, ∞ −1 A = e−At dt ≥ D−1 > 0. 0

Conversely, let A be positively invertible. Put T = −V+ − V− . Then for any λ with Re λ ≥ 0, |(A + λ)h| = |(D + λ − T )h| ≥ |(D + λ)h| − T |h| ≥ D|h| − T |h| = A|h|. Here |h| means the vectors whose coordinates are the absolute values of the coordinates of h. This proves the result. 2 Lemma 3.6.1 and the previous lemma imply Corollary 3.7.3 Let F be deﬁned by (7.1) with a non-negative function f satisfying (7.2). In addition, let A be a Z-matrix and condition (4.1) hold. Then relation (7.3) is valid.

3.8. Notes

3.8

47

Notes

As it was mentioned, although excellent computer softwares are now available for eigenvalue computation, new results on invertibility and spectrum inclusion regions for ﬁnite matrices are still important, since computers are not very useful, in particular, for analysis of matrices dependent on parameters. So the problem of ﬁnding invertibility conditions and spectrum inclusion regions for ﬁnite matrices continues to attract attention of many specialists, cf. (Brualdi, 1982), (Farid, 1995 and 1998), (Gudkov, 1967), (Li and Tsatsomeros, 1997), (Tam et al., 1997) and references given therein. Let A = (ajk ) be a complex n × n-matrix (n ≥ 2) with the nonzero diagonal: akk = 0 (k = 1, ..., n). Put n

Pj =

|ajk |.

k=1, k=j

The well-known Levy-Desplanques theorem states that if |ajj | > Pj (j = 1, ..., n), then A is nonsingular. This theorem has been improved in many ways. For example, each of the following is known to be a suﬃcient condition for non-singularity of A: (i) |aii ||ajj | > Pj Pi (i, j = 1, ..., n) ( Marcus and Minc, 1964, p. 149). (ii) |ajj | ≥ Pj (j = 1, ..., n), provided that at least one inequality is strict and A is irreducible ( Marcus and Minc, 1964, p. 147). (iii) |ajj | ≥ rj mj (j = 1, ..., n), where rj are positive numbers satisfying n

(1 + rk )−1 ≤ 1 and mj = max |ajk | (see (Bailey and Crabtree, 1969). k=j

k=1

(j = 1, ..., n) where (iv) |ajj | > Pj Q1− j 0 ≤ ≤ 1, Qj =

n

|akj | ( Marcus and Minc, 1964, p. 150) .

k=1, k=j

Theorems 3.1.1, 3.3.1 and 3.4.1 yield new invertibility conditions which improve the mentioned results, when the considered matrices are close to triangular ones. Moreover, they give us estimates for diﬀerent norms of the inverse matrices. Note that Theorems 3.1.1, 2.1.1 and 2.14.1 allow us to derive additional invertibility conditions in the terms of the Euclidean norm. The material in Chapter 3 is based on the papers (Gil’, 1997), (Gil’, 1998) and (Gil’, 2001).

References [1] Bailey D. W. and D. E. Crabtree, (1969), Bounds for determinants, Linear Algebra and Its Applications, 2, 303-309.

48

3. Invertibility of Finite Matrices

[2] Brualdi, R.A. (1982), Matrices, eigenvalues and directed graphs, Linear and Multilinear Algebra, 11, 143-165 [3] Collatz, L. (1966). Functional Analysis and Numerical Mathematics. Academic press, New York and London. [4] Farid, F.O. (1995), Criteria for invertibility of diagonally dominant matrices, Linear Algebra and Its Applications, 215, 63-93. [5] Farid, F.O. (1998), Topics on a generalization of Gershgorin’s theorem, Linear Algebra and Its Applications, 268, 91-116. [6] Gil’, M.I. (1997), A nonsingularity criterion for matrices, Linear Algebra and Its Applications, 253, 79-87. [7] Gil’, M.I. (1998), On positive invertibility of matrices, Positivity, 2, 165170. [8] Gil’, M.I. (2001), Invertibility and positive invertibility of matrices, Linear Algebra and and Its Applications, 327, 95-104 [9] Gudkov, V. V. (1967). On a certain test for nonsingularity of matrices, Latvian Math. Yearbook, 1965, 385-390, (Math. Reviews, 33), review 1323) [10] Horn, R. A. and Johnson Ch. R. (1991). Topics in Matrix Analysis, Cambridge, Cambridge University Press. [11] Krasnosel’skii, M. A., Lifshits, J. and A. Sobolev. (1989). Positive Linear Systems. The Method of Positive Operators, Heldermann Verlag, Berlin. [12] Li B. and Tsatsomeros, M.J. (1997), Doubly diagonally dominant matrices, Linear Algebra and Its Applications, 261, 221-235. [13] Marcus M. and Minc, H. (1964). A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston. [14] Tam, B.S., Yang, S. and Zhang. X. (1997) Invertibility of irreducible matrices, Linear Algebra and Its Applications, 259, 39-70

4. Localization of Eigenvalues of Finite Matrices

The present chapter is concerned with perturbations of ﬁnite matrices and bounds for their eigenvalues. In particular, we improve the classical Gershgorin result for matrices, which are ”close” to triangular ones. In addition, we derive upper and lower estimates for the spectral radius. Under some restrictions, these estimates improve the Frobenius inequalities. Moreover, we present new conditions for the stability of matrices, which supplement the Rohrbach theorem.

4.1

Deﬁnitions and Preliminaries

In this chapter, . is an arbitrary norm in Cn , A and B are n × n-matrices having eigenvalues λ1 (A), ..., λn (A) and λ1 (B), ..., λn (B), respectively, and q = A − B. We recall some well-known deﬁnitions from matrix perturbation theory (see (Stewart and Sun, 1990)). The spectral variation of B with respect to A is svA (B) := max min |λi (B) − λj (A)|. i

j

The Hausdorﬀ distance between the spectra of A and B is hd(A, B) := max{svA (B), svB (A)}. The matching distance between eigenvalues of A and B is md(A, B) := min max |λπ(i) (B) − λi (A)|, π

M.I. Gil’: LNM 1830, pp. 49–63, 2003. c Springer-Verlag Berlin Heidelberg 2003

i

50

4. Eigenvalues of Finite Matrices

where π is taken over all permutations of {1, 2, ..., n}. Recall also that σ(A) denotes the spectrum of A. Lemma 4.1.1 For any µ ∈ σ(B), we have either µ ∈ σ(A), or qRµ (A) ≥ 1. Proof:

(1.1)

Suppose that the inequality qRµ (A) < 1.

(1.2)

holds. We can write Rµ (A) − Rµ (B) = Rµ (B)(B − A)Rµ (A). This yields Rµ (A) − Rµ (B) ≤ Rµ (B)qRµ (A).

(1.3)

Thus, (1.2) implies Rµ (B) ≤ Rµ (A)(1 − qRµ (A))−1 . That is, µ is a regular point of B. This contradiction proves the result. 2 Lemma 4.1.2 Assume that Rλ (A) ≤ φ(ρ−1 (A, λ)) for all regular λ of A,

(1.4)

where φ(x) is a monotonically increasing non-negative function of a nonnegative variable x, such that φ(0) = 0 and φ(∞) = ∞. Then the inequality svA (B) ≤ z(φ, q)

(1.5)

is true, where z(φ, q) is the extreme right-hand (positive) root of the equation 1 = qφ(1/z). Proof:

(1.6)

Due to Lemma 4.1.1 and condition (1.4), 1 ≤ qφ(ρ−1 (A, λ)) for all λ ∈ σ(B).

Since φ(x) monotonically increases, z(φ, q) is a unique positive root of (1.6) and ρ(A, λ) ≤ z(φ, q). Thus, the inequality (1.5) is valid. 2

4.2

Perturbations of Multiplicities and Matching Distance

Put Ω(c, r) ≡ {z ∈ C : |z − c| ≤ r} (c ∈ C, r > 0).

4.2. Perturbations of Multiplicities

51

Lemma 4.2.1 Under condition (1.4), let A have an eigenvalue λ(A) of the algebraic multiplicity ν and the distance from λ(A) to the rest of σ(A) is equal to 2d, i.e. distance{λ(A), σ(A)/λ(A)} = 2d. (2.1) In addition, for a positive number a ≤ d, let qφ(1/a) < 1.

(2.2)

Then in Ω(λ(A), a) operator B has eigenvalues whose total algebraic multiplicity is equal to ν. Proof: This result is a particular case of the well-known Theorem 3.18 (Kato, 1966, p. 215). 2 Since φ is a nondecreasing function, comparing (2.2) with (1.6), we get Corollary 4.2.2 Let A have an eigenvalue λ(A) of the algebraic multiplicity ν. In addition, under conditions (1.4) and (2.1), let the extreme right-hand root z(φ, q) of equation (1.6) satisﬁes the inequality z(φ, q) ≤ d.

(2.3)

Then in Ω(λ(A), z(φ, q)) operator B has eigenvalues whose total algebraic multiplicity is equal to ν. Let θ1 , ..., θn1 (n1 ≤ n) be diﬀerent eigenvalues of A, i.e., ˜ d(A) := min{|θj − θk | : j = k; j, k = 1, .., n1 } > 0 and

n1

(2.4)

νk = n,

k=1

where νk is the multiplicity of the eigenvalue θk . By virtue of Lemma 4.2.1, we arrive at the following result. Lemma 4.2.3 Let condition (1.4) hold. In addition, for a positive number ˜ a ≤ d(A), let condition (2.2) is valid. Then all the eigenvalues of operator B lie in the set 1 ∪nk=1 Ω(θk , a) and thus, md (A, B) ≤ a. Moreover, Corollary 4.2.2 implies Corollary 4.2.4 Under conditions (1.4) and (2.4), let the extreme right˜ hand root z(φ, q) of equation (1.6) satisﬁes the inequality z(φ, q) ≤ d(A). Then all the eigenvalues of operator B lie in the set 1 ∪nk=1 Ω(θk , z(φ, q))

and thus md (A, B) ≤ z(φ, q).

52

4.3

4. Eigenvalues of Finite Matrices

Perturbations of Eigenvectors and Eigenprojectors

Under condition (1.4), let A have an eigenvalue λ(A) of the algebraic multiplicity ν and (2.1) hold. Let ∂Ω be the boundary of Ω(λ(A), d): ∂Ω ≡ {z ∈ C : |z − λ(A)| = d}. Put

1 P (A) = − 2πi

and

1 P (B) = − 2πi

∂Ω

Rλ (A)dλ

(3.1)

Rλ (B)dλ.

(3.2)

∂Ω

That is, both P (A) and P (B) are the projectors onto the eigenspaces of A and B, respectively, corresponding to points of the spectra, which lie in Ω(λ(A), d). Lemma 4.3.1 Let A satisfy condition (1.4) and have an eigenvalue λ(A) of the algebraic multiplicity ν, such that the conditions (2.1) and qφ(1/d) < 1

(3.3)

hold. Then dim P (A) = dim P (B). Moreover, P (A) − P (B) ≤

qdφ2 (1/d) . 1 − qφ(1/d)

(3.4)

Proof: Thanks to Lemma 4.2.1 dim P (A) = dim P (B). From (1.3) and (1.4) it follows that Rλ (B) ≤ Rλ (A)(1 − qφ(1/d))−1 ≤ φ(1/d)(1 − qφ(1/d))−1 (λ ∈ ∂Ω) and

1 Rλ (A) − Rλ (B)|dλ| ≤ P (A) − P (B) ≤ 2π ∂Ω 1 qφ2 (1/d)d , Rλ (B)qφ(1/d)|dλ| ≤ 2π ∂Ω 1 − qφ(1/d)

as claimed. 2 We will say that an eigenvalue of a linear operator is a simple eigenvalue, if its algebraic multiplicity is equal to one. The eigenvector corresponding to a simple eigenvalue will be called a simple eigenvector.

4.4. Perturbations in Euclidean Norm

53

Lemma 4.3.2 Suppose A has a simple eigenvalue λ(A), such that the relations (1.4), (2.1) and q[dφ2 (1/d) + φ(1/d)] < 1

(3.5)

hold. Then for the eigenvector e of A corresponding to λ(A) with e = 1, there exists a simple eigenvector f of B with f = 1, such that e − f ≤ 2δ(1 − δ)−1 , where δ≡

qdφ2 (1/d) . 1 − qφ(1/d)

Proof: Firstly, note that condition (3.5) implies the relations (3.3) and δ < 1, and B has in Ω(λ(A), d) a simple eigenvalue λ(B) due to Lemma 4.2.1. Let P (B) and P (A) be deﬁned by (3.1) and (3.2). Due to Lemma 4.3.1, P (A) − P (B) ≤ δ < 1. Consequently, P (B)e = 0, since P (A)e = e. Thanks to the relation BP (B)e = λ(B)P (B)e, P (B)e is an eigenvector of B. Let N = P (B)e. Then f ≡ N −1 P (B)e is a normed eigenvector of B. So e − f = P (A)e − N −1 P (B)e = e − N −1 e + N −1 (P (A) − P (B))e. But N ≥ P (A)e − (P (A) − P (B))e ≥ 1 − δ. Hence N

−1

≤ (1 − δ)−1 and e − f ≤ (N −1 − 1)e + N −1 P (A) − P (B) ≤ (1 − δ)−1 − 1 + (1 − δ)−1 δ = 2δ(1 − δ)−1 ,

as claimed. 2

4.4 4.4.1

Perturbations of Matrices in the Euclidean Norm Perturbations of eigenvalues

We recall that the quantities .2 , g(A) and γn,k are deﬁned in Sections 1.2 and 2.1. In addition, put q2 := A − B2 . The norm for matrices is understood in the sense of the operator norm.

54

4. Localization of Eigenvalues of Finite Matrices

Theorem 4.4.1 Let A and B be n × n-matrices. Then svA (B) ≤ z(q2 , A),

(4.1)

where z(q2 , A) is the extreme right-hand (unique nonegative) root of the algebraic equation n−1 z n = q2 γn,j z n−j−1 g j (A). (4.2) j=0

Proof:

Theorem 2.1.1 gives us the inequality Rλ (A)2 ≤

n−1 k=0

g k (A)γn,k (λ ∈ σ(A)). ρk+1 (A, λ)

(4.3)

Rewrite (4.2) as 1=

n−1 j=0

γn,j g j (A) . z j+1

Now the required result is due to Lemma 4.1.2 and (4.3). 2 Put n−1 wn = γn,j . j=0

Setting z = g(A)y in (4.2) and applying Lemma 1.6.1, we have the estimate z(q2 , A) ≤ q2 wn , provided (4.4) q2 wn ≥ g(A) and

z(q2 , A) ≤ g 1−1/n (A)[q2 wn ]1/n ,

provided q2 wn ≤ g(A).

(4.5)

Now Theorem 4.4.1 ensures the following result. Corollary 4.4.2 Let condition (4.4) hold. Then svA (B) ≤ qwn . If condition (4.5) holds, then svA (B) ≤ g 1−1/n (A)[qwn ]1/n .

4.4.2

Perturbations of multiplicities and matching distance

For a positive scalar variable x, set Gn (x) ≡

n−1 j=0

γn,j xj+1 g j (A).

4.4. Perturbations in the Euclidean Norm

55

Lemma 4.4.3 Let A have an eigenvalue λ(A) of the algebraic multiplicity ν and (2.1) hold. In addition, for a positive number a < d, let q2 Gn (1/a) < 1.

(4.6)

Then in Ω(λ(A), a) operator B has eigenvalues whose total algebraic multiplicity is equal to ν. Proof:

This result follows from (4.3) and Lemma 4.2.1. 2

Inequality (4.3) and Corollary 4.2.2 imply Corollary 4.4.4 Let A have an eigenvalue λ(A) of the algebraic multiplicity ν. In addition, under condition (2.1), let the extreme right-hand root z(q2 , A) of equation (4.2) satisﬁes the inequality z(q2 , A) ≤ d. Then in Ω(λ(A), z(q2 , A)) operator B has eigenvalues whose total algebraic multiplicity is equal to ν. Let θ1 , ..., θn1 (n1 ≤ n) be the diﬀerent eigenvalues of A, again. By virtue of (4.3) and Lemma 4.2.3 we get the following result. ˜ Lemma 4.4.5 Under (2.4), for a positive number a < d(A), let condition (4.6) is valid. Then all the eigenvalues of operator B lie in the set 1 Ω(θk , a) ∪nk=1

and thus md (A, B) ≤ a. In addition, due to (4.3) and Corollary 4.2.4, we get Corollary 4.4.6 Under (2.4), let the extreme right-hand root z(q2 , A) of ˜ Then all the eigenvalequation (4.2) satisﬁes the inequality z(q2 , A) ≤ d(A). ues of operator B lie in the set 1 ∪nk=1 Ω(θk , z(q2 , A))

and thus md (A, B) ≤ z(q2 , A). To estimate z(q2 , A) we can apply Lemma 1.6.1.

4.4.3

Perturbations of eigenvectors and eigenprojectors in the Euclidean norm

Let A have an eigenvalue λ(A) of the algebraic multiplicity ν and (2.1) hold. Deﬁne P (A) and P (B) as in Section 3.1. Recall that Gn and q2 are deﬁned in the previous two subsections.

56

4. Eigenvalues of Finite Matrices

Lemma 4.4.7 Let A and B be linear operators in Cn . In addition, let A have an eigenvalue λ(A) of the algebraic multiplicity ν, such that the conditions (2.1) and q2 Gn (1/d) < 1 hold. Then dim P (A) = dim P (B). Moreover, P (A) − P (B)2 ≤

q2 Gn (1/d)d . 1 − q2 Gn (1/d)

This result is due to (4.3) and Lemma 4.3.1. Lemma 4.4.8 Let A and B be linear operators in Cn . Suppose A has a simple eigenvalue λ(A), such that the relations (2.1) and q2 [G2n (1/d)d + Gn (1/d)] < 1 hold. Then for the eigenvector e of A corresponding to λ(A) with e2 = 1, there exists the simple eigenvector f of B with f 2 = 1, such that e − f 2 ≤ 2δ2 (1 − δ2 )−1 , where δ2 ≡

q2 dG2n (1/d) . 1 − q2 Gn (1/d)

This result is due to (4.3) and Lemma 4.3.2.

4.5

Upper Bounds for Eigenvalues in Terms of the Euclidean Norm

Let A be an n × n-matrix with entries ajk (j, k = 1, ..., n). Recall that D, V+ and V− are the diagonal, upper nilpotent part and lower nilpotent part of A, respectively, (see Section 3.1), and wn =

n−1

γn,j ,

j=0

where γn,j are deﬁned in Section 2.1. Assume that V+ = 0, V− = 0 and denote µ2 (A) = wn min {N −1 (V+ )V− 2 , N −1 (V− )V+ 2 }, and 1/n

1/n

δ2 (A) = wn1/n min {N 1−1/n (V+ )V− 2 , N 1−1/n (V− )V+ 2 } if µ2 (A) ≤ 1 and δ2 (A) = min {V− 2 , V+ 2 } if µ2 (A) > 1.

4.6. Lower Bounds for the Spectral Radius

57

Theorem 4.5.1 All the eigenvalues of matrix A = (ajk )nj,k=1 lie in the union of the discs (5.1) {λ ∈ C : |λ − akk | ≤ δ2 (A)}, k = 1, ..., n. Proof:

Take A+ = D + V+ . Since A+ is triangular, σ(A+ ) = σ(D) = {akk , k = 1, ..., n}.

(5.2)

Since A − A+ = V− , Corollary 4.4.2 implies svA+ (A) ≤ N 1−1/n (V+ )[V− 2 wn ]1/n ,

(5.3)

provided that wn V− 2 ≤ N (V+ ). Replace A+ by A− = D + V− . Repeating the above procedure, we get svA− (A) ≤ N 1−1/n (V− )[V+ 2 wn ]1/n , provided that wn V+ 2 ≤ N (V− ). In addition, σ(A− ) = σ(D). These relations with (5.2) and (5.3) complete the proof in the case µ2 (A) ≤ 1. Te case µ2 (A) > 1 is similarly considered 2

4.6

Lower Bounds for the Spectral Radius

Let A = (ajk ) be an n × n-matrix. Recall that rs (A) = sup |σ(A)| is the (upper) spectral radius; α(A) = sup Re σ(A) and rl (A) = inf |σ(A)| is the inner (lower) spectral radius. Let V+ , V− be the upper and lower triangular parts of A (see Section 3.1). Denote by z(ν) the unique positive root of the equation z n (A) = ν(A)

n−1

g k (A)γn,k z n−k−1

(6.1)

k=0

where ν(A) = min{V− 2 , V+ 2 }. Theorem 4.6.1 Let A = (ajk )nj,k=1 be an n × n-matrix. Then for any k = 1, ..., n, there is an eigenvalue µ0 of A, such that |µ0 − akk | ≤ z(ν).

(6.2)

Moreover, the following inequalities are true: rs (A) ≥ max{0, max |akk | − z(ν)}, k=1,...,n

rl (A) ≤

min |akk | + z(ν),

k=1,...,n

(6.3) (6.4)

and α(A) ≥ max Re akk − z(ν). k=1,...,n

(6.5)

58

4. Eigenvalues of Finite Matrices

Proof: Take A+ = D + V+ . So relation (5.2) holds and A − A+ = V− . Theorem 4.3.1 gives us the inequality svA (A+ ) ≤ z− ,

(6.6)

where z− is the extreme right-hand root of the equation z n = V− 2

n−1

γn,j z n−j−1 g j (A).

j=0

Replace A+ by A− = D + V− . Repeating the same arguments, we get svA (A− ) ≤ z+ ,

(6.7)

where z+ is the extreme right-hand root of the equation z n = V+ 2

n−1

γn,j z n−j−1 g j (A).

j=0

Relations (6.6) and (6.7) imply (6.2). Furthermore, take µ in such a way that |µ| = rs (D). Then due to (6.2), there is µ0 ∈ σ(A), such that |µ0 | ≥ rs (D) − z(ν). Hence, (6.3) follows. Similarly, inequality (6.4) can be proved. Now take µ in such a way that Re µ = α(D). Due to (6.2) for some µ0 ∈ σ(A), |Re µ0 − α(D)| ≤ z(ν). So, either Re µ0 ≥ α(D), or Re µ0 ≥ α(D) − z(ν). Thus, inequality (6.5) is also proved. The proof is complete. 2 Again put n−1 wn = γn,j . j=0

Setting z = g(A)y in (6.1) and applying Lemma 1.6.1, we obtain the estimate z(ν) ≤ δn (A), where if ν(A)wn ≥ g(A), ν(A)wn δn (A) = . 1−1/n 1/n g (A)[ν(A)wn ] if ν(A)wn ≤ g(A) Now Theorem 4.6.1 ensures the following result. Corollary 4.6.2 For a matrix A = (ajk )nj,k=1 , the inequalities rs (A) ≥ max |akk | − δn (A), k=1,...,n

rl (A) ≤

min |akk | + δn (A),

k=1,...,n

α(A) ≥ max Re akk − δn (A) k=1,...,n

are valid.

(6.8)

(6.9)

4.7. Additional Bounds

4.7

59

Additional Bounds for Eigenvalues

In the present section instead of the Euclidean norm we use the norm .∞ . Besides, we derive additional bounds for eigenvalues. Let A be an n × n-matrix with entries ajk (j, k = 1, ..., n), again. Denote qup = max

j−1

j=2,...,n

|ajk |, qlow =

k=1

max

j=1,...,n−1

n

|ajk |.

j+1

Recall that v˜k , w ˜k , mup (A) and mlow (A) are deﬁned in Section 3.4. Without lossing the generality, assume that min {qlow mup (A), qup mlow (A)} ≤ 1.

(7.1)

Theorem 4.7.1 Under condition (1.1), all the eigenvalues of A lie in the union of the discs {λ ∈ C : |λ − akk | ≤ δ∞ (A)}, k = 1, ..., n,

where δ∞ (A) :=

n

min{qlow mup (A), qup mlow (A)}.

The proof of this theorem is presented in the next section. If A is a triangular matrix, then Theorem 4.7.1 gives the exact relation σ(A) = {akk , k = 1, ..., n}. Moreover, we have Corollary 4.7.2 Under condition (1.1), the spectral radius rs (A) of A satisﬁes the inequality rs (A) ≤ max |akk | + δ∞ (A). k=1,...,n

Furthermore, consider the quantity s(A) ≡ max |λi (A) − λj (A)|. i,j

According to Theorem 4.7.1, for arbitrary λi (A), λj (A), there are akk , amm , such that |λi (A) − akk | ≤ δ∞ (A), |λj (A) − amm | ≤ δ∞ (A). Consequently, |λi (A) − λj (A)| ≤ |λi (A) − akk | + |λj (A) − amm | + s(D) ≤ s(D) + 2δ∞ (A) (i, j ≤ n), where s(D) = maxi,j |aii − ajj |. So we have

60

4. Eigenvalues of Finite Matrices

Corollary 4.7.3 Under condition (1.1), s(A) ≤ s(D) + 2δ∞ (A). According to Theorem 4.7.1, Re λk (A) ≤ maxj Re ajj + δ∞ (A). We thus get Corollary 4.7.4 Under (7.1), let maxj Re ajj + δ∞ (A) < 0. Then A is a Hurwitz matrix. These results supplement the well known theorems by Hirsh and Rochrbach (Marcus and Minc, 1964, Sections III.1.3 and Section III.3.1).

4.8

Proof of Theorem 4.7.1

Recall that V± and D are deﬁned in Section 3.1. Lemma 4.8.1 The following estimate is true: −1

Rλ (D + V+ )∞ ≤ ρ

(D, λ)

n

[1 +

k=2

v˜k ], |akk − λ|

where ρ(D, λ) = mink=1,...n |akk − λ|. Proof: 2.10.1,

Clearly, Rλ (D + V+ ) = Rλ (D)(I + Vup Rλ (D))−1 . Due to Lemma (I + V+ Rλ (D))−1 ∞ ≤

n

[1 +

k=2

v˜k ], |akk − λ|

since V+ Rλ (D) is nilpotent and its entries are ajk (akk − λ)−1 . Hence, the required result follows. 2 Lemma 4.8.2 Each eigenvalue µ of the matrix A = (ajk ) satisﬁes the inequality n v˜k (1 + 1 ≤ qlow ρ−1 (D, µ) ). |akk − µ| k=2

Proof:

Take B = V+ + D. We have A − B∞ = V− ∞ = max

j=2,...,n

j−1

|ajk | = qlow .

k=1

Lemma 4.1.1 and Lemma 4.8.1 imply 1 ≤ qlow Rµ (D + V+ ) ≤ qup ρ−1 (D, µ)

n k=2

for any µ ∈ σ(A), as claimed. 2

(1 +

v˜k ) |akk − λ|

4.8. Proof of Theorem 4.7.1

61

Lemma 4.8.3 All the eigenvalues of matrix A = (ajk )nj,k=1 lie in the set ∪nj=2 Ω(akk , z(A)), where Ω(akk , z(A)) = {λ ∈ C : |λ − akk | ≤ z(A)} (k = 1, ..., n) and z(A) is the extreme right-hand (unique positive and simple) root of the algebraic equation n z n = qlow (z + v˜k ). (8.1) k=2

Proof: Since |akk − λ| ≥ ρ(D, λ) for all k = 1, ..., n, Lemma 4.8.2 implies the inequality 1 ≤ qlow δ −1 (D, µ)

n

(1 + v˜k )ρ−1 (D, µ))

(8.2)

k=2

for any eigenvalues µ of A. Dividing the algebraic equation (8.1) by z n we can write down n (1 + v˜k z −1 ). 1 = qlow z −1 k=2

Comparing this with (8.2) and taking into account that z(A) is the extreme right root of (8.1) we obtain ρ(A, λ) ≤ z(A), as claimed. 2 Proof of Theorem 4.7.1: By Lemma 1.6.1, z(A) ≤ n qlow mup (A) if qlow mup (A) ≤ 1. Then due to Lemma 4.8.3 all the eigenvalues of A lie in the union of the discs {λ ∈ C : |λ − akk | ≤ n qlow mup (A) } , k = 1, ..., n. Replace A by the adjoint matrix A∗ , and repeat our arguments. We can assert that all the eigenvalues of A∗ lie in the union of the discs {λ ∈ C : |λ − akk | ≤ n qup mlow (A) } ; k = 1, ..., n, provided that qup mlow (A) ≤ 1. Since σ(A) = σ(A∗ ), we get the required result. 2

62

4.9

4. Eigenvalues of Finite Matrices

Notes

Recall that, for nonnegative matrices, Frobenius has derived the following lower estimate: n rs (A) ≥ r˜(A) ≡ min ajk , (9.1) j=1,...,n

k=1

cf. (Marcus and Minc, 1964, Chapter 3, Section 3.1). Relation (6.8) improves estimate (9.1) in the case ajk ≥ 0 (j, k = 1, ..., n), provided that maxk akk − δn (A) > r˜(A). That is, (6.8) is sharper than (9.1) for matrices which are close to triangular ones, since δn (A) → 0, when V− → 0 or V+ → 0. Due to inequality (6.9), matrix A is unstable, when max Re akk − z(ν) > 0.

k=1,...,n

The latter result supplements the Rorhbach theorem (Marcus and Minc, 1964, Chapter 3, Section 3.3.3). A lot of papers and books are devoted to bounds for m(A, B), cf. (Stewart and Sun, 1990), (Bhatia, 1987), (Elsner, 1985), (Phillips, 1990 and 1991) and references therein. One of the recent results belongs to R. Bhatia, L. Elsner and G. Krause (1990). They have proved that m(A, B) ≤ 22−1/n q 1/n (A2 + B2 )1−1/n .

(9.2)

In the paper (Farid, 1992) that inequality was improved in the case when both A and B are normal matrices with spectra on two intersecting lines. Corollary 4.4.6 improves (9.2), if A is close to a triangular matrix. The contents of Sections 4.2 and 4.3 are taken from (Gil’, 1997) while the results of Sections 4.4 and 4.5 are adapted from (Gil’, 2001). The material in Section 4.6 is taken from (Gil’, 1995). About other perturbation results see, for instance, the books (Kato, 1966) and (Baumgartel, 1985).

References [1] Baumgartel, H. (1985). Analytic Perturbation Theory for Matrices and Operators. Operator Theory, Advances and Appl., 52. Birkhauser Verlag, Basel, Boston, Stuttgart [2] Bhatia, R. (1987). Perturbation Bounds for Matrix Eigenvalues, Pitman Res. Notes Math. 162, Longman Scientiﬁc and Technical, Essex, U.K. [3] Bhatia R., Elsner, L. and Krause, G. (1990). Bounds for variation of the roots of a polynomial and the eigenvalues of a matrix. Linear Algebra and Appl. 142, 195-209

4.9. Notes

63

[4] Elsner, L. (1985). On optimal bound for the spectral variation of two matrices. Linear Algebra and Appl. 71, 77-80. [5] Farid, F. 0. (1992). The spectral variation for two matrices with spectra on two intersecting lines, Linear Algebra and Appl. 177: 251-273. [6] Gil’, M.I. (1995). Norm Estimations for Operator-Valued Functions and Applications. Marcel Dekker, Inc. New York. [7] Gil’, M.I. (1997). A nonsingularity criterion for matrices, Linear Algebra Appl. 253, 79-87. [8] Gil’, M.I. (2001). Invertibility and positive invertibility of matrices, Linear Algebra and Appl., 327, 95-104 [9] Kato, T. (1966). Perturbation Theory for Linear Operators, SpringerVerlag. New York. [10] Marcus M. and Minc, H. (1964). A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston. [11] Phillips, D. (1990). Improving spectral-variation bound with Chebyshev polynomials, Linear Algebra Appl., 133, 165-173 [12] Phillips, D. (1991). Resolvent bounds and spectral variation, Linear Algebra Appl., 149, 35-40 [13] Stewart, G. W. and Ji-guang Sun, (1990). Matrix Perturbation Theory, Academic Press, New York.

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5. Block Matrices and π-Triangular Matrices

The present chapter is devoted to block matrices. In particular, we derive the invertibility conditions, which supplement the generalized Hadamard criterion and some other well-known results for block matrices.

5.1

Invertibility of Block Matrices

Let an n × n-matrix A = (ajk )nj,k=1 be partitioned in the following manner: 

A11  A21 A=  . Am1

A12 A22 ... Am2

 . . . A1m . . . A2m  , . .  . . . Amm

(1.1)

where m < n, Ajk are matrices. Again I = In is the unit operator in Cn and A∞ = max

j=1,...,n

n

|ajk |.

k=1

Let the diagonal blocks Akk be invertible. Denote vkup =

max

j=1,2,...,k−1

Ajk A−1 kk ∞ (k = 2, ..., m),

and vklow =

max

j=k+1,...,m

Ajk A−1 kk ∞ (k = 1, ..., m − 1).

M.I. Gil’: LNM 1830, pp. 65–74, 2003. c Springer-Verlag Berlin Heidelberg 2003

66

5. Block Matrices

Theorem 5.1.1 Let the diagonal blocks Akk (k = 1, ..., m) be invertible. In addition, with the notations (1 + vkup ), Mlow ≡ (1 + vklow ), Mup ≡ 2≤k≤m

1≤k≤m−1

let the condition Mlow Mup < Mlow + Mup

(1.2)

hold. Then matrix A deﬁned by (1.1) is invertible. Moreover, A−1 ∞ ≤

maxk A−1 kk ∞ Mlow Mup . Mlow + Mup − Mlow Mup

(1.3)

The proof of this theorem is divided into a series of lemmas, which are presented in Sections 5.2-5.5. Theorem 5.1.1 is exact in the following sense. Let the matrix A in (1.1) be upper block triangular and Akk be invertible. Then Mlow = 1 and condition (1.2) takes the form Mup < 1 + Mup . Thus, due to Theorem 5.1.1, A is invertible. We have the same result if the matrix in (1.1) is lower block triangular. Consider a block matrix with m = 2: A11 A12 A= . (1.4) A21 A22 Then In addition, Assume that

low = A21 A−1 v2up = A12 A−1 22 ∞ , v1 11 ∞ .

Mup = 1 + v2up and Mlow = 1 + v1low . (1 + v2up )(1 + v1low ) < 2 + v2up + v1low ,

or, equivalently, that −1 A12 A−1 22 ∞ A21 A11 ∞ < 1.

(1.5)

Then due to Theorem 5.1.1, the matrix in (1.4) is invertible. Moreover, A−1 ∞ ≤

maxk=1,2 A−1 kk ∞ Mlow Mup . Mlow + Mup − Mlow Mup

Now assume that m is even: m = 2m0 with a natural m0 , and Ajk are 2 × 2 matrices: a2j−1,2k−1 a2j−1,2k Ajk = a2j−1,2k a2j,2k

5.2. π-Triangular Matrices

67

with j, k = 1, ..., m0 . Take into account that −a2k−1,2k a2k,2k −1 = d A−1 kk k −a2k,2k−1 a2k−1,2k−1 with dk = a2k−1,2k−1 a2k,2k − a2k,2k−1 a2k−1,2k . Thus, the quantities vkup , vklow are simple to calculate. Now relation (1.2) yields the invertibility, and (1.3) gives the estimate for the inverse matrix. In Section 5.4 below, we show that the matrix in (1.4) is invertible provided −1 (1.6) A12 A−1 22 A21 A11 < 1 with an arbitrary matrix norm . in Cn .

5.2

π-Triangular Matrices

Let B(Cn ) be the set of all linear operators in Cn . In what follows π = {Pk , k = 0, ..., m ≤ n} is a chain of orthogonal projectors Pk in Cn , such that 0 = P0 ⊂ P1 ⊂ .... ⊂ Pm = In . The relation Pk−1 ⊂ Pk means that Pk−1 Cn ⊂ Pk Cn (k = 1, ..., m). Let operators A, D, V ∈ B(Cn ) satisfy the relations APk = Pk APk (k = 1, ..., m),

(2.1)

DPk = Pk D (k = 1, ..., m),

(2.2)

V Pk = Pk−1 V Pk (k = 2, ..., m); V P1 = 0.

(2.3)

Then A, D and V will be called a π-triangular operator, a π- diagonal one and a π-nilpotent one, respectively Since V m = V m Pm = V m−1 Pm−1 V = V m−2 Pm−2 V Pm−1 V = ... = V P1 ...V Pm−2 V Pm−1 V, we have

V m = 0.

That is, every π-nilpotent operator is a nilpotent operator. Denote ∆Pk = Pk − Pk−1 ; Vk = V ∆Pk (k = 1, ..., m).

(2.4)

68

5. Block Matrices

Lemma 5.2.1 Let A be π-triangular. Then A = D + V,

(2.5)

where V is a π-nilpotent operator and D is a π-diagonal one. Proof:

Clearly, A=

m

∆Pj A

j=1

m

∆Pk =

k m

∆Pj A∆Pk =

k=1 j=1

k=1

m

Pk A∆Pk .

k=1

Hence (2.5) is valid with D=

m

∆Pk A∆Pk ,

(2.6)

k=1

and V =

m

Pk−1 A∆Pk =

k=2

2

m

Vk .

(2.7)

k=2

Deﬁnition 5.2.2 Let A ∈ B(Cn ) be a π-triangular operator and suppose (2.5) holds. Then the π-diagonal operator D and π-nilpotent operator V will be called the π-diagonal part and π-nilpotent part of A, respectively. n Lemma 5.2.3 Let π = {Pk }m k=1 be a chain of orthogonal projectors in C . If V˜ is a π-nilpotent operator, and A is a π-triangular one, then both operators AV˜ and V˜ A are π-nilpotent ones.

Proof:

By (2.1) and (2.3) we get

V˜ APk = V˜ Pk APk = Pk−1 V˜ Pk APk = Pk−1 V˜ APk , k = 1, ..., m. That is, V˜ A is indeed a π-nilpotent operator. Similarly we can prove that AV˜ is a π-nilpotent operator. 2 Lemma 5.2.4 Let A ∈ B(Cn ) be a π- triangular operator and D be its π-diagonal part. Then the spectrum of A coincides with the spectrum of D. Proof:

Due to (2.5)

Rλ (A) = (A − λI)−1 = (D + V − λI)−1 = Rλ (D)(I + V Rλ (D))−1 . (2.8) According to Lemma 5.2.3, V Rλ (D) is π- nilpotent if λ is not an eigenvalue of D. Therefore, Rλ (A) = Rλ (D)(I + V Rλ (D))−1 =

m−1 k=0

Rλ (D)(−V Rλ (D))k .

5.3. Multiplicative Representation of Resolvents

69

This relation implies that λ is a regular point of A if it is a regular point of D. Conversely, let λ ∈ σ(A). Due to (2.5) Rλ (D) = (D − λI)−1 = (A − V − λI)−1 = Rλ (A)(I − V Rλ (A))−1 . According to Lemma 5.2.3, V Rλ (A) is π- nilpotent. Therefore, Rλ (D) =

m−1

Rλ (A)(V Rλ (A))k .

k=0

So λ ∈ σ(D), provided let λ ∈ σ(A). The lemma is proved. 2

5.3

Multiplicative Representation of Resolvents of π-Triangular Operators

For X1 , X2 , ..., Xm ∈ B(Cn ) and j < m, again denote →

Xk ≡ Xj Xj+1 ...Xm .

j≤k≤m n (m ≤ Lemma 5.3.1 Let π = {Pk }m k=1 be a chain of orthogonal projectors in C n), V be a π-nilpotent operator. Then

(I − V )−1 =

→

(I + V ∆Pk ).

(3.1)

2≤k≤m

Proof:

According to (2.4) (I − V )−1 =

m−1

V k.

(3.2)

k=0

On the other hand, →

(I + V ∆Pk ) = I +

2≤k≤m

m

Vk +

k=2

V k1 V k 2

2≤k1
+... + V2 V3 ...Vm . Here, as above, Vk = V ∆Pk . However, V k1 V k2 = V 2≤k1
2≤k1
∆Pk1 V ∆Pk2 =

70

5. Block Matrices

V

3≤k2 ≤m

Similarly,

Pk1 −1 V ∆Pk2 = V 2

∆Pk2 = V 2 .

3≤k2 ≤m

Vk1 Vk2 ...Vkj = V j

2≤k1
for j < m. Thus from (3.2) follows (3.1). This is the desired conclusion. 2. Theorem 5.3.2 For any π- triangular operator A and a regular λ ∈ C →

Rλ (A) = (D − λI)−1

(I − V ∆Pk (D − λI)−1 ∆Pk ),

2≤k≤m

where D and V are the π-diagonal and π-nilpotent parts of A, respectively. Proof: gives

Due to Lemma 5.2.3, V Rλ (D) is π-nilpotent. Now Lemma 5.3.1 −1

(I + V Rλ (D))

=

→

(I − V Rλ (D)∆Pk ).

2≤k≤m

But Rλ (D)∆Pk = ∆Pk Rλ (D). This proves the result. 2

5.4

Invertibility with Respect to a Chain of Projectors

Again, let π = {Pk , k = 0, ..., m} be a chain of orthogonal projectors Pk . Denote the chain of the complementary projectors by π ˜: P˜k = In − Pm−k and π ˜ = {In − Pm−k , k = 0, ..., m}. In the present section, V is a π-nilpotent operator, D is a π-diagonal one, and W is a π ˜ -nilpotent operator. Lemma 5.4.1 Any operator A ∈ B(Cn ) admits the representation A = D + V + W. Proof:

Clearly, A=

m j=1

∆Pj A

m k=1

∆Pk .

(4.1)

5.4. Invertibility with Respect to a Chain

71

Hence, (4.1) holds, where D and V are deﬁned by (2.6) and (2.7), and W =

m m

∆Pj A∆Pk =

k=1 j=k+1

m

P˜k−1 A∆P˜k

k=2

with ∆P˜k = P˜k − P˜k−1 . 2 Let . be an arbitrary norm in Cn . Lemma 5.4.2 Let the π-diagonal matrix D be invertible. In addition, with the notations VA ≡ V D−1 , WA ≡ W D−1 , let the condition θ := VA (I + VA )−1 (I + WA )−1 WA < 1

(4.2)

hold. Then the operator A deﬁned by (4.1) is invertible. Moreover, A−1 ≤ (I + VA )−1 D−1 (I + WA )−1 (1 − θ)−1 . Proof:

(4.3)

Due to Lemma 5.4.1,

A = D + V + W = (I + VA + WA )D = [(I + VA )(I + WA ) − VA WA ]D. Clearly, VA and WA are nilpotent matrices and, consequently, the matrices, I + VA and I + WA are invertible. Thus, A = (I + VA )(I − BA )(I + WA )D, where

(4.4)

BA = (I + VA )−1 VA WA (I + WA )−1 .

Condition (4.2) yields (I − BA )−1 ≤ (1 − θ)−1 . Therefore, (4.2) provides the invertibility of A. Moreover, according to (4.4), inequality (4.3) is valid. 2 Corollary 5.4.3 Under condition (1.6), the matrix in (1.4) is invertible. Indeed, under the consideration, VA2 = 0, WA2 = 0. So VA (I + VA )−1 = VA and WA (I + WA )−1 = WA . Hence θ = VA WA . Now Lemma 5.4.2 yields the required result. Furthermore, Lemmas 5.3.1 and 5.2.3 yield the following:

72

5. Block Matrices

Lemma 5.4.4 The inequalities

(I + VA )−1 ≤ M (VA ) :=

(1 + V D−1 ∆Pk )

2≤k≤m

and

(I + WA )−1 ≤ M (WA ) :=

(1 + W D−1 ∆Pk )

1≤k≤m−1

are valid.

5.5

Proof of Theorem 5.1.1

In the present section, D, V and W are the diagonal, upper diagonal and lower diagonal parts of the matrix in (1.1), respectively, that is,     0 A12 . . . A1m A11 0 ... 0   0 0  0 . . . A2m  A22 . . . , V =  0  , (5.1) D=  . ...  . . .  ... . .  0 0 ... 0 0 0 . . . Amm and



0  A21  W = . Am1

0 0 ... Am2

 ... 0 ... 0  . . .  ... 0

(5.2)

Recall that VA ≡ V D−1 , WA ≡ W D−1 . Lemma 5.5.1 Let D be invertible. Then the inequalities

and

(I + VA )−1 ∞ ≤ Mup

(5.3)

(I + WA )−1 ∞ ≤ Mlow

(5.4)

are valid. Proof: Let π ˆ = {Pˆk , k = 1, .., m} be the chain of the projectors onto the standard basis. That is, for an h = (hk ) ∈ Cn , Pˆk h = (h1 , ...., hνk , ..., 0), ˆ -diagonal and where νk = dim Pˆk . Then according to (5.1) D and V are π π ˆ -nilpotent operators, respectively. Moreover, ˆ V D−1 ∆Pˆk = Vk A−1 kk ∆Pk =

k−1 j=1

ˆ Ajk A−1 kk ∆Pk .

5.5. Proof of Theorem 5.1.1

73

But, clearly,

k−1 j=1

up −1 ˆ Ajk A−1 kk ∆Pk ∞ = max Ajk Akk ∞ = vk . j

Therefore inequality (5.3) is due to the previous lemma. Inequality (5.4) can be similarly proved. 2 Lemma 5.5.2 Let D be invertible. Then the inequalities

and

VA (I + VA )−1 ∞ ≤ Mup − 1

(5.5)

WA (I + WA )−1 ∞ ≤ Mlow − 1

(5.6)

are valid. Proof:

Let B = (bjk )nk=1 be a positive matrix with the property Bh ≥ h

(5.7)

for any nonnegative h ∈ Cn . Then B − I∞ = max [ j=1,...,n

n

bjk − δjk ] = B∞ − 1.

(5.8)

k=1

Here δjk is the Kronecker symbol. Furthermore, since VA is nilpotent, VA (I + VA )−1 ∞ ≤

n−1

|VA |k ∞ = (I − |VA |)−1 − I∞ ,

k=1

where |VA | is the matrix whose entries are the absolute values of the entries of VA . Moreover, n−1 |VA |k h ≥ h k=0

for any nonnegative h ∈ Cn . So according to (5.7) and (5.8) VA (I + VA )−1 ∞ ≤ (I − |VA |)−1 ∞ − 1. Since |VA |∆Pˆk ∞ = VA ∆Pˆk ∞ , inequality (5.3) with −|VA | instead of VA yields inequality (5.5). Inequality (5.6) can be proved similarly. 2 The assertion of Theorem 5.1.1 follows from Lemmas 5.4.2, 5.5.1 and 5.5.2.

74

5.6

5. Block Matrices

Notes

Many books and papers are devoted to the invertibility of block matrices, cf. (Feingold and Varga, 1962), (Fiedler, 1960), (Gantmacher, 1967), (Ostrowski, 1961), etc. In these works, the Hadamard theorem is mainly generalized to block matrices. Note that the generalized Hadamard theorem does not assert that a block triangular matrix with nonsingular diagonal blocks is invertible. But it is not hard to check that such a matrix is always invertible. Theorem 5.1.1 gives us the invertibility conditions which improve wellknown results for matrices that are ”close” to block triangular matrices. Moreover, we derive an estimate for the norm of the inverse matrices. Chapter 5 is based on the paper (Gil’, 2002).

References [1] Feingold D.G. and Varga, R.S. (1962). Block diagonally dominant matrices and generalization of the Gershgorin circle theorem. Paciﬁc J. of Mathematics, 12, 1241-1250. [2] Fiedler, M. (1960) Some estimates of spectra of matrices. Symposium of the Numerical Treatment of Ordinary Diﬀerential Equations, Integral and Integro- Diﬀerential Equations, Birkh´ auser Verlag, Rome, 33-36. [3] Gantmacher, F. R. (1967). Theory of Matrices, Nauka, Moscow (in Russian). [4] Gil’, M.I. (2002). Invertibility and spectrum localization of nonselfadjoint operators, Adv. Appl. Mathematics, 28, 40-58. [5] Horn, R. A. and Johnson, C. R. (1991). Topics in Matrix Analysis , Cambridge University Press, Cambridge. [6] Ostrowski, A.M. (1961). On some metrical properties of operator matrices and matrices partitioned into blocks. J. Math. Ann. Appl., 2, 161209.

6. Norm Estimates for Functions of Compact Operators in a Hilbert Space

The present chapter contains the estimates for the norms of the resolvents and analytic functions of Hilbert-Schmidt operators and resolvents of von Neumann-Schatten operators.

6.1

Bounded Operators in a Hilbert Space

In this section we recall very brieﬂy some basic notions of the theory of operators in a Hilbert space. More details can be found in any textbook on Hilbert spaces (e.g. (Ahiezer and Glazman, 1981), (Dunford and Schwartz, 1963) ). In the sequel H denotes a separable Hilbert space with a scalar product (., .) and the norm h = (h, h) (h ∈ H). A sequence {hn } of elements of H converges strongly (in the norm) to h ∈ H if hn − h → 0 as n → ∞. Any separable Hilbert space possesses an orthonormal basis. This means that there is a sequence {ek ∈ H}, such that (ek , ej ) = 0 if j = k and (ek , ek ) = 1 (j, k = 1, 2, ...) M.I. Gil’: LNM 1830, pp. 75–96, 2003. c Springer-Verlag Berlin Heidelberg 2003

76

6. Functions of Compact Operators

and any h ∈ H can be represented as h=

∞

ck ek

k=1

with ck = (h, ek ) (k = 1, 2, . . .). In addition, this series strongly converges. If the closed linear span of vectors {vk ∈ H}∞ k=1 coincides with H, then the set of these vectors is said to be complete in H. A linear operator A acting in H is called a bounded one, if there is a constant a such that Ah ≤ ah for all h ∈ H. The quantity A = sup

h∈H

Ah h

is called the norm of A. A sequence {An } of bounded linear operators converges strongly to an operator A, if the sequence of elements {An h} strongly converges to Ah for every h ∈ H. {An } converges in the uniform operator topology (in the operator norm ) to an operator A, if An − A → 0 as n → ∞. A bounded linear operator A∗ is called adjoint to A, if (Af, g) = (f, A∗ g) for every h, g ∈ H. The relation A = A∗ is true. A bounded operator A is a selfadjoint one, if A = A∗ . A is a unitary operator, if AA∗ = A∗ A = I. Here and below I ≡ IH is the identity operator in H. A selfadjoint operator A is positive (negative) deﬁnite, if (Ah, h) ≥ 0 ((Ah, h) ≤ 0) for every h ∈ H. A selfadjoint operator A is strongly positive (strongly negative) deﬁnite, if there is a constant c > 0, such that (Ah, h) ≥ c (h, h) ((Ah, h) < −c (h, h)) for every h ∈ H. A bounded linear operator satisfying the relation AA∗ = A∗ A is called a normal operator. It is clear that unitary and selfadjoint operators are examples of normal ones. The operator B ≡ A−1 is the inverse one to A, if AB = BA = I. An operator P is called a projector if P 2 = P . If, in addition, P ∗ = P , then it is called an orthogonal projector (an orthoprojector). A point λ of the complex plane is said to be a regular point of an operator A, if the operator Rλ (A) ≡ (A − Iλ)−1 (the resolvent) exists and is bounded. The complement of all regular points of A in the complex plane is the spectrum of A. The spectrum of A is denoted by σ(A). The spectrum of a selfadjoint operator is real, the spectrum of a unitary operator lies on the unit circle.

6.2. Compact Operators in a Hilbert Space

77

The quantity rs (A) = sups∈σ(A) |s| is the spectral radius of A. An operator V is called a quasinilpotent one, if its spectrum consists of zero, only. If there is a nontrivial solution e of the equation Ae = λ(A)e, where λ(A) is a number, then this number is called an eigenvalue of operator A, and e ∈ H is an eigenvector corresponding to λ(A). Any eigenvalue is a point of the spectrum. An eigenvalue λ(A) has the (algebraic) multiplicity r ≤ ∞ if k dim(∪∞ k=1 ker(A − λ(A)I) ) = r. In the sequel λk (A), k = 1, 2, ... are the eigenvalues of A repeated according to their multiplicities. A vector v satisfying (A − λ(A)I)n v = 0 for a natural n, is a root vector of operator A corresponding to λ(A).

6.2

Compact Operators in a Hilbert Space

All the results, presented in this section can be found, for instance, in (Gohberg and Krein, 1969, Chapters 2 and 3). The set of all linear completely continuous (compact) operators in H is deﬁned by C∞ . Recall that the spectrum of an operator from C∞ is either ﬁnite, or the sequence of the eigenvalues of A converges to zero, any nonzero eigenvalue has the ﬁnite multiplicity. Moreover, any normal operator A ∈ C∞ can be represented in the form ∞ A= λk (A)Ek , (2.1) k=1

where Ek are eigenprojectors of A, i.e. the projectors deﬁned by Ek h = (h, dk )dk for all h ∈ H. Here dk are the normal eigenvectors of A. Recall that eigenvectors of normal operators are mutually orthogonal. A completely continuous positive deﬁnite selfadjoint operator has non-negative eigenvalues, only. Let A ∈ C∞ be positive deﬁnite and represented by (2.1). Then we write ∞ β A := λβk (A)Ek (β > 0). k=1

A completely continuous quasinilpotent operator is called a Volterra operator. Let {ek } be an orthogonal normal basis in H, and the series ∞ k=1

(Aek , ek ) (A ∈ C∞ )

78

6. Functions of Compact Operators

converges. Then the sum of this series is called the trace of A: T race A = T r A =

∞

(Aek , ek ).

k=1

An operator A satisfying the condition T r (A∗ A)1/2 < ∞ is called a nuclear operator. An operator A, satisfying the relation T r (A∗ A) < ∞ is said to be a Hilbert-Schmidt operator. The eigenvalues λk ((A∗ A)1/2 ) (k = 1, 2, ...) of the operator (A∗ A)1/2 are called the singular numbers (s-numbers) of A and are denoted by sk (A). That is, sk (A) ≡ λk ((A∗ A)1/2 ) (k = 1, 2, ...). Enumerate singular numbers of A taking into account their multiplicity and in decreasing order. The set of completely continuous operators acting in a Hilbert space and satisfying the condition Np (A) := [

∞ k=1

spk (A) ]1/p < ∞,

for some p ≥ 1, is called the von Neumann - Schatten ideal and is denoted by Cp . Np (.) is called the norm of the ideal Cp . It is not hard to show that Np (A) = p T r (AA∗ )p/2 . Thus, C1 is the ideal of nuclear operators (the Trace class) and C2 is the ideal of Hilbert-Schmidt operators. N2 (A) is called the Hilbert-Schmidt norm. Sometimes we will omit index 2 of the Hilbert-Schmidt norm, i.e. N (A) := N2 (A) = T r (A∗ A). For any orthogonal normal basis {ek } we can write N2 (A) = (

∞

Aek 2 )1/2 .

k=1

This equality is equivalent to the following one: N2 (A) = (

∞

j,k=1

|ajk |2 )1/2 ,

(2.2)

6.3. Triangular Representations

79

where ajk = (Aek , ej ) (j, k = 1, 2, . . .) are entries of a Hilbert-Schmidt operator A in basis {ek }. For all p ≥ 1, the following propositions are true (the proofs can be found in the books (Gohberg and Krein, 1969, Section 3.7), and (Pietsch, 1988)): If A ∈ Cp , then also A∗ ∈ Cp . If A ∈ Cp and B is a bounded linear operator, then both AB and BA belong to Cp . Moreover, Np (AB) ≤ Np (A)B and Np (BA) ≤ Np (A)B. In addition, the inequality n j=1

|λj (A)|p ≤

n j=1

spj (A) (n = 1, 2, . . .)

(2.3)

is valid, cf. (Gohberg and Krein, 1969, Theorem II.3.1). Lemma 6.2.1 If A ∈ Cp and B ∈ Cq (1 < p, q < ∞), then AB ∈ Cs with 1 1 1 = + . s p q Moreover, Ns (AB) ≤ Np (A)Nq (B).

(2.4)

For the proof of this lemma see (Gohberg and Krein, 1969, Section III.7). We need also the following result (Lidskij’s theorem). Theorem 6.2.2 The trace of A ∈ C1 does not depend on a choice of an orthogonal normal basis and Tr A =

∞

λk (A).

k=1

The proof of this theorem can be found in (Gohberg and Krein, 1969, Section III.8).

6.3

Triangular Representations of Compact Operators

Let R0 be a set in the complex plane and let > 0. By S(R0 , ) we denote the -neighborhood of R0 . That is, dist{R0 , S(R0 , )} ≤ .

80

6. Functions of Compact Operators

Lemma 6.3.1 Let A be a bounded operator and let > 0. Then there is a δ > 0, such that, if a bounded operator B satisﬁes the condition A − B ≤ δ, then σ(B) lies in S(σ(A), ) and Rλ (A) − Rλ (B) ≤ for any λ, which does not belong to S(σ(A), ). For the proof of this lemma we refer the reader to the book (Dunford and Schwartz, 1963, p. 585). Lemma 6.3.2 Let V ∈ Cp , p > 1 be a Volterra operator. Then there is a sequence of nilpotent operators, having ﬁnite dimensional ranges and converging to V in the norm Np (.). Proof: Let T = V − V ∗ . Due to the well-known Theorems 22.1 and 16.3 from the book (Brodskii, 1971), for an > 0, there is a ﬁnite chain {Pk }nk=0 of orthogonal projectors onto invariant subspaces of V : 0 = Range(P0 ) ⊂ Range(P1 ) ⊂ .... ⊂ Range(Pn ) = H, such that with the notation n Pk−1 T ∆Pk (∆Pk = Pk − Pk−1 ), Wn = k=1 (k)

the inequality Np (Wn − V ) < is valid. Furthermore, let {em }∞ m=1 be an orthonormal basis in ∆Pk H. Put (k)

Ql (k)

Clearly, Ql

=

l

(k) (., e(k) m )em (k = 1, ..., n; l = 1, 2, ....).

m=1

strongly converge to ∆Pk as l → ∞. Moreover, (k)

(k)

Ql ∆Pk = ∆Pk Ql Since, Wn =

n k−1

(k)

= Ql .

∆Pj T ∆Pk ,

k=1 j=1

the operators Wnl =

n k−1 k=1 j=1

(j)

(k)

Ql T Ql

have ﬁnite dimensional ranges and tend to Wn in the norm Np as l → ∞, since T ∈ Cp . Thus, Wnl tend to V in the norm Np as l, n → ∞. Put (l)

Lk =

k j=1

(j)

Ql

(k = 1, ..., n).

6.3. Triangular Representations (l)

(l)

81

(l)

n Then Lk−1 Wnl Lk = Wnl Lk . Hence we easily have Wnl = 0. This proves the lemma. 2

We recall the following well-known result, cf. (Gohberg and Krein, 1969, Lemma I.4.2). Lemma 6.3.3 Let M = H be the closed linear span of all the root vectors of an operator A ∈ C∞ and let QA be the orthogonal projector of H onto M ⊥ , where M ⊥ is the orthogonal complement of M in H. Then QA AQA is a Volterra operator. The previous lemma means that A can be represented by the matrix BA A12 A= 0 V1

(3.1)

acting in M ⊕ M ⊥ . Here BA = A(I − QA ), V1 = QA AQA is a Volterra operator in QA H and A12 = (I − QA )AQA . Theorem 6.3.4 Let A ∈ C∞ . Then there are a normal operator D and a Volterra operator V , such that A = D + V and σ(D) = σ(A).

(3.2)

Moreover, A, D and V have the same invariant subspaces. Proof: Let M be the linear closed span of all the root vectors of A, and PA is the projector of H onto M . So the system of the root vectors of the operator BA = APA is complete in M . Thanks to the well-known Lemma I.3.1 from (Gohberg and Krein, 1969), there is an orthonormal basis (Schur’s basis) {ek } in M , such that BA ej = Aej = λj (BA )ej +

j−1

ajk ek (j = 1, 2, ...).

(3.3)

k=1

We have BA = DB + VB , where DB ek = λk (BA )ek , k = 1, 2, ... and VB = BA − DB is a quasinilpotent operator. But according to (3.1) λk (BA ) = λk (A), since V1 is a quasinilpotent operator. Moreover DB and VB have the same invariant subspaces. Take the following operator matrix acting in M ⊕ M ⊥: DB 0 VB A12 D= and V = . 0 V1 0 0 Since the diagonal of V contains VB and V1 only, σ(V ) = σ(VB )∪σ(V1 ) = {0}. So V is quasinilpotent and (3.2) is proved. From (3.1) and (3.3) it follows that A, D and V have the same invariant subspace, as claimed. 2

82

6. Functions of Compact Operators

Deﬁnition 6.3.5 Equality (3.2) is said to be the triangular representation of A. Besides, D and V will be called the diagonal part and nilpotent part of A, respectively. Lemma 6.3.6 Let A ∈ Cp , p ≥ 1. Let V be the nilpotent part of A. Then there exists a sequence {An } of operators, having n-dimensional ranges, such that σ(An ) ⊆ σ(A), (3.4) and

n

|λ(An )|p →

k=1

∞

|λ(A)|p as n → ∞.

(3.5)

k=1

Moreover, Np (An − A) → 0 and Np (Vn − V ) → 0 as n → ∞,

(3.6)

where Vn are the nilpotent parts of An (n = 1, 2, ...). Proof: Again, let M be the linear closed span of all the root vectors of A, and PA the projector of H onto M . So the system of root vectors of the operator BA = APA is complete in M . Let DB and VB be the nilpotent parts of BA , respectively. According to (3.3), put Pn =

n

(., ek )ek .

k=1

Then σ(BA Pn ) = σ(DB Pn ) = {λ1 (A), ..., λn (A)}.

(3.7)

In addition, DB Pn and VB Pn are the diagonal and nilpotent parts of BA Pn , respectively. Due to Lemma 6.3.2, there exists a sequence {Wn } of nilpotent operators having n-dimensional ranges and converging in Np to the operator V1 . Put BA Pn Pn A12 . An = 0 Wn Then the diagonal part of An is Dn = and the nilpotent part is

Vn =

DB Pn 0

V B Pn 0

0 0

Pn A12 Wn

.

So relations (3.6) are valid. According to (3.7), relation (3.5) holds. Moreover Np (Dn −DB ) → 0. So relation (3.5) is also proved. This ﬁnishes the proof. 2

6.4. Resolvents of Hilbert-Schmidt Operators

6.4

83

Resolvents of Hilbert-Schmidt Operators

Let A be a Hilbert-Schmidt operator. The following quantity plays a key role in this section: ∞ g(A) = [N22 (A) − |λk (A)|2 ]1/2 , (4.1) k=1

where N2 (A) is the Hilbert-Schmidt norm of A, again. Since ∞

2

|λk (A)| ≥ |

k=1

one can write

∞

λ2k (A)| = |T race A2 |,

k=1

g 2 (A) ≤ N22 (A) − |T race A2 |.

(4.2)

If A is a normal Hilbert-Schmidt operator, then g(A) = 0, since N22 (A)

=

∞

|λk (A)|2

k=1

in this case. Let AI = (A − A∗ )/2i. We will also prove the inequality g 2 (A) ≤

N22 (A − A∗ ) = 2N22 (AI ) 2

(4.3)

(see Lemma 6.5.2 below). Again put ρ(A, λ) := inf t∈σ(A) |λ − t|. Theorem 6.4.1 Let A be a Hilbert-Schmidt operator. Then Rλ (A) ≤

∞

g k (A) √ (λ ∈ σ(A)). ρk+1 (A, λ) k! k=0

(4.4)

Proof: Due to Lemma 6.3.6 there exists a sequence {An } of operators, having n-dimension ranges, such that the relations (3.4), N2 (An ) → N2 (A) and g(An ) → g(A) as n → ∞

(4.5)

are valid. But due to Corollary 2.1.2, Rλ (An ) ≤

n−1 k=0

g k (An ) √ (λ ∈ σ(An )). ρk+1 (An , λ) k!

According to (3.4) ρ(An , λ) ≥ ρ(A, λ). Now, letting n → ∞ in the latter relation, we arrive at the stated result. 2 An additional proof of this theorem can be found in (Gil’, 1995, Chapter 2).

84

6. Functions of Compact Operators

Theorem 6.4.1 is precise. Inequality (4.4) becomes the equality Rλ (A) = ρ−1 (A, λ), if A is a normal operator, since g(A) = 0 in this case. Note that for an arbitrary constant c > 1, Schwarz’s inequality implies the relations ∞ ∞ √ k k ∞ k 2k ∞ 2 xk c x c x 1 1/2 c √ = √ ecx /2 (x ≥ 0). = ≤ j k k! c c − 1 k! k=0 c k! j=0 k=0 k=0 (4.6) With c = 2, we have ∞ √ 2 xk √ ≤ 2 ex (x ≥ 0). k! k=0

Now Theorem 6.4.1 implies the inequality Rλ (A) ≤

b0 g 2 (A) a0 exp [ 2 ] for all regular λ, ρ(A, λ) ρ (A, λ)

(4.7)

where according to (4.6), one can take. √ c c and b0 = for any c > 1. In particular, a0 = 2 and b0 = 1. a0 = c−1 2 (4.8) Moreover, letting n → ∞ in Theorem 2.14.1, we get (4.7) with a0 = e1/2 and b0 = 1/2.

(4.9)

We thus have proved Theorem 6.4.2 Let A ∈ C2 . Then there are nonnegative constants a0 , b0 , such that estimate (4.7) is valid. Moreover, a0 and b0 can be taken as in (4.8) or in (4.9). In particular, if V ∈ C2 is a quasinilpotent operator, then Rλ (V ) ≤

6.5

a0 b0 N22 (V ) ] for all λ = 0. exp [ |λ| |λ|2

Equalities for Eigenvalues of a Hilbert-Schmidt Operator

Lemma 6.5.1 Let V be a Volterra operator and VI ≡ (V − V ∗ )/2i ∈ C2 . Then V ∈ C2 . Moreover, N22 (V ) = 2N22 (VI ).

6.5. Equalities for Eigenvalues

85

Proof: By Theorem 6.2.2 we have T race V 2 = T race (V ∗ )2 = 0, because V is a Volterra operator. Hence, N22 (V − V ∗ ) = T race (V − V ∗ )2 = T race (V 2 + V V ∗ + V ∗ V + (V ∗ )2 ) = T race (V V ∗ + V ∗ V ) = 2T race (V V ∗ ). We arrive at the result. 2 Lemma 6.5.2 Let A ∈ C2 . Then N22 (A) −

∞

|λk (A)|2 = 2N22 (AI ) − 2

k=1

∞

|Im λk (A)|2 = N22 (V ),

k=1

where V is the nilpotent part of A. Proof: Let D be the diagonal part of A. By (3.3), it is simple to check that V D∗ is a Volterra operator (see also Lemma 7.3.4). By Theorem 6.2.2, T race V D∗ = T race V ∗ D = 0.

(5.1)

From the triangular representation (3.2) it follows that T r AA∗ = T r (D + V )(D∗ + V ∗ ) = T r AA∗ = T r (DD∗ ) + T r (V V ∗ ). Besides, due to (3.2) σ(A) = σ(D). Thus, N22 (D)

=

∞

|λk (A)|2 .

k=1

So the relation N22 (V ) = N22 (A) −

∞

|λk (A)|2

k=1

is proved. Furthermore, from the triangular representation (3.2) it follows that −4T r A2I = T r (A − A∗ )2 = T r (D + V − D∗ − V ∗ )2 . Hence, thanks to (5.1), we obtain −4T r A2I = T r (D − D∗ )2 + T r (V − V ∗ )2 . That is, N22 (AI ) = N22 (VI ) + N22 (DI ), where VI = (V − V ∗ )/2i and DI = (D − D∗ )/2i. Taking into account Lemma 6.5.1, we arrive at the equality 2N22 (AI ) − 2N22 (DI ) = N22 (V ).

86

6. Functions of Compact Operators

Besides, due to (3.2) σ(A) = σ(D). Thus, N 2 (DI ) =

∞

|Im λk (A)|2 ,

k=1

and we arrive at the required result. 2 Replace in Lemma 6.5.2, operator A by Aeit and Aeiτ with real numbers t, τ . Then we get Corollary 6.5.3 Let A ∈ C2 . Then 2

it

∗ −it

N (Ae − A e

)−

∞

|eit λk (A) − e−it λk (A)|2 =

k=1

N 2 (Aeiτ − A∗ e−iτ ) −

∞

|eiτ λk (A) − e−iτ λk (A)|2 (t, τ ∈ R).

k=1

In particular, take t = 0 and τ = π/2. Then due to Corollary 6.5.3, N 2 (AI ) −

∞

|Im λk (A)|2 = N 2 (AR ) −

k=1

∞

|Re λk (A)|2

(5.2)

k=1

with AR = (A + A∗ )/2 .

6.6

Operators Having Hilbert-Schmidt Powers

Assume that for some positive integer p > 1, Ap is a Hilbert-Schmidt operator.

(6.1)

Note that under (6.1) A can, in general, be a noncompact operator. Below in this section we will give a relevant example. Theorem 6.6.1 Let (6.1) hold for some integer p > 1. Then Rλ (A) ≤ Tλ,p

∞

g k (Ap ) √ (λp ∈ σ(Ap )), k+1 (Ap , λp ) k! ρ k=0

where Tλ,p =

p−1

Ak λp−k−1 ,

k=0

and

ρ(Ap , λp ) = inf |tp − λp | t∈σ(A)

is the distance between σ(Ap ) and the point λp .

(6.2)

(6.3)

6.7. Resolvents of Neumann-Schatten Operators

Proof:

87

We use the identity Ap − Iλp = (A − Iλ)

p−1

Ak λp−k−1 = (A − Iλ)Tλ,p .

k=0

This implies (A − Iλ)−1 = Tλ,p (Ap − Iλp )−1 .

(6.4)

(A − Iλ)−1 ≤ Tλ,p (Ap − Iλp )−1 .

(6.5)

Thus, Applying Theorem 6.4.1 to the resolvent (Ap −Iλp )−1 = Rλp (Ap ), we obtain: Rλp (Ap ) ≤

∞ k=0

g k (Ap )

√

ρk+1 (Ap , λp )

k!

(λp ∈ σ(Ap )).

This and (6.4) complete the proof. 2 According to (6.5) Theorem 6.4.2 gives us Corollary 6.6.2 Let condition (6.1) hold for some integer p > 1. Then Rλ (A) ≤

b0 g 2 (Ap ) a0 Tλ,p exp [ ] (λp ∈ σ(Ap )), ρ(Ap , λp ) ρ2 (Ap , λp )

where constants a0 and b0 can be taken from (4.8) or from (4.9). Example 6.6.3 Consider a noncompact operator satisfying condition (6.1). Let H be an orthogonal sum of Hilbert spaces H1 and H2 : H = H1 ⊕ H2 , and let A be a linear operator deﬁned in H by the formula A=

B1 T 0 B2

,

where B1 and B2 are bounded linear operators acting in H1 and H2 , respectively, and a bounded linear operator T maps H2 into H1 . Evidently A2 is deﬁned by the matrix 2

A =

B12 0

B1 T + T B2 B22

.

If B1 , B2 ∈ C2 and T is a noncompact one, then A2 ∈ C2 , while A is a noncompact operator.

88

6. Functions of Compact Operators

6.7

Resolvents of Neumann-Schatten Operators

Let A ∈ C2p for some integer p > 1.

(7.1)

Then due to (2.4), condition (6.1) holds. So we can directly apply Theorem 6.6.1, but in appropriate situations the following result is more convenient. Theorem 6.7.1 Let A ∈ C2p (p = 2, 3, ...). Then Rλ (A) ≤

p−1 ∞ (2N2p (A))pk+m √ (λ ∈ σ(A)). pk+m+1 (A, λ) k! m=0 k=0 ρ

(7.2)

The proofs of this theorem and the next one are presented in the next section. An additional proof of Theorem 6.7.1 can be found in (Gil’, 1995, Section 2.6). Put 1 (p) θj = , (7.3) [j/p]! where [x] means the integer part of a real number x. Now the previous theorem implies Corollary 6.7.2 Let A ∈ C2p (p = 2, 3, ...). Then Rλ (A) ≤

(p) ∞ θj (2N2p (A))j j=0

ρj+1 (A, λ)

(λ ∈ σ(A)).

(7.4)

Theorem 6.7.3 Under condition (7.1) there are constants a0 , b0 > 0, such that the estimate Rλ (A) ≤ a0

p−1 (2N2p (A))m b0 (2N2p (A))2p exp [ ] (λ ∈ σ(A)) ρm+1 (A, λ) ρ2p (A, λ) m=0

(7.5)

holds. These constants can be taken as in (4.8) or in (4.9). Since, condition (7.1) implies AI ≡ (A − A∗ )/2i ∈ C2p , additional estimates for the resolvent under condition (7.1) are derived in Section 7.9 below.

6.8

Proofs of Theorems 6.7.1 and 6.7.3

We need the following result.

6.8. Proof of Theorems 6.7.1 and 6.7.3

89

Lemma 6.8.1 Let A be a linear operator acting in a Euclidean space Cn with n = jp and integers p ≥ 1, j > 1. Then Rλ (A) ≤

j p−1

kp+m N2p (V ) √ (λ ∈ σ(A)) pk+m+1 (A, λ) k! m=0 k=0 ρ

(8.1)

where V is the nilpotent part of A, . is the Euclidean norm. Proof:

Since A = D + V , where D is the diagonal part of A, (A − λI)−1 = (D + V − λI)−1 = (D − λI)−1 (I + Bλ )−1

(8.2)

where Bλ = −V Rλ (D). By the identity (I − Bλ )(I + Bλ + . . . + Bλp−1 ) = I − Bλp we have

(A − λI)−1 = (D + V − λI)−1 = (D − λI)−1 (I + Bλ + . . . + Bλp−1 )(I − Bλp )−1 .

Clearly, Bλ is a nilpotent operator. So Bλn = (I − Bλp )−1 =

j k=0

Bλpj

(8.3)

= 0 and

Bλkp .

Thus, (I − Bλ )−1 = (I + Bλ + . . . + Bλp−1 )(I − Bλp )−1 = Hence, Rλ (A) = Rλ (D)

p−1 j m=0 k=0

p−1 j m=0 k=0

Bλkp+m .

Bλkp+m .

(8.4)

Taking into account that σ(A) = σ(D), we can assert that Rλ (D) is a bounded operator for all regular points λ of A. But p N2 (Bλp ) ≤ N2p (Bλ )

(see relation (2.4)). Now let us use Theorem 2.5.1. It gives Bλpk ≤ γn,k N2k (Bλp ) ≤

N2k (Bλp ) √ . k!

Thus, (8.5) ensures the estimate Bλpk ≤

kp N2p (Bλ ) √ . k!

(8.5)

90

6. Functions of Compact Operators

Furthermore, it is clear that N2p (Bλ ) = N2p (V Rλ (D)) ≤ N2p (V )Rλ (D). Since D is normal and σ(A) = σ(D), Rλ (D) = ρ−1 (D, λ) = ρ−1 (A, λ).

(8.6)

Hence, N2p (Bλ ) ≤ Thus, Bλpk ≤

N2p (V ) . ρ(A, λ)

(8.7)

kp N2p (V ) √ . ρkp (A, λ) k!

m Evidently, Bλm ≤ N2p (Bλ ). Now relation (8.7) implies

Bλm ≤ Consequently, Bλpk+m ≤

m (V ) N2p . m ρ (A, λ)

kp+m N2p (V ) √ . ρkp+m (A, λ) k!

Taking into account (8.4), we have j p−1

kp+m N2p (V ) √ . Rλ (A) ≤ Rλ (D) kp+m (A, λ) k! m=0 k=0 ρ

Now (8.6) yields the required result. 2 Letting j → ∞ in the last lemma, we easily get Corollary 6.8.2 Let A ∈ C2p (p = 1, 2, ...). Then p−1 ∞

kp+m N2p (V ) √ Rλ (A) ≤ (λ ∈ σ(A)), pk+m+1 (A, λ) k! m=0 k=0 ρ

where V is the nilpotent part of A. Proof of Theorem 6.7.1: Due to inequality (2.3) N2p (D) ≤ N2p (A). Thus, the triangular representation implies N2p (V ) ≤ N2p (A) + N2p (D) ≤ 2N2p (A). Now, the required result follows from the previous lemma. 2 To prove Theorem 6.7.3, we need the gollowing

(8.8)

6.9. Functions of Hilbert-Schmidt Operators

91

Lemma 6.8.3 Under condition (7.1), let V be the nilpotent part of A. Then there are constants a0 , b0 > 0, such that the inequality Rλ (A) ≤ a0

p−1 b0 (N2p (V ))2p (N2p (V ))m exp [ ] (λ ∈ σ(A)) ρm+1 (A, λ) ρ2p (A, λ) m=0

holds. These constants can be taken as in (4.8) or in (4.9). Proof:

According to (8.3), (A − λI)−1 ≤ (D − λI)−1

p−1 k=0

Bλ k (I − Bλp )−1 .

(8.9)

It is not hard to check that Bλp is a Volterra operator (see also Lemma 7.3.4) below. Moreover, Bλp ∈ C2 . Due to Theorem 6.4.2, 2

p

(I − Bλp )−1 ≤ e1/2 eN2 (Bλ )/2 . But

(8.10)

p p N2 (Bλp ) ≤ N2p (Bλ ) ≤ N2p (V )Rλ (D)p .

p (A)ρ−p (A, λ). Now relations In addition, (8.6) implies that N2 (Bλp ) ≤ N2p (8.7), (8.9) and (8.10) yield the required result. 2

The assertion of Theorem 6.7.3 follows from the previous lemma and (8.8).

6.9

Regular Functions of Hilbert-Schmidt Operators

Let A be a bounded linear operator acting in a separable Hilbert space H and f be a scalar-valued function, which is analytic on a neighborhood of σ(A). Let a contour C consist of a ﬁnite number of rectiﬁable Jordan curves, oriented in the positive sense customary in the theory of complex variables. Suppose that C is the boundary of an open set M ⊃ σ(A) and M ∪ C is contained in the domain of analycity of f . We deﬁne f (A) by the equality 1 f (λ)Rλ (A)dλ (9.1) f (A) = − 2πi C (see the book by Dunford and Schwartz (1966, p. 568)). Theorem 6.9.1 Let A be a Hilbert-Schmidt operator and let f be a holomorphic function on a neighborhood of the closed convex hull co(A) of the spectrum of A. Then f (A) ≤

∞

sup |f (k) (λ)|

k=0 λ∈co(A)

g k (A) . (k!)3/2

(9.2)

92

6. Functions of Compact Operators

Proof: Thanks to Corollary 6.3.6, there is a sequence {An } of operators having n-dimensional ranges, such that relations (4.5) hold. Corollary 2.7.2 implies n−1 g k (An ) sup |f (k) (λ)| . (9.3) f (An ) ≤ (k!)3/2 k=0 λ∈co(An ) Due to the well-known Lemma VII.6.5 from (Dunford and Schwartz, 1966) we have f (An ) − f (A) → 0 as n → ∞. Letting n → ∞ in (9.3), due to Lemma 6.3.6, we arrive at the stated result. 2 Theorem 6.9.1 is precise: inequality (9.2) becomes the equality f (A) = sup |f (µ)|, µ∈σ(A)

if A is a normal operator and sup |f (λ)| = sup |f (λ)|,

λ∈co(A)

λ∈σ(A)

because g(A) = 0 in this case. Corollary 6.9.2 Let A be a Hilbert-Schmidt operator. Then At

α(A)t

e ≤ e

∞ k k t g (A) k=0

(k!)3/2

for all t ≥ 0,

where α(A) = sup Re σ(A). In addition, Am ≤

m m!rm−k (A)g k (A) s

k=0

(m − k)!(k!)3/2

(m = 1, 2, ...).

Recall that rs (A) is the spectral radius of operator A. In particular, if V ∈ C2 is a Volterra operator, then V m ≤

N2m (V ) √ (m = 1, 2, ...). m!

(9.4)

Note that an independent proof of inequality (9.4) can be found in Section 2.3 of the book (Gil’, 1995). In addition, that inequality allows us to estimate a power of a Volterra von Neumann - Schatten operator. Lemma 6.9.3 For some integer p ≥ 1, let V ∈ C2p be a Volterra operator. Then V kp+m ≤

pk+m N2p (V ) √ (k = 0, 1, 2, ...; m = 0, ..., p − 1). k!

(9.5)

6.10. A Relation for Determinants

Proof: (9.4)

93

p Relation (2.4) implies N2 (V p ) ≤ N2p (V ). Thus V p ∈ C2 . Due to

V pk ≤

kp N2p (V ) N2k (V p ) √ ≤ √ . k! k!

m (V ), Since V m ≤ N2p kp kp+m N2p N2p (V ) (V ) √ V pk+m ≤ V m √ ≤ , k! k!

as claimed. 2 Inequality (9.5) can be rewritten in the following way: Corollary 6.9.4 For some integer p ≥ 1, let V ∈ C2p be a Volterra operator. Then (p) j (V ) (j = 1, 2, ...). V j ≤ θj N2p (p)

Recall that θj

6.10

is given in Section 6.7.

A Relation between Determinants and Resolvents

Let A ∈ C2 . Then the generalized determinant det2 (I − A) :=

∞

(1 − λk )eλk (λk ≡ λk (A))

k=1

is ﬁnite (Dunford and Schwartz, 1963, p. 1038). The following theorem is due to Carleman (see the next section), but we suggest a new proof and correct a misprint in Theorem XI.6.27 of the book (Dunford and Schwartz, 1963). Theorem 6.10.1 Let A ∈ C2 . Then (I − A)−1 det2 (I − A) ≤ exp [(N22 (A) + 1)/2]. Proof:

(10.1)

Thanks to Theorem 2.11.1,

(I −An )−1 det (I −An ) ≤ [1+

1 (N22 (An )−2Re T race (An )+1)](n−1)/2 n−1

for any n-dimensional operator An . Hence, (I − An )−1 det (I − An ) ≤ exp [(N22 (An ) − 2Re T race (An ) + 1)/2]. Rewrite this relation as (I − An )−1 det (I − An )exp [Re T race (An )] ≤ exp [(N22 (An ) + 1)/2].

94

6. Functions of Compact Operators

Or (I − An )−1

n

|(1 − λk (An ))eλk (An ) | ≤ exp [(N22 (An ) + 1)/2].

k=1

Hence,

(I − An )−1 det2 (I − An ) ≤ exp [(N22 (An ) + 1)/2].

(10.2)

Let An , n = 1, 2, ... converge to A in the norm N2 (.) and satisfy conditions (3.4). Then det2 (I − An ) → det2 (I − A). This ﬁnishes the proof. 2 Replacing in (10.1) A by λ−1 A, we get Corollary 6.10.2 Let A ∈ C2 . Then (λI − A)−1 det2 (I − λ−1 A) ≤

1 N 2 (A) 1 exp [ + 2 2 ] (λ ∈ σ(A)). (10.3) |λ| 2 2|λ|

In particular, if V ∈ C2 is quasinilpotent, then (λI − V )−1 ≤

1 1 N 2 (V ) exp [ + 2 2 ] (λ = 0). |λ| 2 2|λ|

(10.4)

Moreover, relation (10.4) implies. Corollary 6.10.3 Let A ∈ C2 . Then Rλ (A) ≤

1 g 2 (A) 1 exp [ + 2 ] (λ ∈ σ(A)). ρ(A, λ) 2 2ρ (A, λ)

(10.5)

Indeed, due to (3.2), Rλ (A) = Rλ (D)(I + V Rλ (D))−1 . Lemma 7.3.4 below yields that V Rλ (D) is a Volterra operator. So according to (10.4), 1 N 2 (V Rλ (D)) ]≤ (I + V Rλ (D))−1 (I + V Rλ (D))−1 ≤ exp [ + 2 2 2 1 N 2 (V )Rλ (D)2 ]. exp [ + 2 2 2 But due to Lemma 6.5.2, N2 (V ) = g(A). Thus, Rλ (A) ≤ Rλ (D)(I + V Rλ (D))−1 ≤

1 g 2 (A) 1 exp [ + 2 ], ρ(A, λ) 2 2ρ (A, λ)

as claimed. Note that Corollary 6.10.3 gives us an additional proof of Theorem 6.4.2.

6.11. Notes

6.11

95

Notes

Theorems 6.4.1, 6.7.1 and 6.9.1 were derived in the papers (Gil’, 1979a), (Gil’, 1992) and (Gil’, 1979b), respectively (see also (Gil’, 1995, Chapter 2)), but in the present chapter we suggest the new proofs. Theorems 6.4.2 and 6.7.3 are probably new. In the book (Dunford and Schwartz, 1963, p. 1038), instead of (10.3), it is erroneously stated that 1 N 2 (A) (λI − A)−1 det2 (I − λ−1 A) ≤ |λ| exp [ + 2 2 ]. 2 2|λ| Note that the very interesting estimates for the resolvents of operators from Cp are established in the papers (Dechevski and Persson, 1994 and 1996).

References [1] Ahiezer, N. I. and Glazman, I. M. (1981). Theory of Linear Operators in a Hilbert Space. Pitman Advanced Publishing Program, Boston. [2] Brodskii, M. S. (1971). Triangular and Jordan Representations of Linear Operators, Transl. Math. Mongr., v. 32, Amer. Math. Soc., Providence, R. I. [3] Dechevski, L. T. and Persson, L. E. (1994). Sharp generalized Carleman inequalities with minimal information about the spectrum, Math. Nachr., 168, 61-77. [4] Dechevski, L. T. and Persson, L. E. (1996). On sharpness, applications and generalizations of some Carleman type inequalities, Tˆ ohuku Math. J., 48, 1-22. [5] Dunford, N. and Schwartz, J. T. (1966). Linear Operators, part I. General Theory. Interscience publishers, New York. [6] Dunford, N. and Schwartz, J. T. (1963). Linear Operators, part II. Spectral Theory. Interscience publishers, New York, London. [7] Gil’, M. I. (1979a). An estimate for norms of resolvent of completely continuous operators, Mathematical Notes, 26 , 849-851. [8] Gil’, M. I. (1979b). Estimates for norms of functions of a Hilbert-Schmidt operator (in Russian), Izvestiya VUZ, Matematika, 23, 14-19. English translation in Soviet Math., 23, 13-19. [9] Gil’ , M. I. (1992). On estimate for the norm of a function of a quasihermitian operator, Studia Mathematica, 103(1), 17-24.

96

6. Functions of Compact Operators

[10] Gil’, M. I. (1995). Norm Estimations for Operator-valued Functions and Applications. Marcel Dekker, Inc., New York. [11] Gohberg, I. C. and Krein, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Mathem. Monographs, v. 18, Amer. Math. Soc., Providence, R. I. [12] Gohberg, I. C. and Krein, M. G. (1970) . Theory and Applications of Volterra Operators in Hilbert Space, Trans. Mathem. Monographs, v. 24, Amer. Math. Soc., Providence, R. I. [13] Pietsch, A. (1988). Eigenvalues and s-numbers, Cambridge University Press, Cambridge.

7. Functions of Non-compact Operators

The present chapter is concerned with the estimates for the norms of resolvents and analytic functions of so called P -triangular operators. Roughly speaking, a P -triangular operator is a sum of a normal operator and a compact quasinilpotent one, having a suﬃciently rich set of invariant subspaces. In particular, we consider the following classes of P -triangular operators: operators whose Hermitian components are compact operators, and operators, which are represented as sums of unitary operators and compact ones.

7.1

Terminology

Let A be a linear operator acting in a separable Hilbert space H. Let there be a linear manifold Dom (A), such that the relation f ∈ Dom (A) implies Af ∈ H. Then the set Dom (A) is called the domain of A. Let Dom (A) be dense in H. Then the set of vectors g satisfying |(Af, g)| ≤ cf for all f ∈ Dom (A) with a constant c is the domain of the adjoint operator A∗ and, besides, (Af, g) = (f, A∗ g) for all f ∈ Dom (A) and g ∈ Dom (A∗ ). An unbounded operator A is selfadjoint, if Dom (A) = Dom (A∗ ) and Ah = A∗ h (h ∈ Dom (A)). An unbounded selfadjoint operator possesses an unbounded real spectrum. An unbounded operator A is normal, if Dom (AA∗ ) = Dom (A∗ A) and AA∗ h = A∗ Ah (h ∈ Dom (A∗ A)). M.I. Gil’: LNM 1830, pp. 97–121, 2003. c Springer-Verlag Berlin Heidelberg 2003

98

7. Functions of Non-compact Operators

An unbounded normal operator has an unbounded spectrum. Deﬁnition 7.1.1 Let A be a linear operator in H. Then A is said to be a quasi-normal operator, if it is a sum of a normal operator and a compact one. Operator A is said to be a quasi-Hermitian operator, if it is a sum of a selfadjoint operator and a compact one. Let A be bounded and AI ≡ (A − A∗ )/2i ∈ Cp (p ≥ 1). Then, as it is well-known (see e.g. (Gohberg and Krein, 1969, Section II.6)), the nonreal spectrum consists of the eigenvalues having ﬁnite multiplicities, and ∞ k=1

|Im λj (A)|p ≤

∞

|λj (AI )|p (n = 1, 2, . . .)

(1.1)

k=1

where λj (A) and λj (AI ) are the eigenvalues with their multiplicities of A and AI , respectively.

7.2

P -Triangular Operators

A family of orthogonal projectors P (t) in H (i.e. P 2 (t) = P (t) and P ∗ (t) = P (t)) deﬁned on a (ﬁnite or inﬁnite) segment [a, b] of the real axis is a an orthogonal resolution of the identity if for all t, s ∈ [a, b], P (a) = 0, P (b) = I ≡ the unit operator and P (t)P (s) = P (min(t, s)) An orthogonal resolution of the identity P (t) is left-continuous, if P (t − 0) = P (t) for all t ∈ (a, b] in the sense of the strong topology. Deﬁnition 7.2.1 Let P (t) be a left-continuous orthogonal resolution of the identity in H deﬁned on a (ﬁnite or inﬁnite) real segment [a, b]. Then P (.) will be called a maximal resolution of the identity (m.r.i.), if its every gap P (t0 + 0) − P (t0 ) (if it exists) is one-dimensional. Moreover, we will say that an m.r.i. P (.) belongs to a linear operator A (or A has an m.r.i. P (.)), if P (t)AP (t)h = AP (t)h for all t ∈ [a, b] and h ∈ Dom (A).

(2.1)

Recall that a linear operator V is called a Volterra one, if it is compact and quasinilpotent. Deﬁnition 7.2.2 Let a linear generally unbounded operator A have an m.r.i. P (.) deﬁned on [a, b]. In addition, let A = D + V,

(2.2)

7.3. Properties of Volterra Operators

99

where D is a normal operator and V is a Volterra one, having the following properties: P (t)V P (t) = V P (t) (t ∈ [a, b]) (2.3) and DP (t)h = P (t)Dh (t ∈ [a, b], h ∈ Dom (A)).

(2.4)

Then A will be called a P -triangular operator. In addition, equality (2.2), and operators D and V will be called the triangular representation, diagonal part and nilpotent part of A, respectively. Clearly, this deﬁnition is in accordance with the deﬁnition of the triangular representation of compact operators (see Section 6.3).

7.3

Some Properties of Volterra Operators

Lemma 7.3.1 Let a compact operator V in H, have a maximal orthogonal resolution of the identity P (t) (a ≤ t ≤ b) (that is, condition (2.3) hold). If, in addition, (P (t0 + 0) − P (t0 ))V (P (t0 + 0) − P (t0 )) = 0

(3.1)

for every gap P (t0 + 0) − P (t0 ) of P (t) (if it exists), then V is a Volterra operator. Proof: Since the set of the values of P (.) is a maximal chain of projectors, the required result is due to Corollary 1 to Theorem 17.1 of the book by Brodskii (1971). 2 In particular, if P (t) is continuous in t in the strong topology and (2.3) holds, then V is a Volterra operator. We also need the following Lemma 7.3.2 Let V be a Volterra operator in H, and P (t) a maximal orthogonal resolution of the identity satisfying equality (2.3). Then equality (3.1) holds for every gap P (t0 + 0) − P (t0 ) of P (t) (if it exists). Proof: Since the set of the values of P (.) is a maximal chain of projectors, the required result is due to the well-known equality (I.3.1) from the book by Gohberg and Krein (1970). 2 Lemma 7.3.3 Let V and B be bounded linear operators in H having the same m.r.i. P (.). In addition, let V be a Volterra operator. Then V B and BV are Volterra operators, and P (.) is their m.r.i. Proof:

It is obvious that P (t)V BP (t) = V P (t)BP (t) = V BP (t).

(3.2)

100

7. Functions of Non-compact Operators

Now let Q = P (t0 + 0) − P (t0 ) be a gap of P (t). Then according to Lemma 7.3.2 equality (3.1) holds. Further, we have QV BQ = QV B(P (t0 + 0) − P (t0 )) = QV [P (t0 + 0)BP (t0 + 0) − P (t0 )BP (t0 )] = QV [(P (t0 ) + Q)B(P (t0 ) + Q) − P (t0 )BP (t0 )] = QV [P (t0 )BQ + QBP (t0 )]. Since QP (t0 ) = 0 and P (t) projects onto the invariant subspaces, we obtain QV BQ = 0. Due to Lemma 7.3.1 this relation and equality (3.2) imply that V B is a Volterra operator. Similarly we can prove that BV is a Volterra one. 2 Lemma 7.3.4 Let A be a (generally unbounded) P -triangular operator. Let V and D be the nilpotent and diagonal parts of A, respectively. Then for any regular point λ of D, the operators V Rλ (D) and Rλ (D)V are Volterra ones. Besides, A, V Rλ (D) and Rλ (D)V have the same m.r.i. Proof:

Due to (2.4) P (t)Rλ (D) = Rλ (D)P (t) for all t ∈ [a, b].

Now Lemma 7.3.3 ensures the required result. 2

7.4

Powers of Volterra Operators

Let Y be a a norm ideal of compact linear operators in H. That is, Y is algebraically a two- sided ideal, which is complete in an auxiliary norm | · |Y for which |CB|Y and |BC|Y are both dominated by C|B|Y . In the sequel we suppose that there are positive numbers θk (k ∈ N), with 1/k θk → 0 as k → ∞, such that

V k ≤ θk |V |kY

(4.1)

for an arbitrary Volterra operator V ∈ Y.

(4.2)

Recall that C2p (p = 1, 2, ...) is the von Neumann-Schatten ideal of compact operators with the ﬁnite ideal norm N2p (K) ≡ [T race (K ∗ K)p ]1/2p (K ∈ C2p ).

7.5. Resolvents of P -Triangular Operators

101

Let V ∈ C2p be a Volterra operator. Then due to Corollary 6.9.4, (p)

j (V ) (j = 1, 2, ...) V j ≤ θj N2p

where (p)

θj

=

(4.3)

1 [j/p]!

and [x] means the integer part of a positive number x. Inequality (4.3) can be written as V kp+m ≤

7.5

pk+m N2p (V ) √ (k = 0, 1, 2, ...; m = 0, ..., p − 1). k!

(4.4)

Resolvents of P -Triangular Operators

Lemma 7.5.1 Let A be a P -triangular operator. Then σ(A) = σ(D), where D is the diagonal part of A. Proof: Let λ be a regular point of the operator D. According to the triangular representation (2.2) we obtain Rλ (A) = (D + V − λI)−1 = Rλ (D)(I + V Rλ (D))−1 .

(5.1)

Operator V Rλ (D) for a regular point λ of the operator D is a Volterra one due to Lemma 7.3.4. Therefore, (I + V Rλ (D))−1 =

∞

(V Rλ (D))k (−1)k

k=0

and the series converges in the operator norm. Thus, Rλ (A) = Rλ (D)

∞

(V Rλ (D))k (−1)k .

(5.2)

k=0

Hence, it follows that λ is the regular point of A. Conversely let λ ∈ σ(A). According to the triangular representation (2.2) we obtain Rλ (D) = (A − V − λI)−1 = Rλ (A)(I − V Rλ (A))−1 . Operator V Rλ (A) for a regular point λ of operator A is a Volterra one due to Lemma 7.3.3. So (I − V Rλ (A))−1 =

∞ k=0

(V Rλ (A))k

102

7. Functions of Non-compact Operators

and the series converges in the operator norm. Thus, Rλ (D) = Rλ (A)

∞

(V Rλ (A))k .

k=0

Hence, it follows that λ is the regular point of D. This ﬁnishes the proof. 2 Furthermore, for a natural number m and a z ∈ (0, ∞), under (4.2), put JY (V, m, z) =

m−1 k=0

θk |V |kY . z k+1

(5.3)

Deﬁnition 7.5.2 A number ni(V ) is called the nilpotency index of a nilpotent operator V , if V ni(V ) = 0 = V ni(V )−1 . If V is quasinilpotent but not nilpotent we write ni(V ) = ∞. Everywhere below one can replace ni(V ) by ∞. Theorem 7.5.3 Let A be a P -triangular operator and let its nilpotent part V belong to a norm ideal Y with the property (4.1). Then ν(λ)−1

Rλ (A) ≤ JY (V, ν(λ), ρ(A, λ)) :=

k=0

θk |V |kY k+1 ρ (A, λ)

(5.4)

for all regular λ of A. Here ν(λ) = ni(V Rλ (D)), and D is the diagonal part of A. Proof: Due to Lemma 7.3.4, V Rλ (D) ∈ Y is a Volterra operator. So according to (4.1), (V Rλ (D))k ≤ θk |V Rλ (D)|kY . But |V Rλ (D)|Y ≤ |V |Y Rλ (D), and thanks to Lemma 7.5.1, Rλ (D) =

1 1 = . ρ(D, λ) ρ(A, λ)

So (V Rλ (D))k ≤

θk |V |kY . ρk (A, λ)

Relation (5.2) implies ν(λ)−1

Rλ (A) ≤ Rλ (D)

k=0

(V Rλ (D))k ≤

7.5. Resolvents of P -Triangular Operators

103

ν(λ)−1 θk |V |k 1 Y , ρ(A, λ) ρk (A, λ) k=0

as claimed. 2 For a natural number m, a Volterra operator V ∈ C2p and a z ∈ (0, ∞), put m−1 k θk(p) N2p (V ) J˜p (V, m, z) := . (5.5) k+1 z k=0

Theorem 7.5.3 and inequality (4.3) yield Corollary 7.5.4 Let A be a P -triangular operator and its nilpotent part V ∈ C2p for an integer p ≥ 1. Then ν(λ)−1

Rλ (A) ≤ J˜p (V, ν(λ), ρ(A, λ)) ≡

k θk(p) N2p (V ) (λ ∈ σ(A)). k+1 ρ (A, λ)

(5.6)

k=0

In particular, let A be a P -triangular operator, whose nilpotent part V is a Hilbert-Schmidt operator. Then due to the previous corollary ν(λ)−1

Rλ (A) ≤ J˜1 (V, ν(λ), ρ(A, λ)) ≡

k=0

N2k (V ) √ (λ ∈ σ(A)). (5.7) ρk+1 (A, λ) k!

Furthermore, inequality (5.6) implies Rλ (A) ≤

p−1 ∞

pk+j N2p (V ) √ . pk+j+1 ρ (A, λ) k! j=0 k=0

(5.8)

Theorem 7.5.5 Let A be a P -triangular operator and its nilpotent part V ∈ C2p for some integer p ≥ 1. Then there are constants a0 , b0 > 0, such that Rλ (A) ≤ a0

p−1 j 2p b0 N2p N2p (V ) (V ) exp [ ] (λ ∈ σ(A)). j+1 2p ρ (A, λ) ρ (A, λ) j=0

(5.9)

One can take √ c c and b0 = for any c > 1; in particular, a0 = 2 and b0 = 1, a0 = c−1 2 (5.10) or (5.11) a0 = e1/2 and b0 = 1/2.

104

7. Functions of Non-compact Operators

Proof:

Due to Lemma 6.8.3, for any Volterra operator V ∈ C2p , (I − V )−1 ≤ a0

p−1

j 2p N2p (V ) exp [ b0 N2p (V )].

j=0

But V Rλ (D) is a Volterra operator due to Lemma 7.3.4. Hence, (I + V Rλ (D))−1 ≤ a0

p−1

j 2p N2p (V Rλ (D)) exp [ b0 N2p (V Rλ (D))] ≤

j=0 p−1 j 2p N2p (V ) N2p (V ) exp [ b0 2p ]. ≤ a0 j (A, λ) ρ ρ (A, λ) j=0

Now (5.1) implies the required result. 2

7.6

Triangular Representations of Quasi-Hermitian Operators

Theorem 7.6.1 Let a linear generally unbounded operator A satisfy the conditions Dom (A∗ ) = Dom (A) and AI = (A − A∗ )/2i ∈ Cp (1 < p < ∞).

(6.1)

Then A admits the triangular representation (2.2). Proof: First, let A be bounded. Then, as it is proved by L. de Branges (1965a, p. 69), under condition (6.1), there are a maximal orthogonal resolution of the identity P (t) deﬁned on a ﬁnite real segment [a, b] and a real nondecreasing function h(t), such that A=

b

h(t)dP (t) + 2i a

a

b

P (t)AI dP (t).

(6.2)

The second integral in (6.2) is understood as the limit in the operator norm of the operator Stieltjes sums n

Ln =

1 [P (tk−1 ) + P (tk )]AI ∆Pk , 2 k=1

where (n)

tk = tk ; ∆Pk = P (tk ) − P (tk−1 ); a = t0 < t1 . . . < tn = b.

7.6. Representations of Quasi-Hermitian Operators

105

We can write Ln = Wn + Tn with Wn =

n

n

P (tk−1 )AI ∆Pk and Tn =

k=1

1 ∆Pk AI ∆Pk . 2

(6.3)

k=1

The sequence {Tn } converges in the operator norm due to the well-known Lemma I.5.1 from the book (Gohberg and Krein, 1970). We denote its limit by T . Clearly, T is selfadjoint and P (t)T = T P (t) for all t ∈ [a, b]. Put b h(t)dP (t) + 2iT. (6.4) D= a

Then D is normal and satisﬁes condition (2.4). Furthermore, it can be directly checked that Wn is a nilpotent operator: (Wn )n = 0. Besides, the sequence {Wn } converges in the operator norm, because the second integral in (6.2) and {Tn } converge in this norm. We denote the limit of the sequence {2iWn } by V . It is a Volterra operator, since the limit in the operator norm of a sequence of Volterra operators is a Volterra one (see for instance Lemma 2.17.1 from the book (Brodskii, 1971)). From this we easily obtain relations (2.1)-(2.4). So, for bounded operators the theorem is proved. Now let A be unbounded. Due to De Branges (1965a, p. 69), there is a maximal orthogonal resolution of the identity P (t), −∞ ≤ t ≤ ∞ and nondecreasing functions h(t), such that under (6.1), the representation ∞ ∞ h(t)dP (t) + 2i P (t)AI dP (t) (6.5) A= −∞

−∞

holds. The integrals in (6.5) are understood as the limits of corresponding integrals from (6.2) when a → −∞ and b → ∞. Besides, the ﬁrst integral in (6.5) is the strong limit on Dom(A) of the ﬁrst integral in (6.2), and the second integral in (6.5) is the limit in the uniform operator topology of the second integral in (6.2). Take An = A(P (n) − P (−n)).

(6.6)

Clearly, An is bounded and has the property (6.1). So as it was proved above, according to (6.5) it has a triangular representation with the m.r.i. P (.). Hence, letting n → ∞, we get the required result. 2 Corollary 7.6.2 Let A be an unbounded operator with the property (6.1). Then it is P -triangular and there is a sequence of bounded P -triangular operators An , such that An − A∗n ∈ Cp , σ(An ) ⊆ σ(A), Np (An − A∗ ) → 0 and Np (Vn − V ) → 0 as n → 0, where Vn and V are the nilpotent part of An and A, respectively. Moreover, operators A and An (n = 1, 2, ...) have the same m.r.i. Indeed, taking An as in (6.6), we arrive at the result due to (2.2) and the previous theorem.

106

7.7

7. Functions of Non-compact Operators

Resolvents of Operators with Hilbert-Schmidt Hermitian Components

In this section we obtain an estimate for the norm of the resolvent of a generally unbounded quasi-Hermitian operator under the conditions Dom (A) = Dom (A∗ ) and AI = (A − A∗ )/2i is a Hilbert-Schmidt operator.

(7.1)

Let us introduce the quantity gI (A) ≡

√

2 [N22 (AI ) −

∞

(Im λk (A))2 ]1/2 .

(7.2)

k=1

Theorem 7.7.1 Let condition (7.1) hold. Then Rλ (A) ≤

∞

gIk (A) √ (λ ∈ σ(A)). ρk+1 (A, λ) k! k=0

(7.3)

Moreover, there are constants a0 , b0 > 0, such that Rλ (A) ≤

b0 g 2 (A) a0 exp [ 2 I ] (λ ∈ σ(A)). ρ(A, λ) ρ (A, λ)

(7.4)

These constants can be taken from (5.10) or from (5.11). Here ρ(A, λ) is the distance between the spectrum σ(A) of A and a complex point λ, again. To prove this theorem we need the following Lemma 7.7.2 Let an operator A satisfy the condition (7.1). Then it admits the triangular representation (due to Theorem 7.6.1). Moreover, N2 (V ) = gI (A), where V is the nilpotent part of A. Proof: First, assume that A is a bounded operator. Let D be the diagonal part of A. From the triangular representation (2.2) it follows that −4T r A2I = T r (A − A∗ )2 = T r (D + V − D∗ − V ∗ )2 . By Theorem 6.2.2 and Lemma 7.3.3, T race V D∗ = T race V ∗ D = 0. Hence, omitting simple calculations, we obtain −4T r A2I = T r (D − D∗ )2 + T r (V − V ∗ )2 . That is, N22 (AI ) = N22 (VI ) + N22 (DI ), where VI = (V − V ∗ )/2i and DI = (D − D∗ )/2i. Taking into account Lemma 6.5.1, we arrive at the equality 2N 2 (AI ) − 2N 2 (DI ) = N 2 (V ).

7.7. Operators with Property Ap − (A∗ )p ∈ C2

107

Recall that the nonreal spectrum of a quasi-Hermitian operator consists of isolated eigenvalues. Besides, due to Lemma 7.5.1 σ(A) = σ(D). Thus, N 2 (DI ) =

∞

|Im λk (A)|2 ,

k=1

and we arrive at the result, if A is bounded. If A is unbounded, then the required assertion is due to Corollary 7.6.2 and the just proved result for bounded operators. 2 Proof of Theorem 7.7.1: Inequality (7.3) follows from Corollary 7.5.4 and Lemma 7.7.2. Inequality (7.4) follows from Theorem 7.5.5 and Lemma 7.7.2. 2

7.8

Operators with the Property Ap − (A∗ )p ∈ C2

Theorem 7.8.1 Let a bounded linear operator A satisfy the condition Ap − (A∗ )p ∈ C2 (p = 2, 3, ...). Then Rλ (A) ≤ Tλ,p

(8.1)

∞

gIk (Ap ) √ (λp ∈ σ(Ap )). k+1 (Ap , λp ) k! ρ k=0

Here ρ(Ap , λp ) is the distance between the spectrum σ(Ap ) of Ap and λp , and Tλ,p =

p−1

Ak λp−k−1 .

k=0

Indeed, this result follows from Theorem 7.7.1 and the obvious relation Rλ (A) = Tλ,p Rλp (Ap ).

(8.2)

Moreover, relations (7.4) and (8.2) yield Corollary 7.8.2 Let condition (8.1) hold. Then there are constants a0 , b0 > 0, such that Rλ (A) ≤

a0 Tλ,p b0 gI2 (Ap ) exp [ ] (λp ∈ σ(Ap )). ρ(Ap , λp ) ρ2 (Ap , λp )

These constants can be taken from (5.10) or from (5.11).

108

7. Functions of Non-compact Operators

7.9

Resolvents of Operators with Neumann - Schatten Hermitian Components

In this section we obtain estimates for the resolvent of an operator A, assuming that Dom (A) = Dom (A∗ ) and AI := (A − A∗ )/2i ∈ C2p for some integer p > 1. That is,

(9.1)

∞ 2p λj (AI ) < ∞. N2p (AI ) = 2p j=1

Put β˜p :=

π )) 2(1 + ctg ( 4p 2p 2(1 + exp(2/3)ln2 )

if p = 2m−1 , m = 1, 2, ... . otherwise

(9.2)

Theorem 7.9.1 Let condition (9.1) hold. Then there is a constant βp , depending on p, only, such that p−1 ∞ (βp N2p (AI ))kp+m √ (λ ∈ σ(A)). Rλ (A) ≤ pk+m+1 (A, λ) k! m=0 k=0 ρ

(9.3)

βp ≤ β˜p .

(9.4)

Besides, Moreover, there are constants a0 , b0 > 0, such that p−1 (βp N2p (AI ))m b0 (βp N2p (AI ))2p Rλ (A) ≤ a0 exp [ ] (λ ∈ σ(A)). ρm+1 (A, λ) ρ2p (A, λ) m=0 (9.5) Constants a0 , b0 can be taken from (5.10) or from (5.11).

In order to prove this theorem we need the following result. Lemma 7.9.2 Let A satisfy condition (9.1). Then it admits the triangular representation (due to Theorem 7.6.1), and the nilpotent part V of A satisﬁes the relation N2p (V ) ≤ βp N2p (AI ). Proof: Let V ∈ Cp (2 ≤ p < ∞) be a Volterra operator. Then due to the well-known Theorem III.6.2 from the book by Gohberg and Krein (1970, p. 118), there is a constant γp , depending on p, only, such that Np (VI ) ≤ γp Np (VR ) (VI = (V + V ∗ )/2i, VR := (V + V ∗ )/2).

(9.6)

7.10. Regular Functions of Bounded Operators

109

Besides, as it is proved in (Gohberg and Krein, 1970, pages 123 and 124), p p ≤ γp ≤ . 2π exp(2/3)ln2 Moreover, γp = ctg

π if p = 2n (n = 1, 2, ...), 2p

cf. (Gohberg and Krein, 1970, page 124). Take βp = 2(1 + γ2p ). Now let D be the diagonal part of A. Let VI , DI be the imaginary components of V and D, respectively. According to (2.2) AI = VI + DI . First, assume that A is a bounded operator. Due to (1.1), the condition AI ∈ C2p entails the inequality N2p (DI ) ≤ N2p (AI ). Therefore, N2p (VI ) ≤ N2p (AI ) + N2p (DI ) ≤ 2N2p (AI ). Hence, due to (9.6), N2p (V ) ≤ N2p (VR ) + N2p (VI ) ≤ N2p (VI )(1 + γ2p ) = βp N2p (AI ), as claimed. Now let A be unbounded. To obtain the stated result in this case it is suﬃcient to apply Corollary 7.6.2 and the just obtained result for bounded operators. 2 The assertion of Theorem 7.9.1 follows from (5.8), Theorem 7.5.5 and Lemma 7.9.2.

7.10

Regular Functions of Bounded Quasi-Hermitian Operators

Let A be a bounded linear operator in a H and let f (z) be a scalar-valued function which is analytic on some neighborhood of σ(A). Again, put 1 f (λ)Rλ (A)dλ, (10.1) f (A) = − 2πi C where C is a closed Jordan contour, surrounding σ(A) and having the positive orientation with respect to σ(A). Below we also consider unbounded operators. Theorem 7.10.1 Let a bounded linear operator A satisfy the condition AI is a Hilbert-Schmidt operator.

(10.2)

110

7. Functions of Non-compact Operators

In addition, let f be a holomorphic function on a neighborhood of the closed convex hull co(A) of σ(A). Then f (A) ≤

∞

sup |f (k) (λ)|

k=0 λ∈co(A)

gIk (A) . (k!)3/2

(10.3)

Recall that the quantity gI (A) is deﬁned by the equality (7.2). Theorem 7.10.1 is precise: the inequality (10.3) becomes the equality f (A) = sup |f (µ)|, µ∈σ(A)

(10.4)

if A is a normal operator and sup |f (λ)| = sup |f (λ)|,

λ∈co(A)

λ∈σ(A)

(10.5)

because gI (A) = 0 for a normal operator A. Example 7.10.2 Let a bounded operator A satisfy condition (10.2). Then Theorem 7.10.1 gives us the inequality Am ≤

m m!rm−k (A)g k (A) s

k=0

I

(m − k)!(k!)3/2

for any integer m ≥ 1. Recall that rs (A) is the spectral radius of A. Example 7.10.3 Let a bounded operator A satisfy the condition (10.2). Then Theorem 7.10.1 gives us the inequality eAt ≤ eα(A)t

∞ k k t g (A) I

k=0

(k!)3/2

for all t ≥ 0,

(10.6)

where α(A) = sup Re σ(A).

7.11

Proof of Theorem 7.10.1

Let Pk (k = 0, ..., n) be a ﬁnite chain of orthogonal projectors: 0 = Range(P0 ) ⊂ Range(P1 ) ⊂ .... ⊂ Range(Pn ) = H. We need the following Lemma 7.11.1 Let a bounded operator A in H have the representation A=

n k=1

φk ∆Pk + V (∆Pk = Pk − Pk−1 ),

7.11. Proof of Theorem 7.10.1

111

where φk (k = 1, ..., n) are numbers and V is a Hilbert-Schmidt operator satisfying the relations Pk−1 V Pk = V Pk (k = 1, ..., n). In addition, let f be holomorphic on a neighborhood of the closed convex hull co(A) of σ(A). Then f (A) ≤ Proof:

∞

sup |f (k) (λ)|

k=0 λ∈co(A)

Put D=

n

N2k (V ) . (k!)3/2

φk ∆Pk .

k=1

Clearly, the spectrum of D consists of numbers φk (k = 1, ..., n). It is simple to check that V n = 0. That is, V is a nilpotent operator. Due to Lemma 7.5.1, σ(D) = σ(A). Consequently, φk (k = 1, ..., n) are eigenvalues of A. (k) Furthermore, let {em }∞ m=1 be an orthogonal normal basis in ∆Pk H. Put (k)

Ql (k)

Clearly, Ql

=

l

(k) (., e(k) m )em (k = 1, ..., n; l = 1, 2, ....).

m=1

strongly converge to ∆Pk as l → ∞. Moreover, (k)

(k)

Ql ∆Pk = ∆Pk Ql Then the operators Dl =

n k=1

(k)

= Ql .

(k)

φk Ql

strongly tend to D as l → ∞. We can write out, V =

n k−1

∆Pi V ∆Pk .

k=1 i=1

Introduce the operators Wl =

n k−1 k=1 i=1

(k)

(i)

(k)

Ql V Ql .

Since projectors Ql strongly converge to ∆Pk as l → ∞, and V is compact, operators Wl converge to V in the operator norm. So the ﬁnite dimensional operators Tl := Dl + Wl

112

7. Functions of Non-compact Operators

strongly converge to A and f (Tl ) strongly converge to f (A). Thus, f (A) ≤ sup f (Tl ),

(11.1)

l

thanks to the Banach-Steinhaus theorem (see e.g. (Dunford and Schwartz, 1966)). But Wl are nilpotent, and Wl and Dl have the same invariant subspaces. Consequently, due to Lemma 7.5.1, σ(Dl ) = σ(Tl ) ⊆ σ(A) = {φk }. The dimension of Tl is nl. Due to Lemma 2.8.2 and Corollary 6.9.2, we have the inequality f (Tl ) ≤

ln−1

sup |f (k) (λ)|

k=0 λ∈co(A)

N2k (Wl ) . (k!)3/2

Letting l → ∞ in this inequality, we prove the stated result. 2 Lemma 7.11.2 Let A be a bounded P -triangular operator, whose nilpotent part V ∈ C2 . Let f be a function holomorphic on a neighborhood of co(A). Then ∞ N k (V ) sup |f (k) (λ)| 2 3/2 . f (A) ≤ (k!) k=0 λ∈co(A) Proof: Let D be the diagonal part of A. According to (2.4) and the von Neumann Theorem (Ahiezer and Glazman, 1981, Section 92), there exists a bounded measurable function φ, such that b D= φ(t)dP (t). a

Deﬁne the operators Vn =

n

P (tk−1 )V ∆Pk and Dn =

k=0

n

φ(tk )∆Pk

k=1

(n)

(tk = tk , a = t0 ≤ t1 ≤ ... ≤ tn = b; ∆Pk = P (tk ) − P (tk−1 ), k = 1, ..., n). Besides, put Bn = Dn +Vn . Then the sequence {Bn } strongly converges to A due to the triangular representation (2.2). According to (10.1) the sequence {f (Bn )} strongly converges to f (A). The inequality f (A) ≤ sup f (Bn )

(11.2)

n

is true thanks to the Banach-Steinhaus theorem. Since the spectral resolution of Bn consists of n < ∞ projectors, Lemma 7.11.1 yields the inequality f (Bn ) ≤

∞

sup

k=0 λ∈co(Bn )

|f (k) (λ)|

N2k (Vn ) . (k!)3/2

(11.3)

7.12. Functions of Unbounded Operators

113

Thanks to Lemma 7.5.1 we have σ(Bn ) = σ(Dn ). Clearly, σ(Dn ) ⊆ σ(D). Hence, σ(Bn ) ⊆ σ(A). (11.4) Due to the well-known Theorem III.7.1 from (Gohberg and Krein, 1970), {N2 (Vn )} tends to N2 (V ) as n tends to inﬁnity. Thus (11.2), (11.3) and (11.4) imply the required result. 2 The assertion of Theorem 7.10.1 immediately follows from Lemmas 7.11.2 and 7.7.2. 2

7.12

Regular Functions of Unbounded Operators

Let an operator A be unbounded with a dense domain Dom (A) = Dom (A∗ ). In addition, let the conditions (10.2) and β(A) := inf Re σ(A) > −∞

(12.1)

hold. Let f be analytic on some neighborhood of σ(A) and M an open set containing σ(A) whose boundary C consists of a ﬁnite number of Jordan arcs such that f is analytic on M ∪ C. Let C have positive orientation with respect to M . Then we deﬁne the function of the operator A by the equality 1 f (λ)Rλ (A)dλ, (12.2) f (A) = f (∞)I − 2πi C cf. (Dunford and Schwartz, 1966, p. 601). Without loss of generality we assume that f (∞) = 0. In the other case we can consider the function f (λ) − f (∞). Theorem 7.12.1 Under the conditions D(A) = D(A∗ ), (10.2) and (12.1), let f be regular on a neighborhood of co(A) and f (∞) = 0. Then inequality (10.3) is true. Proof: Due to Theorem 7.6.1, under (10.2), A admits the triangular representation. According to (2.4) and the von Neumann Theorem (Ahiezer and Glazman, 1981, Section 92), there exists a P -measurable scalar function φ, such that ∞ D= φ(t)dP (t). −∞

In addition, as it was shown in the proof of Theorem 7.6.1, the function Re φ(t) nondecreases as t increases. Thus due to Lemma 7.5.1, inf Re φ(t) = β(D) = β(A) > −∞ and sup |Im φ(t)| < ∞. t

t

114

7. Functions of Non-compact Operators

So the operators An = AP (n) are bounded. Moreover, relations σ(An ) ⊆ σ(A) and (12.3) (Iλ − A)−1 P (n) = (Iλ − An )−1 P (n) hold. Hence, due to (12.2) 1 f (A)P (n) = f (λ)(Iλ − An )−1 dλP (n) = f (An )P (n). 2πi C Due to Theorem 7.10.1, f (An ) ≤

∞

sup

k=0 λ∈co(An )

|f (k) (λ)|

gIk (An ) . (k!)3/2

Letting in this relation n → ∞, we get inequality (10.3). Thus, the theorem is proved. 2 Theorem 12.1 is exact. Indeed, under (12.1), let A be normal and (10.5) hold. Then we have equality (10.4), provided f (∞) = 0. Furthermore, under conditions (12.1) and (10.2), put −At

e

1 := 2πi

c0 +i∞

c0 −i∞

etλ (Iλ + A)−1 dλ (c0 > −β(A)).

(12.4)

Since the non-real spectrum of A is bounded, the integral in (12.4) converges in the sense of the Laplace transformation. Clearly, function ezt is nonanalytic at inﬁnity. Let An = AP (n). According to (12.4) and (12.3), e−At P (n) =

1 2πi

c0 +i∞

c0 −i∞

etλ (Iλ + An )−1 P (n)dλ = e−An t P (n).

Due to Theorem 7.10.1 exp [−An t] ≤ e−tβ(An )

∞ k k t g (An ) I

k=0

(k!)3/2

,

since An is bounded. Letting in this relation n → ∞, we get Theorem 7.12.2 Let conditions (10.2) and (12.1) hold. Then exp [−At] ≤ e−tβ(A)

∞ k k t g (A) I

k=0

(k!)3/2

(t ≥ 0).

This theorem is exact. Indeed, let A be normal. Then we have exp [−At] = e−tβ(A) , t ≥ 0.

7.13. Triangular Representations of Functions

115

For a scalar-valued function h deﬁned on [0, ∞), let the integral ∞ Φ(A) = e−At h(t)dt 0

strongly converges. Denote,

g k (A) θk (Φ, A) := I 3/2 (k!)

∞

0

e−tβ(A) |h(t)|tk dt (k = 0, 1, ...).

Then due to Theorem 7.12.2, Φ(A) ≤

∞

θk (Φ, A),

k=0

provided the integrals and series converge. In particular, let β(A) > 0. Then −m

A

1 = (m − 1)!

∞

e−At tm−1 dt (m = 1, 2, ...).

0

In this case gIk (A) (m − 1)!(k!)3/2

θk (Φ, A) = Therefore

A−m ≤

∞

e−tβ(A) tm+k−1 dt.

0

∞

(k + m − 1)!gIk (A) . (m − 1)!(k!)3/2 β(A)m+k k=0

In addition, under (12.5) A−ν =

(12.5)

1 Γ(ν)

∞

(12.6)

e−At tν−1 dt (0 < ν < 1),

0

where Γ(.) is the Euler Gamma-function. Hence, A−ν ≤

7.13

∞

Γ(ν + k)gIk (A) . Γ(ν)(k!)3/2 β(A)k+ν k=0

(12.7)

Triangular Representations of Regular Functions

Lemma 7.13.1 Let a bounded operator A admit a triangular representation with some maximal resolution of the identity. In addition, let f be a function analytic on a neighborhood of σ(A). Then the operator f (A) admits the triangular representation with the same maximal resolution of the identity. Moreover, the diagonal part Df of f (A) is deﬁned by Df = f (D), where D is the diagonal part of A.

116

7. Functions of Non-compact Operators

Proof:

Due to representations (10.1) and (2.2), 1 1 f (λ)Rλ (A)dλ = − f (λ)Rλ (D)(I + V Rλ (D))−1 dλ. f (A) = − 2πi C 2πi C

Consequently, f (A) = − where

1 2πi

1 W =− 2πi

C

C

f (λ)Rλ (D)dλ + W = f (D) + W,

(13.1)

f (λ)Rλ (D)[(I + V Rλ (D))−1 − I]dλ.

But (I + V Rλ (D))−1 − I = −V Rλ (D)(I + V Rλ (D))−1 (λ ∈ C). We thus get W = where

1 2πi

f (λ)ψ(λ)dλ, C

ψ(λ) = Rλ (D)V Rλ (D)(I + V Rλ (D))−1 .

Let P (.) be a maximal resolution of the identity of A. Due to Lemma 7.3.3, for each λ ∈ C, ψ(λ) is a Volterra operator with the same m.r.i. Since P (t) is a bounded operator, we have by Lemma 7.3.2

1 2πi

C

(P (t0 + 0) − P (t0 ))W (P (t0 + 0) − P (t0 )) = f (λ)(P (t0 + 0) − P (t0 ))ψ(λ)(P (t0 + 0) − P (t0 ))dλ = 0

for every gap P (t0 + 0) − P (t0 ) of P (t). Thus, W is a Volterra operator thanks to Lemma 7.3.1. This and (13.1) prove the lemma. 2

7.14

Triangular Representations of Quasiunitary Operators

A bounded linear operator A is called a quasiunitary operator, if A∗ A − I is a completely continuous operator. Lemma 7.14.1 Let A be a linear operator in H satisfying the condition A∗ A − I ∈ Cp (1 ≤ p < ∞),

(14.1)

and let the operator I − A be invertible. Then the operator B = i(I − A)−1 (I + A)

(14.2)

(Cayley’s transformation of A) is bounded and satisﬁes the condition B − B ∗ ∈ Cp .

7.15. Functions of Quasiunitary Operators

Proof:

117

According to (14.1) we can write down A = U (I + K0 ) = U + K,

where U is a unitary operator and both operators K and K0 belong to Cp and K0 = K0∗ . Consequently, B ∗ = −i(I + U ∗ + K ∗ )(I − U ∗ − K ∗ )−1 = −i(I + U −1 + K ∗ )(I − U −1 − K ∗ )−1 . That is, B ∗ = −i(I + U + K0 )(U − I − K0 )−1 , since K0 = K ∗ U . But (14.2) clearly forces B = i(I + U + K)(I − U − K)−1 . Thus, 2BI = (B − B ∗ )/i = T1 + T2 , where T1 = (I + U )[(I − U − K)−1 + (U − I − K0 )−1 ] and

T2 = K(I − U − K)−1 + K0 (U − I − K0 )−1 .

Since K, K0 ∈ Cp , we conclude that T2 ∈ Cp . It remains to prove that T1 ∈ Cp . Let us apply the identity (I − U − K)−1 + (U − I − K0 )−1 = −(I − U − K)−1 (K0 + K)(U − I − K0 )−1 . Hence, T1 ∈ Cp . This completes the proof. 2 Lemma 7.14.2 Under condition (14.1), let A have a regular point on the unit circle. Then A admits the triangular representation (2.2)-(2.4). Proof: Without any loss of generality we assume that A has on the unit circle a regular point λ0 = 1. In the other case we can consider instead of A the operator Aλ−1 0 . Let us consider in H the operator B deﬁned by (14.2). By the previous lemma it satisﬁes (14.2) and therefore due to Theorem 7.6.1 has a triangular representation. The transformation inverse to (14.2) must be deﬁned by the formula (14.3) A = (B − iI)(B + iI)−1 . Now Lemma 7.13.1 ensures the result. 2

7.15

Resolvents and Analytic Functions of Quasiunitary Operators

Assume that A has a regular point on the unit circle and AA∗ − I is a nuclear operator.

(15.1)

118

7. Functions of Non-compact Operators

Due to Lemma 7.14.1, from (1.1), (14.3) and (15.1) it follows that ∞

(|λk (A)|2 − 1) < ∞,

k=1

where λk (A), k = 1, 2, ... are the nonunitary eigenvalues with their multiplicities, that is, the eigenvalues with the property |λk (A)| = 1. Under (15.1) put ∞ (|λk (A)|2 − 1)]1/2 . ϑ(A) = [T r (A∗ A − I) − k=1

If A is a normal operator, then ϑ(A) = 0. Let A have the unitary spectrum, only. That is, σ(A) lies on the unit circle. Then ϑ(A) = [T r (A∗ A − I)]1/2 . Moreover, if the condition ∞

(|λk (A)|2 − 1) ≥ 0

(15.2)

k=1

holds, then T r (A∗ A − I) =

∞

(s2k (A) − 1) ≥

k=1

∞

(|λk (A)|2 − 1) = T r (D∗ D − I) ≥ 0

k=1

and therefore, under (15.2), ϑ(A) ≤ [T r (A∗ A − I)]1/2 .

(15.3)

Theorem 7.15.1 Under condition (15.1), let an operator A have a regular point on the unit circle. Then Rλ (A) ≤

∞ k=0

√

ϑk (A) (λ ∈ σ(A)). k!ρk+1 (A, λ)

(15.4)

Moreover, there are constants a0 , b0 > 0, independent of λ, such that Rλ (A) ≤

b0 ϑ2 (A) a0 exp [ 2 ] (λ ∈ σ(A)). ρ(A, λ) ρ (A, λ)

(15.5)

These constants can be taken from (5.10) or from (5.11). To prove this theorem we need the following Lemma 7.15.2 Under condition (15.1), let an operator A have a regular point on the unit circle. Then A admits the triangular representation (2.2) (due to Lemma 7.14.2). Moreover, N2 (V ) = ϑ(A), where V is the nilpotent part of A.

7.15. Functions of Quasiunitary Operators

119

Proof: By Lemma 7.3.3 and Theorem 6.2.2 T r (D∗ V ) = T r (V ∗ D) = 0. Employing the triangular representation (2.2) we obtain T r (A∗ A − I) = T r [(D + V )∗ (D + V ) − I)] = T r (D∗ D − I) + T r (V ∗ V ). Since D is a normal operator and the spectra of D and A coincide due to Lemma 7.5.1, then we can write T r (D∗ D − I) =

∞

(|λk (A)|2 − 1).

k=1

This equality implies the required result. 2 Proof of Theorem 7.15.1: Inequality (15.4) follows from relation (5.7) and Lemma 7.15.2. Inequality (15.5) follows from Theorem 7.5.5 and Lemma 7.15.2. 2 Let us extend Theorem 7.15.1 to the case Ap (A∗ )p − I ∈ C1

(15.6)

for some integer number p > 1. Corollary 7.15.3 Under condition (15.6), let an operator A have a regular point on the unit circle. Then Rλ (A) ≤ Tλ,p where Tλ,p =

p−1

∞

ϑk (Ap ) √ (λp ∈ σ(Ap )), k+1 (Ap , λp ) k! ρ k=0

Ak λp−k−1 and ρ(Ap , λp ) = inf |tp − λp |. t∈σ(A)

k=0

Moreover, there are constants a0 , b0 > 0, independent of λ, such that Rλ (A) ≤

b0 ϑ2 (Ap ) a0 Tλ,p exp [ ] (λp ∈ σ(Ap )). ρ2 (Ap , λp ) ρ(Ap , λp )

These constants can be taken from (5.10) or from (5.11). This result is due Theorem 7.15.1 and identity (8.3). Theorem 7.15.4 Let a linear operator A satisfy condition (15.1) and have a regular point on the unit circle. If, in addition, f is a holomorphic function on a neighborhood of closed convex hull co(A) of σ(A), then f (A) ≤

∞

sup |f (k) (λ)|

k=0 λ∈co(A)

ϑk (A) . (k!)3/2

(15.7)

120

7. Functions of Non-compact Operators

Proof: The result immediately follows from Lemmas 7.15.2 and 7.11.2. 2 Theorem 7.15.4 is precise: inequality (15.7) becomes equality (10.4) if A is a unitary operator and (10.5) holds, because ϑ(A) = 0 in this case. Example 7.15.5 Let an operator A satisfy the condition (15.1) and have a regular point on the unit circle. Then the inequality Am ≤

m m!rm−k (A)ϑk (A) s

k=0

(m − k)!(k!)3/2

holds for any integer m ≥ 1. Recall that rs (A) is the spectral radius of A.

7.16

Notes

Notions similar to Deﬁnitions 7.1.1, 7.2.1 and 7.2.2 can be found in the books (Gohberg and Krein, 1970) and (Brodskii, 1971), as well as in the papers (Branges, 1963, 1965a and 1965b) and (Brodskii, Gohberg, and Krein, 1969). Theorem 7.5.3 is probably new. The results presented in Sections 7.6-7.8 are based on Chapter 3 of the book (Gil’, 1995), but the proofs are considerably improved. Theorem 7.9.1 was established in the paper (Gil’, 1993). Theorems 7.10.1 and 7.12.1 were derived in the paper (Gil’, 1992). Triangular representations of quasi-Hermitian and quasiunitary operators can be found, in particular, in the paper by V. Brodskii, I. Gohberg and M. Krein (1969), and references given therein.

References [1] Ahiezer, N. I. and Glazman, I. M. (1981). Theory of Linear Operators in a Hilbert Space. Pitman Advanced Publishing Program, Boston, London, Melburn. [2] Branges, L. de. (1963). Some Hilbert spaces of analytic functions I, Proc. Amer. Math. Soc. 106, 445-467. [3] Branges, L. de. (1965a). Some Hilbert spaces of analytic functions II, J. Math. Analysis and Appl., 11, 44-72. [4] Branges, L. de. (1965b). Some Hilbert spaces of analytic functions III, J. Math. Analysis and Appl., 12, 149-186. [5] Brodskii, M. S. (1971). Triangular and Jordan Representations of Linear Operators, Transl. Math. Mongr., v. 32, Amer. Math. Soc., Providence, R.I.

7.13. Notes

121

[6] Brodskii, V.M., Gohberg, I.C. and Krein M.G. (1969). General theorems on triangular representations of linear operators and multiplicative representations of their characteristic functions, Funk. Anal. i Pril., 3,1-27 (in Russian); English Transl., Func. Anal. Appl. 3, 255-276. [7] Dunford, N and Schwartz, J. T. (1966). Linear Operators, part I. General Theory. Interscience publishers, New York, London. [8] Dunford, N and Schwartz, J. T. (1963). Linear Operators, part II. Spectral Theory. Interscience publishers, New York, London. [9] Gil , M. I. (1992). One estimate for the norm of a function of a quasihermitian operator, Studia Mathematica, 103(1), 17-24. [10] Gil’, M. I. (1993). Estimates for Norm of matrix-valued and operator-value functions, Acta Applicandae Mathematicae 32, 5987. [11] Gil’, M. I. (1995). Norm Estimations for Operator-Valued Functions and Applications. Marcel Dekker, Inc, New York. [12] Gohberg, I. C. and Krein, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Mathem. Monographs, v. 18, Amer. Math. Soc., R.I. [13] Gohberg, I. C. and Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space, Trans. Mathem. Monographs, v. 24, Amer. Math. Soc., Providence, R. I.

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8. Bounded Perturbations of Nonselfadjoint Operators

In the present chapter we consider the operators of the kind A + B, where A is a P -triangular operator and B is a bounded operator. We investigate the invertibility conditions and bounds for the spectra of such operators. In particular, we consider perturbations of the von Neumann - Schatten operators and operators with von Neumann - Schatten Hermitian components.

8.1

Invertibility of Boundedly Perturbed P -Triangular Operators

Throughout the present chapter, A is a P -triangular operator in a separable Hilbert space H with the nilpotent part V and diagonal part D. According to Deﬁnition 7.2.2 and Lemma 7.5.1 this means that A = D + V and σ(D) = σ(A).

(1.1)

Besides, D is a normal operator and V is a Volterra one. Moreover, D and V have the same maximal resolution of the identity. In addition, assume that B is a bounded linear operator in H.

(1.2)

In this section we suppose that D is boundedly invertible: rl (D) = inf |σ(D)| > 0. M.I. Gil’: LNM 1830, pp. 123–134, 2003. c Springer-Verlag Berlin Heidelberg 2003

(1.3)

124

8. Bounded Perturbations of P -Triangular Operators

Then due to Lemma 7.3.3, the operator W := D−1 V is a Volterra one. Let V belong to a norm ideal Y with a norm |·|Y , introduced in Section 7.4. Namely, there are positive numbers θk (k ∈ N), with 1/k

θk such that

→ 0 as k → ∞,

V k ≤ θk |V |kY .

Put

(1.4)

ni(W )−1

JY (W ) =

θk |W |kY .

k=0

Recall that ni(W ) is the nilpotency index of W . Clearly, JY (W ) ≤

∞ θk |V |kY . rk+1 (D) k=0 l

Theorem 8.1.1 Under conditions (1.1)-(1.4), let rl−1 (D)BJY (W ) < 1. Then the operator A + B is invertible. Moreover, (A + B)−1 ≤

JY (W ) . rl (D) − BJY (W )

To prove this theorem we need the following simple result Lemma 8.1.2 Let A1 , A2 be linear operators in H. In addition, let A1 be invertible and B1 A−1 1 < 1, where B1 = A2 − A1 . Then A2 is also invertible, with A−1 2 ≤ Proof:

A−1 1 . 1 − B1 A−1 1

Clearly A2 = A1 + B1 = (I + B1 A−1 1 )A1 . Hence −1 A−1 2 = A1

∞ k=0

This proves the result. 2

k (−A−1 1 B1 ) .

8.1. Invertibility

125

Proof of Theorem 8.1.1: From (1.1) and (1.4) it follows ni(W )−1

A−1 = (I + D−1 V )−1 D−1 =

W k D−1

k=0

So A−1 ≤ rl−1 (D)JY (W ). Now the required result is due to the previous lemma. 2 Assume now that the nilpotent part of A belongs to a Neumann-Schatten ideal: N2p (V ) := [T race (V ∗ V )p ]1/2p < ∞ (1.5) for some integer p ≥ 1. Put ni(W )−1

Jp (W ) =

k=0

(p)

θk |W |kY ,

(p)

where θk are deﬁned in Section 6.7. Lemma 8.1.2 and Corollary 7.5.4 imply Corollary 8.1.3 Under conditions (1.1)-(1.3) and (1.5), let BJp (W ) < rl (D). Then operator A + B is invertible. Moreover, (A + B)−1 ≤

Jp (W ) . rl (D) − BJp (W )

Note that Theorem 7.5.5 implies under (1.5) the inequality A−1 ≤ ψp (V, rl (D)), where ψp (V, rl (D)) = a0

p−1 j N2p (V ) j=0

rlj+1 (D)

exp [

(1.6) 2p (V ) b0 N2p

rl2p (D)

].

(1.7)

Besides, the constants a0 , b0 can be taken as in the relations √ c c and b0 = for any c > 1, in particular, a0 = 2 and b0 = 1, a0 = c−1 2 (1.8) or as in the relations a0 = e1/2 and b0 = 1/2. (1.9) Lemma 8.1.2 and (1.6) yield

126

8. Bounded Perturbations of P -Triangular Operators

Corollary 8.1.4 Under the conditions (1.1)-(1.3) and (1.5), let Bψp (V, rl (D)) < 1. Then operator A + B is invertible. Moreover, ψp (V, rl (D)) . 1 − Bψp (V, rl (D))

(A + B)−1 ≤

8.2

Resolvents of Boundedly Perturbed P -Triangular Operators

We need the following result, which immediately follows from Lemma 8.1.2: Lemma 8.2.1 Let A1 , A2 be linear operators in H. In addition, let λ be a regular point of A1 and B1 (A1 − λI)−1 < 1 (B1 = A1 − A2 ). Then λ is regular also for A2 , and (A2 − λI)−1 ≤

(A1 − λI)−1 . 1 − B1 (A1 − λI)−1

Again put ν(λ) = ni (V Rλ (D)). Recall that one can replace ν(λ) by ∞. Under (1.4), let m−1 θk |V |k Y JY (V, m, z) = (z > 0). z k+1 k=0

Then the previous lemma and Theorem 7.5.3 imply Theorem 8.2.2 Under conditions (1.1), (1.2), (1.4), let BJY (V, ν(λ), ρ(D, λ)) < 1. Then λ is a regular point of A + B. Moreover, (A + B − λI)−1 ≤

JY (V, ν(λ), ρ(D, λ)) . 1 − BJY (V, ν(λ), ρ(D, λ))

Under (1.5) denote J˜p (V, m, z) =

m−1 k=0

k N2p (V ) (z > 0). z k+1

Then Lemma 8.2.1 and Corollary 7.5.4 yield

8.3. Roots of Functional Equations

127

Corollary 8.2.3 Under conditions (1.1), (1.2) and (1.5), let BJ˜p (V, ν(λ), ρ(D, λ)) < 1. Then λ is a regular point of A + B. Moreover, (A + B − λI)−1 ≤

J˜p (V, ν(λ), ρ(D, λ)) . 1 − BJ˜p (V, ν(λ), ρ(D, λ))

Furthermore, put ψp (V, z) = a0

p−1 k N2p (V ) k=0

z k+1

exp [

2p b0 N2p (V ) ] (z > 0, V ∈ C2p ), z 2p

where a0 , b0 do not depend on z and can be taken as in (1.8) or as in (1.9). Now Theorem 7.5.5 and Lemma 8.2.1 yield Corollary 8.2.4 Under conditions (1.1), (1.2) and (1.5), let Bψp (V, ρ(D, λ)) < 1. Then λ is a regular point of A + B. Moreover, (A + B − λI)−1 ≤

8.3

ψp (V, ρ(D, λ)) . 1 − Bψp (V, ρ(D, λ))

Roots of Scalar Equations

Consider the scalar equation ∞

ak z k = 1

(3.1)

k=1

where the coeﬃcients ak , k = 1, 2, ... have the property γ0 ≡ 2 max k |ak | < ∞. k

We will need the following Lemma 8.3.1 Any root z0 of equation (3.1) satisﬁes the estimate |z0 | ≥ 1/γ0 . Proof:

Set in (3.1) z0 = xγ0−1 . We have 1=

∞ k=1

ak γ0−k xk ≤

∞ k=1

|ak |γ0−k |x|k .

(3.2)

128

8. Bounded Perturbations of P -Triangular Operators

But

∞

|ak |γ0−k ≤

k=1

∞

2−k = 1

k=1

and therefore, |x| ≥ 1. Hence, |z0 | =

γ0−1 |x|

≥ γ0−1 . As claimed. 2

Note that the latter lemma generalizes the well-known result for algebraic equations, cf. the book (Ostrowski, 1973, p. 277). Lemma 8.3.2 The extreme right (unique positive) root za of the equation p−1 j=0

1 1 1 exp [ (1 + 2p )] = a (a ≡ const > 0) y j+1 2 y

satisﬁes the inequality za ≤ δp (a), where pe/a δp (a) := [ln (a/p)]−1/2p Proof:

if a ≤ pe, . if a > pe

(3.3)

(3.4)

Assume that pe ≥ a.

(3.5)

Since the function f (y) ≡

p−1 j=0

1 1 1 exp [ (1 + 2p )] y j+1 2 y

is nonincreasing and f (1) = pe, we have za ≥ 1. But due to (3.3), za = 1/a

p−1

za−j exp [(1 + za−2p )/2] ≤ pe/a.

j=0

So in the case (3.5), the lemma is proved. Let now pe < a. Then za ≤ 1. But p−1

xj+1 ≤ pxp ≤ p exp [xp − 1] ≤ p exp [(x2p + 1)/2 − 1]

j=0

= p exp [x2p /2 − 1/2] (x ≥ 1). So f (y) =

p−1 j=0

1 y j+1

1 1 1 exp [ (1 + 2p )] ≤ p exp [ 2p ] (y ≤ 1). 2 y y

(3.6)

8.4. Spectral Variations

129

But za ≤ 1 under (3.6). We thus have a = f (za ) ≤ p exp [ Or

1 ]. za2p

za2p ≤ 1/ln (a/p).

This ﬁnishes the proof. 2

8.4

Spectral Variations

Deﬁnition 8.4.1 Let A and B be linear operators in H. Then the quantity svA (B) := sup

inf |µ − λ|

µ∈σ(B) λ∈σ(A)

is called the spectral variation of a B with respect to A. In addition, hd(A, B) := max{svA (B), svB (A)} is the Hausdorﬀ distance between the spectra of A and B. First, we will prove the following technical lemma Lemma 8.4.2 Let A1 and A2 be linear operators in H with the same domain and q ≡ A1 − A2 < ∞. In addition, let Rλ (A1 ) ≤ F (ρ−1 (A1 , λ)) (λ ∈ σ(A1 ))

(4.1)

where F (x) is a monotonically increasing non-negative function of a nonnegative variable x, such that F (0) = 0 and F (∞) = ∞. Then svA1 (A2 ) ≤ z(F, q), where z(F, q) is the extreme right-hand (positive) root of the equation 1 = qF (1/z). Proof:

(4.2)

Due to (4.1) and Lemma 8.2.1, 1 ≤ qF (ρ−1 (A, µ)) for all µ ∈ σ(B).

(4.3)

Compare this inequality with (4.2). Since F (x) monotonically increases, z(F, q) is a unique positive root of (4.2), and ρ(A, µ) ≤ z(F, q). This proves the required result. 2 The previous lemma and Theorem 7.5.3 imply

130

8. Bounded Perturbations of P -Triangular Operators

Theorem 8.4.3 Let conditions (1.1), (1.2) and (1.4) hold. Then svD (A + B) ≤ zY (V, B) where zY (V, B) is the extreme right-hand (positive) root of the equation 1 = BJY (ν0 , V, z) ≡ B

ν 0 −1 k=0

θk |V |kY z k+1

and ν0 = sup

λ∈σ(D)

ni (V Rλ (D)).

Note that to estimate the root zY (V, B), one can use Lemma 8.3.1. Moreover, Lemma 8.4.2 and Theorem 7.5.5 imply Theorem 8.4.4 Let conditions (1.1), (1.2) and (1.5) hold. Then svD (A+B) ≤ zp (V, B), where zp (V, B) is the extreme right-hand (positive) root of the equation p−1 j 2p N2p (V ) 1 N2p (V ) + exp [ ]. (4.4) 1 = B z j+1 2 2z 2p j=0 Substitute in (4.4) the equality z = xN2p (V ). Then we have equation (3.3) with N2p (V ) . a= B Thanks to Lemma 8.3.2, we get yp (V, B) ≤ δp (V, B), where δp (V, B) := N2p (V )δp (

N2p (V ) ) B

and δp (.) is deﬁned by (3.4). We thus have derived Corollary 8.4.5 Under the hypothesis of Theorem 8.4.4, svD (A + B) ≤ δp (V, B).

8.5

Perturbations of Compact Operators

First assume that A ∈ C2 .

(5.1)

By virtue of Lemma 8.4.2 and Theorem 6.4.2 we arrive at the following result.

8.5. Compact Operators

131

Theorem 8.5.1 Let condition (5.1) hold and B be a bounded operator in H. Then svA (A + B) ≤ z˜1 (A, B), where z˜1 (A, B) is the extreme right-hand (positive) root of the equation 1=

1 g 2 (A) B exp [ + ]. z 2 2z 2

(5.2)

Furthermore, substitute in (5.2) the equality z = xg(A). Then we arrive at the equation 1 1 g(A) 1 exp [ + 2 ] = . z 2 2z B ˜ 1 (A, B), where Applying Lemma 8.3.2 to this equation, we get z˜1 (A, B) ≤ ∆ ˜ 1 (A, B) := g(A)δ1 (g(A)/B) ∆ where δ1 is deﬁned by (3.4) with p = 1. So eB ˜ ∆1 (A, B) := g(A) [ln (g(A)/B)]−1/2

if g(A) ≤ eB, . if g(A) > eB

(5.3)

Now Theorem 8.5.1 yields Corollary 8.5.2 Under the hypothesis of Theorem 8.5.1, for any µ ∈ σ(A + B), there is a µ0 ∈ σ(A), such that |µ − µ0 | ≤ δ˜1 (A, B). In particular, ˜ 1 (A, B), rs (A + B) ≤ rs (A) + ∆

(5.4)

˜ 1 (A, B), 0} and α(A + B) ≤ α(A) + ∆ ˜ 1 (A, B). rl (A + B) ≥ max{rl (A) − ∆ (5.5) Remark 8.5.3 According to Lemma 6.5.2, in Theorem 8.5.1 and its corol√ lary, one can replace g(A) by 2N2 (AI ). Now let A ∈ C2p (p = 2, 3, ...).

(5.6)

Then by virtue of Lemma 8.4.2 and Theorem 6.7.3 we arrive at the following result. Theorem 8.5.4 Let condition (5.6) hold and B be a bounded operator in H. Then svA (A + B) ≤ y˜p (A, B), where y˜p (A, B) is the extreme right-hand (positive) root of the equation 1 = B

p−1 (2N2p (A))m 1 (2N2p (A))2p exp [ + ]. m+1 z 2 2z 2p m=0

(5.7)

132

8. Bounded Perturbations of P -Triangular Operators

Furthermore, substitute in (5.7) the equality z = x2N2p (A) and apply Lemma 8.3.2. Then we get y˜p (A, B) ≤ ∆p (A, B), where ∆p (A, B) := 2N2p (A)δp (2N2p (A)/B). Recall that δp (.) is deﬁned by (3.4). So peB ∆p (A, B) := 2N2p (A) [ln (2N2p (A)/pB)]−1/2p

if 2N2p (A) ≤ epB, . if 2N2p (A) > epB (5.8)

Now Theorem 8.5.4 yields Corollary 8.5.5 Under the hypothesis of Theorem 8.5.4, for any µ ∈ σ(A + B), there is a µ0 ∈ σ(A), such that |µ − µ0 | ≤ ∆p (A, B). In particular, ˜ 1 (A, B). relations (5.4) and (5.5) hold with ∆p (A, B) instead of ∆ Remark 8.5.6 According to Theorem 7.9.1, in Theorem 8.5.4 and its corollary one can replace 2N2p (A) by β˜p N2p (AI ).

8.6

Perturbations of Operators with Compact Hermitian Components

First, let Dom (A) = Dom (A∗ ) and A − A∗ ∈ C2 .

(6.1)

Then by virtue of Lemma 8.4.2 and Theorem 7.7.1 we arrive at the following result. Theorem 8.6.1 Let condition (6.1) hold and B be a bounded operator in H. Then svA (A + B) ≤ x1 (A, B), where x1 (A, B) is the extreme right-hand (positive) root of the equation 1 = B

∞ g k (A) √I . k!z k+1 k=0

(6.2)

√ Recall that gI (A) ≤ 2N2 (AI ). According to Theorem 7.7.1, one can replace (6.2) by the equation 1=

1 g 2 (A) B exp [ + I 2 ]. z 2 2z

(6.3)

Furthermore, substitute in (6.3) the equality z = xgI (A) and apply Lemma 8.3.2. Then we can assert that extreme right-hand root of equation (6.3) is less than τ1 (A, B), where τ1 (A, B) = gI (A)δ1 (gI (A)/B).

8.6. Operators with Compact Components

So, according to (3.4), eB τ1 (A, B) = gI (A) [ln (gI (A)/B)]−1/2

133

if gI (A) ≤ eB, . if gI (A) > eB

Hence, Theorem 8.6.1 yields Corollary 8.6.2 Let condition (6.1) hold and B be a bounded operator in H. Then for any µ ∈ σ(A + B), there is a µ0 ∈ σ(A), such that |µ − µ0 | ≤ τ1 (A, B). In particular, rs (A + B) ≤ rs (A) + τ1 (A, B), rl (A + B) ≥ max{rl (A) − τ1 (A, B), 0} and α(A + B) ≤ α(A) + τ1 (A, B). Now let

A − A∗ ∈ C2p (p = 2, 3, ...).

(6.4)

Then by virtue of Lemma 8.4.2 and Theorem 7.9.1 we arrive at the following result. Theorem 8.6.3 Let condition (6.4) hold and B be a bounded operator in H. Then svD (A + B) ≤ x ˜p (A, B), where x ˜p (A, B) is the extreme right-hand (positive) root of the equation p−1 ∞ wpk+m (A) √p . 1 = B k!z pk+m+1 m=0 k=0

where

(6.5)

wp (A) = β˜p N2p (AI ).

According to Theorem 7.9.1, one can replace (6.5) by the equation 1 = B

p−1 (wp (A))m 1 (wp (A))2p exp [ + ]. m+1 z 2 2z 2p m=0

(6.6)

Furthermore, substitute in (6.6) the equality z = xwp (A) and apply Lemma 8.3.2. Then we can assert that the extreme right-hand root of equation (6.6) is less than mp (A, B) := wp (A)δp (wp (A)/B), where δp (a) is deﬁned by (3.4). That is, peB mp (A, B) = wp (A) [ln (wp (A)/pB)]−1/2p

if wp (A) ≤ epB . if wp (A) > epB

Now Theorem 8.6.3 implies Corollary 8.6.4 Let conditions (1.2) and (6.6) hold. Then for any µ ∈ σ(A + B), there is a µ0 ∈ σ(A), such that |µ − µ0 | ≤ mp (A, B). In particular, rs (A + B) ≤ rs (A) + mp (A, B), rl (A + B) ≥ max{rl (A) − τ˜p (A, B), 0} and α(A + B) ≤ α(A) + τ˜p (A, B).

134

8.7

8. Bounded Perturbations of P -Triangular Operators

Notes

The material of this chapter is based on the papers (Gil’, 2002) and (Gil’, 2003). About the well-known perturbations results see (Kato, 1966), (Baumg´artel, 1985) and references therein.

References [1] Baumg´ artel, H. (1985). Analytic Perturbation Theory for Matrices and Operators. Operator Theory, Advances and Appl., 52. Birkh´ auser Verlag, Basel, Boston, Stuttgart. [2] Gil’, M. I. (2002). Invertibility and spectrum localization of nonselfadjoint operators, Advances in Applied Mathematics, 28, 40-58. [3] Gil’, M. I. (2003). Inner bounds for spectra of linear operators, Proceedings of the American Mathematical Society , (to appear) [4] Kato, T. (1966). Perturbation Theory for Linear Operators, Springer-Verlag. New York. [5] Ostrowski, A. M. (1973). Solution of Equations in Euclidean and Banach spaces. Academic Press, New York - London.

9. Spectrum Localization of Nonself-adjoint Operators

In the present chapter we consider operators of the form A = D + V+ + V− , where D is a normal operator and V± are Volterra (quasinilpotent compact) operators. Numerous integral, integro-diﬀerential operators and inﬁnite matrices can be represented in such a form. We investigate the invertibility conditions and bounds for the spectra of the mentioned operators.

9.1

Invertibility Conditions

Let P (t) (a ≤ t ≤ b; −∞ ≤ a < b ≤ ∞) be a maximal resolution of the identity in a separable Hilbert space H (see Section 7.2). Let A = D + V + + V− ,

(1.1)

where D is a normal operator and V± are Volterra operators satisfying the conditions P (t)V+ P (t) = V+ P (t); P (t)V− P (t) = P (t)V− , P (t)Dh = DP (t)h (a ≤ t ≤ b; h ∈ Dom (D)).

(1.2)

As above, Y is a norm ideal of compact operators in H, which is complete in an auxiliary norm | · |Y . In addition, there are positive numbers θk (k ∈ N) 1/k with θk → 0 (k → ∞), for which, for arbitrary Volterra operators V ∈ Y , k V ≤ θk |V |kY (k = 1, 2, ...). It is assumed that V± ∈ Y . That is, V±k ≤ θk |V± |kY (k = 1, 2, ...). M.I. Gil’: LNM 1830, pp. 135–149, 2003. c Springer-Verlag Berlin Heidelberg 2003

(1.3)

136

9. Spectrum Localization

In addition, in this section it is assumed that D is boundedly invertible: rl (D) = inf |σ(D)| > 0.

(1.4)

Since D is normal and V± are Volterra operators, due to Lemma 7.3.3 conditions (1.2) are enough to guarantee that W± := D−1 V± are also Volterra operators. Under (1.2)-(1.4), put ni(W± )−1

JY (W± ) ≡

θk |W± |kY ,

(1.5)

k=0

where ni(V ) again denotes the “nilpotency index” of a quasinilpotent operator V (see Section 7.5). With this notation we have Theorem 9.1.1 Under conditions (1.1)-(1.4), let ζY (A) := max {

1 1 − W+ , − W− } > 0. JY (W− ) JY (W+ )

(1.6)

Then A is boundedly invertible, and A−1 ≤

1 . rl (D)ζY (A)

(1.7)

The proof of Theorem 9.1.1 is presented in the next section. Note that in Theorem 9.1.1, one can replace JY (W± ) by IY (W± ) :=

∞

θk |W± |kY .

k=0

Consider operators, whose oﬀ-diagonal parts belong to the Neumann-Schatten ideal C2p with some integer p ≥ 1: V± ∈ C2p . Again put (p)

θj

(1.8)

1 , = [j/p]!

where [x] means the integer part of a real number x. For a Volterra operator V ∈ C2p denote ni(V )−1 (p) k θk N2p (V ). Jp (V ) = k=0

Now Theorem 9.1.1 and Corollary 6.9.4 imply

9.2. Proof of Theorems 9.1.1 and 9.1.3

137

Corollary 9.1.2 Let relations (1.2), (1.4) and (1.8) hold. In addition, let ζ2p (A) ≡ max{

1 1 − W+ , − W− } > 0. Jp (W− ) Jp (W+ )

(1.9)

Then operator A represented by (1.1) is boundedly invertible. Moreover, A−1 ≤

1 . ζ2p (A)rl (D)

It is simple to see that, in this corollary, one can replace Jp (W± ) by I2p (W± ), where p−1 j+pk ∞ N2p (W± ) √ I2p (W± ) := . k! j=0 k=0 Moreover, put ψp (W± ) := a0

p−1

j 2p N2p (W± ) exp [b0 N2p (W± )].

j=0

Besides, the constants a0 , b0 can be taken as in the relations √ c c and b0 = for any c > 1, in particular, a0 = 2 and b0 = 1, a0 = c−1 2 (1.10) or as in the relations (1.11) a0 = e1/2 and b0 = 1/2. In the next section we also prove Theorem 9.1.3 Let relations (1.2), (1.4) and (1.8) hold. In addition, let ζ˜p (A) := max{

1 1 − W+ , − W− } > 0. ψp (W− ) ψp (W+ )

Then operator A represented by (1.1) is boundedly invertible. Moreover, A−1 ≤

9.2

1 . ˜ ζp (A)rl (D)

Proofs of Theorems 9.1.1 and 9.1.3

We need the following simple Lemma 9.2.1 Under conditions (1.2) and (1.4), let θ0 := (D + V− )−1 V+ < 1.

(2.1)

Then operator A represented by (1.1) is boundedly invertible. Moreover, A−1 ≤

(D + V− )−1 . 1 − θ0

(2.2)

138

9. Spectrum Localization

Proof:

According to (1.1) we have A = (D + V− )(I + (D + V− )−1 V+ ).

(2.3)

Thanks to (1.2) and Lemma 7.5.1, σ(D + V± ) = σ(D).

(2.4)

So according to (1.4), D + V± is invertible. Moreover, under condition (2.1), the operator I + (D + V− )−1 V+ is invertible and (I + (D + V− )−1 V+ )−1 ≤

∞

((D + V− )−1 V+ )k ≤

k=0 ∞

θ0k = (1 − θ0 )−1 .

k=0

So due to (2.3) A−1 ≤ (I + (D + V− )−1 V+ )−1 (D + V− )−1 . This proves the required result. 2 Proof of Theorem 9.1.1: Since V± ∈ Y , (D + V− )−1 V+ = (I + D−1 V− )−1 D−1 V+ = (I + W− )−1 W+ ≤ W+

∞

W−k = W+

k=0

W−k ≤

k=0

ni(W− )−1

W+

ni(W− )−1

θk |W− |kY = W+ JY (W− ).

k=0

Hence θ0 ≤ W+ JY (W− ). But condition (1.6) implies that at least one of the following inequalities W+ JY (W− ) < 1

(2.5)

W− JY (W+ ) < 1

(2.6)

or

9.3. Resolvents of Quasinormal Operators

139

are valid. If condition (2.5) holds, then (2.1) is valid. Moreover, since D is a normal operator, D−1 = rl (D)−1 . Thus, (D + V− )−1 = (I + W− )−1 D−1 ≤ D−1

∞

W−k =

k=0

rl (D)−1

ni(W− )−1

W−k ≤ rl (D)−1

ni(W− )−1

k=0

θk |W− |kY = rl (D)−1 JY (W− ).

k=0

Thus, under (2.5), Lemma 9.2.1 yields the inequality A−1 0 ≤

1 JY (W− ) = . rl (D)(1 − W+ JY (W− )) rl (D)(JY−1 (W− ) − W+ )

(2.7)

Interchanging W− and W+ , under condition (2.6), we get A−1 0 ≤

1 rl (D)(JY−1 (W+ )

− W− )

.

This relation and (2.7) yield the required result. 2 Proof of Theorem 9.1.3: Due to Theorem 6.7.3, (D + V± )−1 = (I + W± )−1 D−1 ≤ ψp (W± )ρ−1 l (D). In addition, (D + V− )−1 V+ = (I + W− )−1 W+ ≤ ψp (W− )W+ and

(D + V+ )−1 V− ≤ ψp (W+ )W− .

Now the required result is due to Lemma 9.2.1. 2

9.3

Resolvents of Quasinormal Operators

For a V ∈ Y , denote JY (V, m, z) :=

m−1

z −1−k θk |V |kY (z > 0).

k=0 −1

Due to Lemma 7.3.4, (D − λI) V± is a quasinilpotent operator for any λ ∈ σ(D). Put ν± (λ) ≡ ni((D − λI)−1 V± ). Everywhere below we can replace ν± (λ) by ∞. Again, Rλ (A) is the resolvent and ρ(D, λ) = inf z∈σ(D) |s − z|.

140

9. Spectrum Localization

Lemma 9.3.1 Under conditions (1.2), (1.3), for a λ ∈ σ(D), let ζ(A, λ) ≡ max{

1 − V+ , JY (V− , ν− (λ), ρ(D, λ))

1 − V− } > 0. JY (V+ , ν+ (λ), ρ(D, λ))

(3.1)

Then λ is a regular point of operator A represented by (1.1). Moreover, Rλ (A) ≤ Proof:

1 . ζ(A, λ)ρ(D, λ)

(3.2)

Since, D is a normal operator, (D − λI)−1 = ρ−1 (λ, D). Thus, |(D − λI)−1 V± |Y ≤ (D − λI)−1 |V± |Y = ρ−1 (λ, D)|V± |Y .

Hence, (D − λI)−1 V+

ν− (λ)−1

θk |(D − λI)−1 V− |kY ≤

k=0 ν− (λ)−1

V+

θk ρ−1−k (λ, D)|V− |kY = V+ JY (V− , ν− (λ), ρ(D, λ)).

k=0

Similarly, (D − λI)−1 V−

ν+ (λ)−1

θk |(D − λI)−1 V+ |kY ≤ V− JY (V+ , ν+ (λ), ρ(D, λ)).

k=0

Now Theorem 9.1.1 with A − λI = D + V+ + V− − λI instead of A, yields the required result. 2 Furthermore, Lemma 9.3.1 implies Corollary 9.3.2 Under conditions (1.1), (1.2) and (1.3), for any µ ∈ σ(A), there is a µ0 ∈ σ(D), such that, either µ = µ0 , or both the inequalities V+ JY (V− , ν− (µ), |µ − µ0 |) ≥ 1 and V− JY (V+ , ν+ (µ), |µ − µ0 |) ≥ 1 are true.

(3.3)

9.4. Upper Bounds for Spectra

141

This result is exact in the following sense: if either V− = 0, or (and) V+ = 0, then due to the latter corollary, σ(A) = σ(D).

(3.4)

Now let condition (1.8) hold. Put J˜p (V± , m, z) =

m−1 k=0

(p)

k θk N2p (V± ) (z > 0). k+1 z

Theorem 6.7.1 and Lemma 9.3.1 imply Corollary 9.3.3 Under conditions (1.2) and (1.8), for a λ ∈ σ(D), let ζ2p (λ, A) ≡ max{

1 − V+ , ˜ Jp (V− , ν− (λ), ρ(D, λ))

1 − V− } > 0. ˜ Jp (V+ , ν+ (λ), ρ(D, λ)) Then λ is a regular point of operator A, represented by (1.1). Moreover, Rλ (A) ≤

1 . ρ(D, λ)ζ2p (λ, A)

Furthermore, thanks to Theorem 9.1.3 with A−λ instead of A, we can replace J˜p by the function ψp (V± , z) = a0

p−1 j N2p (V± )

z j+1

j=0

exp [

2p b0 N2p (V± ) ], z 2p

where a0 , b0 can be taken from (1.10) or from (1.11). Then we have Corollary 9.3.4 Under conditions (1.2) and (1.8), for a λ ∈ σ(D), let ζ˜2p (λ, A) ≡ max{

1 − V+ , ψp (V− , ρ(D, λ))

1 − V− } > 0. ψp (V+ , ρ(D, λ)) Then λ is a regular point of operator A, represented by (1.1). Moreover, Rλ (A) ≤

1 . ρ(D, λ)ζ2p (λ, A)

142

9.4

9. Spectrum Localization

Upper Bounds for Spectra

Recall that svD (A) denotes the spectral variatin of A with respect to D. Put τ (A) := min{V− , V+ }, V+ if V+ ≥ V− ˜ V := V− if V− > V+ and ν˜0 = sup

λ∈σ(D)

(4.1) (4.2)

ni ((D − λI)−1 V˜ ).

In the sequel one can replace ν˜0 by ∞. Theorem 9.4.1 Under conditions (1.1), (1.2), let V˜ ∈ Y and V˜ k ≤ θk |V˜ |kY (k = 1, 2, ...).

(4.3)

τ (A)JY (V˜ , ν˜0 , z) = 1

(4.4)

Then the equation has a unique positive root zY (A). Moreover, svD (A) ≤ zY (A). Proof:

Due to Corollary 9.3.2, τ (A)JY (V˜ , ν˜0 , ρ(D, µ)) ≥ 1

for any µ ∈ σ(A). Comparing this inequality with (4.4), we have ρ(D, µ) ≤ zY (A). This inequality proves the theorem. 2 Lemma 9.4.2 Under the conditions (1.1), (1.2) and V˜ ∈ C2p (p = 1, 2, ...),

(4.5)

the equation τ (A)

p−1 j ˜ N2p (V ) j=0

z j+1

2p ˜ (V ) N2p 1 exp [ (1 + )] = 1. 2p 2 z

(4.6)

has a unique positive root z2p (A). Moreover, svD (A) ≤ z2p (A). Proof:

Due to Corollary 9.3.4, τ (A)

p−1 j ˜ 2p ˜ N2p N2p (V ) (V ) 1 exp [ )] ≥ 1 (1 + j+1 2p ρ (D, µ) 2 ρ (D, µ) j=0

(4.7)

9.5. Inner Bounds for Spectra

143

for any µ ∈ σ(A). Comparing this inequality with (4.6), we get ρ(D, µ) ≤ z2p (A). This inequality proves the required result. 2 To estimate z2p (A), substitute z = xN2p (V˜ ) in (4.6) and use Lemma 8.3.2. Then z2p (A) ≤ φp (A) = N2p (V˜ )δp (a), where a = N2p (V˜ )/τ (A) and pe/a if a ≤ pe δp (a) := . (4.8) [ln (a/p)]−1/2p if a > pe That is, φp (A) :=

peτ (A) N2p (V˜ )[ln (N2p (V˜ )/pτ (A))]−1/2p

if N2p (V˜ ) ≤ τ (A)pe . if N2p (V˜ ) > τ (A)pe (4.9)

Now the previous lemma yields Corollary 9.4.3 Under conditions (1.1), (1.2) and (4.5), svD (A) ≤ φp (A). In particular, rs (A) ≤ rs (D) + φp (A), provided D is bounded. In the case V˜ ∈ C2 we have δ1 (a) := and

φ1 (A) :=

e/a [ln a]−1/2

eτ (A) N2 (V˜ ) [ln (N2 (V˜ )/τ (A))]−1/2

if a ≤ e if a > e

if N2 (V˜ ) ≤ τ (A)e . if N2 (V˜ ) > τ (A)e

(4.10)

(4.11)

Now (4.7) implies svD (A) ≤ z2 (A) ≤ φ1 (A).

(4.12)

Remark 9.4.4 Everywhere below V± can be replaced by their upper bounds, since the right roots of equations (4.4) and (4.6) increase, when the coeﬃcients of these equations increase.

9.5

Inner Bounds for Spectra

Again, let there be a monotonically increasing continuous scalar-valued function F (z) (z ≥ 0) with the properties F (0) = 0, F (∞) = ∞

(5.1)

(λI − A)−1 ≤ F (ρ−1 (A, λ))

(5.2)

such that the inequality

144

9. Spectrum Localization

holds, where ρ(A, λ) is the distance between σ(A) and a regular point λ ∈ C of A. Recall that τ (A) := min{V− , V+ } and denote by y(τ, F ) the unique positive root of the equation τ (A)F (1/z) = 1 (z > 0).

(5.3)

Now we are in a position to formulate the main result of the present section. Theorem 9.5.1 Let A be deﬁned by (1.1) and conditions (1.2) and (5.2) hold. Then (5.4) svA (D) ≤ y(τ, F ). The proof of this theorem is presented below. Recall that rl (A) := inf |σ(A)|, α(A) := sup Re σ(A). Corollary 9.5.2 Under the hypothesis of Theorem 9.5.1, the following inequalities are true: rs (A) ≥ max{0, rs (D) − y(τ, F )} if D is bounded,

(5.5)

rl (A) ≤ rl (D) + y(τ, F ) and

(5.6)

α(A) ≥ α(D) − y(τ, F ) if α(D) < ∞.

(5.7)

Indeed, take µ in such a way that |µ| = rs (D). Then due to (5.4), there is µ0 ∈ σ(A), such that |µ0 | ≥ rs (D) − y(τ, F ). Hence, (5.5) follows. Similarly, inequality (5.6) can be proved. Furthermore, take µ in such a way that Re µ = α(D). Due to (1.6) for some µ0 ∈ σ(A), |Re µ0 − α(D)| ≤ y(τ, F ). So, either Re µ0 ≥ α(D), or Re µ0 ≥ α(D) − y(τ, F ). Thus, inequality (5.7) is also proved. Proof of Theorem 9.5.1: Take the operator B+ = D + V+ . Then due to (2.4) σ(B+ ) = σ(D). Due to to Lemma 8.4.2, for any µ ∈ σ(D), there is µ0 ∈ σ(A), such that (5.8) |µ0 − µ| ≤ z− , where z− is the unique positive root of the equation V− F (1/z) = 1 (z ≥ 0). Now, take B− = D + V− . Then due to relation (2.4), we get σ(B− ) = σ(D). Similarly to (5.8), we have that for any µ ∈ σ(D), there is µ0 ∈ σ(A), such that (5.9) |µ0 − µ| ≤ z+ , where z+ is the unique positive root of the equation V+ F (1/z) = 1, since A − B− = V+ . Relations (5.8) and (5.9) prove the required result. 2

9.7. Hilbert-Schmidt Operators

9.6

145

Bounds for Spectra of Hilbert-Schmidt Operators

Assume that A ∈ C2 .

(6.1)

Recall that g(A) is deﬁned in Section √ 6.4. According to Lemma 6.5.1 everywhere below one can replace g(A) by 2N2 (AI ). Under (6.1), denote by y˜2 (A, τ ) the unique non-negative root of the equation τ (A) 1 g 2 (A) ] = 1, (6.2) exp [ + z 2 2z 2 where τ (A) is deﬁned by (4.1). Theorem 9.6.1 Let conditions (1.1), (1.2) and (6.1) hold. Then the relations (4.12) and svA (D) ≤ y˜2 (A, τ ) are valid. Proof: The required result is due to Theorems 6.4.2 and 9.5.1, and Corollary 9.4.3. 2 Substitute z = g(A)x in (6.2) and use Lemma 8.3.2. Then we get ˜ 2 (A) := g(A)δ1 ( y˜2 (A, τ ) ≤ ∆ where δ1 (a) is deﬁned by (4.10). That is, eτ (A) ˜ ∆2 (A) := g(A)[ln (g(A)/τ (A))]−1/2

g(A) ), τ (A)

if g(A) ≤ eτ (A) . if g(A) > eτ (A)

Thus Theorem 9.6.1 and relations (4.12) imply Corollary 9.6.2 Let relations (1.1), (1.2) and (6.1) hold. Then ˜ 2 (A). svD (A) ≤ φ1 (A) and svA (D) ≤ ∆ In particular, ˜ 2 (A)} ≤ rs (A) ≤ rs (D) + φ1 (A), max{0, rs (D) − ∆ ˜ 2 (A) max{0, rl (D) − φ1 (A)} ≤ rl (A) ≤ rl (D) + ∆ ˜ 2 (A) ≤ α(A) ≤ α(D) + φ1 (A). and α(D) − ∆

(6.3)

146

9.7

9. Spectrum Localization

Von Neumann-Schatten Operators

Assume that A ∈ C2p for an integer p > 1

(7.1)

and denote by yp (A, τ ) the unique non-negative root of the equation τ (A)

p−1 (2N2p (A))m 1 (2N2p (A))2p + exp [ ] = 1, z m+1 2 2z 2p m=0

(7.2)

where τ (A) is deﬁned by (4.1). Theorem 9.7.1 Let conditions (1.1), (1.2) and (7.1) hold. Then the inequalities (4.7) and svA (D) ≤ yp (A, τ ) are valid. Proof: The required result is due to Theorems 6.7.4 and 9.5.1, and Corollary 9.4.4. 2 Substitute z = 2N2p (A)x in (7.2) and use Lemma 8.3.2. Then we arrive at the inequality yp (A, τ ) ≤ ∆p (A) := N2p (A)δp ( where δp (a) is deﬁned by (4.8). That is, p e τ (A) ∆p (A) := 2N2p (A)[ln (2N2p (A)/τ (A))]−1/2p

2N2p (A) ), τ (A)

if 2N2p(A) ≤ p e τ (A) . if 2N2p (A) > p e τ (A) (7.3)

Thus Theorem 9.7.1 and Corollary 9.4.3 imply Corollary 9.7.2 Let relations (1.1), (1.2) and (7.1) hold. Then svD (A) ≤ φp (A) and svA (D) ≤ ∆p (A). In particular, max{0, rs (D) − ∆p (A)} ≤ rs (A) ≤ rs (D) + φp (A), max{0, rl (D) − φp (A)} ≤ rl (A) ≤ rl (D) + ∆p (A) and α(D) − ∆p (A) ≤ α(A) ≤ α(D) + φp (A).

9.8. Operators with Hilbert-Schmidt Components

9.8

147

Operators with Hilbert-Schmidt Hermitian Components

In this section it is assumed that Dom (A) = Dom (A∗ ) and A has the Hilbert-Schmidt imaginary component AI ≡ (A − A∗ )/2i: N22 (AI ) = T race (AI )2 < ∞.

(8.1)

Recall that √ gI (A) is deﬁned in Section 7.7. Everywhere below on can replace gI (A) by 2N2 (AI ). Under (8.1), denote by yH (A, τ ) the unique non-negative root of the equation τ (A) 1 g 2 (A) (8.2) exp [ + I 2 ] = 1. z 2 2z Theorem 9.8.1 Let conditions (1.1), (1.2) and (8.1) hold. Then the relations (4.12) and svA (D) ≤ yH (A, τ ) are valid. Proof: The required result is due to Theorems 7.7.1 and 9.5.1, and Corollary 9.4.3. 2 Substitute z = gI (A)x in (8.2) and apply Lemma 8.3.2. Then yH (A, τ ) ≤ ∆H (A) := gI (A)δ1 ( where δ1 (a) is deﬁned by (4.10). That is, e τ (A) ∆H (A) := gI (A)[ln (gI (A)/τ (A))]−1/2

gI (A) ), τ (A)

if gI (A) ≤ eτ (A) . if gI (A) > e τ (A)

Thanks to (4.12), Theorem 9.8.1 implies Corollary 9.8.2 Let relations (1.1), (1.2) and (8.1) hold. Then svD (A) ≤ φ1 (A) and svA (D) ≤ ∆H (A). In particular, max{0, rs (D) − ∆H (A)} ≤ rs (A) ≤ rs (D) + φ1 (A), max{0, rl (D) − φ1 (A)} ≤ rl (A) ≤ rl (D) + ∆H (A) and α(D) − ∆H (A) ≤ α(A) ≤ α(D) + φ1 (A).

148

9. Spectrum Localization

9.9

Operators with Neumann-Schatten Hermitian Components

In this section it is assumed that the Hermitian component AI = (A−A∗ )/2i belongs to the Neumann-Schatten ideal C2p with some integer p > 1: 1/2p Np (AI ) = [T race A2p < ∞. I ]

(9.1)

Recall that β˜p is deﬁned in Section 7.9. Set wp (A) := β˜p N2p (AI ). Let xp (A, τ ) be the unique positive root of the equation p−1 wpm (A) 1 wp2p (A) + τ (A) exp [ ] = 1, z m+1 2 2z 2p m=0

(9.2)

where τ (A) is deﬁned by (4.1). Theorem 9.9.1 Let conditions (1.1), (1.2) and (9.1) hold. Then the relations (4.7) and svA (D) ≤ xp (A, τ ) are valid. Proof: The required result is due to Theorems 7.9.1 and 9.5.1, and Corollary 9.4.3. 2 Substitute z = wp (A)x in (9.2) and apply Lemma 8.3.2. Then xp (A, τ ) ≤ mp (A) := wp (A)δp ( where δp (a) is deﬁned by (4.8). That is, eτ (A) mp (A) := wp (A)[ln (wp (A)/pτ (A))]−1/2p

wp (A) ), τ (A)

if wp (A) ≤ peτ (A) . (9.4) if wp (A) > peτ (A)

Thus Theorem 9.9.1 and Corollary 9.4.3 imply Corollary 9.9.2 Let relations (1.1), (1.2) and (9.1) hold. Then svD (A) ≤ φp (A) and svA (D) ≤ mp (A). In particular, max{0, rs (D) − mp (A)} ≤ rs (A) ≤ rs (D) + φp (A), max{0, rl (D) − φp (A)} ≤ rl (A) ≤ rl (D) + mp (A) and α(D) − mp (A) ≤ α(A) ≤ α(D) + φp (A).

(9.3)

9.10. Notes

9.10

149

Notes

The present chapter is based on the papers (Gil’, 2002) and (Gil’, 2003). Theorem 9.1.1 supplements the well-known results on the invertibility of linear operators, cf. (Harte, 1988). As it was above mentioned, a lot of papers and books have been devoted to the spectrum of linear operators. Mainly, the asymptotic distributions of the eigenvalues are considered, cf. the books (Pietsch, 1987), (K¨onig, 1986), and references therein. But the bounds and invertibility conditions have been investigated considerably less than the asymptotic distributions. At the same time, in particular, Theorems 9.6.1, 9.7.1 and 9.8.1 and their corollaries give us explicit bounds for the spectrum of the considered operators.

References [1] Gil’, M.I. (2002). Invertibility and spectrum localization of nonselfadjoint operators, Adv. Appl. Mathematics, 28, 40-58. [2] Gil’, M. I. (2003). Inner bounds for spectra of linear operators, Proceedings of the American Mathematical Society (to appear). [3] Harte R. (1988). Invertibility and Singularity for Bounded Linear Operators. Marcel Dekker, Inc. New York. [4] K¨ onig, H. (1986). Eigenvalue Distribution of Compact Operators, Birkh¨ auser Verlag, Basel- Boston-Stuttgart. [5] Pietsch, A. (1987). Eigenvalues and s-Numbers, Cambridge University Press, Cambridge.

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10. Multiplicative Representations of Resolvents

In the present chapter we introduce the notion of the multiplicative operator integral in a separable Hilbert space H. By virtue of the multiplicative operator integral, we derive spectral representations for resolvents of various classes of P -triangular operators. These representations are generalizations of the classical spectral representation for the resolvent of a normal operator. If the maximal resolution of the identity is discrete, then the multiplicative integral is an operator product.

10.1

Operators with Finite Chains of Invariant Projectors

Recall that I is the unit operator in H. Lemma 10.1.1 Let P be a projector onto an invariant subspace of a bounded linear operator A in H, P = 0 and P = I. Then λRλ (A) = −(I − AP Rλ (A)P )(I − A(I − P )Rλ (A)(I − P )) (λ ∈ σ(A)). Proof:

Denote E = I − P . Since A = (E + P )A(E + P ) and EAP = 0,

we have A = P AE + P AP + EAE. M.I. Gil’: LNM 1830, pp. 151–161, 2003. c Springer-Verlag Berlin Heidelberg 2003

(1.1)

152

10. Multiplicative Representations of Resolvents

Let us check the equality Rλ (A) = P Rλ (A)P − P Rλ (A)P AERλ (A)E + ERλ (A)E.

(1.2)

In fact, multiplying this equality from the left by A − Iλ and taking into account the equalities (1.1), AP = P AP and P E = 0, we obtain the relation ((A − Iλ)P + (A − Iλ)E + P AE)(P Rλ (A)P − P Rλ (A)P AERλ (A)E + ERλ (A)E) = P − P AERλ (A)E + E + P AERλ (A)E = I. Similarly, multiplying (1.2) by A − Iλ from the right and taking into account (1.1), we obtain I. Therefore, (1.2) is correct. Due to (1.2) I − ARλ (A) = (I − ARλ (A)P )(I − AERλ (A)E).

(1.3)

I − ARλ (A) = −λRλ (A).

(1.4)

But We thus arrow at the result. 2 Let Pk (k = 1, . . . , n) be a chain of projectors onto the invariant subspaces of a bounded linear operator A. That is, Pk APk = APk (k = 1, ..., n)

(1.5)

0 = P0 H ⊂ P1 H ⊂ ... ⊂ Pn−1 H ⊂ Pn H = H.

(1.6)

and

For bounded linear operators X1 , X2 , ..., Xn again put →

Xk := X1 X2 ...Xn .

1≤k≤n

I.e. the arrow over the symbol of the product means that the indexes of the co-factors increase from left to right. Lemma 10.1.2 Let a bounded linear operator A have properties (1.5) and (1.6). Then λRλ (A) = −

→

(I − A∆Pk Rλ (A)∆Pk ) (λ ∈ σ(A)),

1≤k≤n

where ∆Pk = Pk − Pk−1 (1 ≤ k ≤ n).

10.1. Operators with Finite Chains

Proof:

153

Due to the previous lemma

λRλ (A) = −(I − APn−1 Rλ (A)Pn−1 )(I − A(I − Pn−1 )Rλ (A)(I − Pn−1 )). But I − Pn−1 = ∆Pn . So equality (1.4) implies. I − ARλ (A) = (I − APn−1 Rλ (A)Pn−1 )(I − A∆Pn Rλ (A)∆Pn ). Applying this relation to APn−1 , we get I − APn−1 Rλ (A)Pn−1 = (I − APn−2 Rλ (A)Pn−2 )(I − A∆Pn−1 Rλ (A)∆Pn−1 ). Consequently, I − ARλ (A) = (I − APn−2 Rλ (A)Pn−2 )(I − A∆Pn−1 Rλ (A)∆Pn−1 )(I− A∆Pn Rλ (A)∆Pn ). Continuing this process, we arrive at the required result. 2. Let us consider an operator of the form A=

n

ak ∆Pk + V

(1.7)

k=1

where ak are some numbers, {Pk } is a chain of projectors deﬁned by (1.6) and V is a nilpotent operator with the property Pk−1 V Pk = V Pk (k = 1, ..., n). Lemma 10.1.3 Under conditions (1.7) and (1.8), the relation λRλ (A) = −

→

(I +

1≤k≤n

A∆Pk ) λ − ak

is valid for any λ = ak (k = 1, ..., n). Proof:

It is not hard to check that ∆Pk Rλ (A)∆Pk =

∆Pk . ak − λ

Now the required result is due to the previous lemma. 2

(1.8)

154

10. Multiplicative Representations of Resolvents

10.2

Complete Compact Operators

Let A be a compact operator in H whose system of all the root vectors is complete in H. Then there is an orthogonal normed basis (Schur’s basis) {ek }, such that k Aek = ajk ej , (2.1) j=1

cf. (Gohberg and Krein, 1969, Chapter 5). Moreover akk = λk (A) are the eigenvalues of A with their multiplicities. Introduce the orthogonal projectors Pk =

k

(., ej )ej (k = 1, 2, ...).

j=1

If there exists a limit in the operator norm of the products →

(I + Xk ) ≡ (I + X1 )(I + X2 )...(I + Xn ).

1≤k≤n

as n → ∞, then we denote this limit by →

(I + Xk ).

1≤k≤∞

That is,

→

(I +

1≤k≤∞

A∆Pk ) λ − λk (A)

is a limit in the operator norm of the sequence the operators Πn (λ) :=

→

(I +

1≤k≤n

(I +

A∆Pk ) := λ − λk (A)

A∆P1 A∆P2 A∆Pn )(I + )...(I + ) λ − λ1 (A) λ − λ2 (A) λ − λn (A)

for λ = λk (A). Here ∆Pk = Pk − Pk−1 , k = 1, 2, ... ; P0 = 0, again. Lemma 10.2.1 Suppose that the system of all the root vectors of a compact linear operator A is complete in H. Then

λRλ (A) = −

→ 1≤k≤∞

(I +

A∆Pk ) (λ ∈ σ(A)). λ − λk (A)

(2.2)

10.2. Complete Compact Operators

Proof:

155

Let An = APn . Lemma 10.1.3 implies the equality λRλ (An ) = −Πn (λ).

(2.3)

Since A is compact, An tends to A in the operator norm as n tends to ∞. Besides, (An − λI)−1 Pn → (A − λI)−1 in the operator norm for any regular λ . We arrive at the result. 2 Let A be a normal compact operator. Then A=

∞

λk (A)∆Pk .

k=1

Hence, A∆Pk = λk (A)∆Pk . Since ∆Pk ∆Pj = 0 for j = k, Lemma 10.2.1 gives us the equality −λRλ (A) = I +

∞

(I +

k=1

But I=

∞

A∆Pk ). λ − λk (A)

∆Pk .

k=1

Thus, λRλ (A) = − −

∞

[1 + (λ − λk (A))−1 λk (A)]∆Pk =

k=1 ∞

λ∆Pk (λ − λk (A))−1 .

k=1

Or Rλ (A) =

∞ k=1

∆Pk . λk (A) − λ

Thus, Lemma 10.2.1 generalizes the well-known spectral representation for the resolvent of a normal completely continuous operator. Furthermore, according to (2.1), the nilpotent part V of A can be deﬁned as k−1 V ek = ajk ej . (2.4) j=1

Therefore, Pk−1 V ∆Pk = Pk−1 A∆Pk = V ∆Pk and A∆Pk = Pk A∆Pk = ∆Pk A∆Pk + Pk−1 A∆Pk = λk (A)∆Pk + V ∆Pk . Now Lemma 10.2.1 implies the relation

λRλ (A) = −

→ 1≤k≤∞

(I +

(λk (A) + V )∆Pk ) (λ ∈ σ(A)). λ − λk (A)

(2.5)

156

10. Multiplicative Representations of Resolvents

10.3

The Second Representation for Resolvents of Complete Compact Operators

Let V be a Volterra operator, deﬁned by (2.4). Then due to (2.5) (I − V )−1 =

→

(I + V ∆Pk ).

(3.1)

2≤k≤∞

Furthermore, according to (2.1) A = D + V , where D is deﬁned by Dek = λk (A)ek . Clearly, (A − λI)−1 = (D + V − Iλ)−1 = (D − Iλ)−1 (I + Bλ )−1 ,

(3.2)

where Bλ = V (D − Iλ)−1 . Due to Lemma 7.3.4 Bλ is a Volterra operator. Moreover, Pk−1 Bλ Pk = Bλ Pk . Thus relation (3.1) implies (I + Bλ )−1 =

→

(I − Bλ ∆Pk ).

2≤k≤∞

But

V ∆Pk . λk − λ

Bλ ∆Pk = Therefore,

→

(I + Bλ )−1 =

(I +

2≤k≤∞

V ∆Pk ). λ − λk (A)

Now (3.2) yields Theorem 10.3.1 Suppose that the system of all the root vectors of a compact linear operator A is complete in H. Then Rλ (A) = Rλ (D)

→

(I +

2≤k≤∞

V ∆Pk ) (λ ∈ σ(A)), λ − λk (A)

where V is the nilpotent part of A and Rλ (D) =

∞ k=1

∆Pk . λk (A) − λ

10.4. Operators with Compact Inverse Ones

10.4

157

Operators with Compact Inverse Ones

Let a linear operator A in H have a compact inverse one A−1 . Let the system of the root vectors of A−1 (and therefore of A) is complete in H. Then due to (2.1), there is an orthogonal normed basis (Schur’s basis) {ek }, such that A−1 ek =

k

bjk ej

(4.1)

j=1

with entries bjk . The nilpotent part V0 and diagonal one D0 of A−1 are deﬁned by k−1 bjk ej . (4.2) V0 ek = j=1 −1

and D0 ek = bkk ek = λk (A

Pk =

)ek . As above, put k

(., ej )ej (k = 1, 2, ...).

j=1

Theorem 10.4.1 Let operator A have the compact inverse one A−1 . Let the system of the root vectors of A−1 is complete in H. Then λ Rλ (A) =

→

(I +

1≤k≤∞

λ(1 + λk (A)V0 )∆Pk ) − I (λ ∈ σ(A)). λk (A) − λ

The product converges in the operator norm. Proof:

Thanks to Lemma 10.2.1, (A − λI)−1 = A−1 (I − λA−1 )−1 = A−1

→

(I +

1≤k≤∞

λA−1 ∆Pk ) 1 − λk (A−1 )λ

for any regular λ of A. But D0 ∆Pk = λk (A−1 )∆Pk . Hence, →

(A − λI)−1 = A−1

(I +

λ(λk (A−1 ) + V0 )∆Pk ). 1 − λk (A−1 )λ

(I +

λ(1 + λk (A)V0 )∆Pk ). λk (A) − λ

1≤k≤∞

Thus, we have derived the relation (A − λI)−1 = A−1

→ 1≤k≤∞

Taking into account that A(A − λI)−1 = I + λ(A − λI)−1 , we arrive at the required result. 2

158

10. Multiplicative Representations of Resolvents

10.5

Multiplicative Integrals

Let F be a function deﬁned on a ﬁnite real segment [a, b] whose values are bounded linear operators in H. We deﬁne the right multiplicative integral as the limit in the uniform operator topology of the sequence of the products →

(n)

1≤k≤n

(n)

(n)

(1 + δF (tk )) := (1 + δF (t1 ))(I + δF (t2 ))...(I + δF (t(n) n ))

(n)

(n)

as maxk |tk − tk−1 | tends to zero. Here (n)

(n)

(n)

δF (tk ) = F (tk ) − F (tk−1 ) for k = 1, ..., n (n)

(n)

and a = t0 denote by

< t1

(n)

< ... < tn

= b. The right multiplicative integral we

→

(1 + dF (t)). [a,b]

In particular, let P be an orthogonal resolution of the identity deﬁned on [a, b], φ be a function integrable in the Riemann-Stieljes with respect to P , and A be a compact linear operator. Then the right multiplicative integral → (I + φ(t)AdP (t)) [a,b]

is the limit in the uniform operator topology of the sequence of the products →

(n)

(n)

1≤k≤n (n)

(n)

(n)

(n)

(I + φ(tk )A∆P (tk )) (∆P (tk ) = P (tk ) − P (tk−1 )) (n)

as maxk |tk − tk−1 | tends to zero.

10.6. Volterra Operators

10.6

159

Resolvents of Volterra Operators

Lemma 10.6.1 Let V be a Volterra operator with a m.r.i. P (t) deﬁned on a ﬁnite real segment [a, b]. Then the sequence of the operators Vn =

n k=1

(n)

(n)

P (tk−1 )V ∆P (tk )

(6.1) (n)

tends to V in the uniform operator topology as maxk |tk zero. Proof:

(n)

− tk−1 | tends to

We have V − Vn =

n k=1

(n)

(n)

∆P (tk )V ∆P (tk ).

But thanks to the well known Lemma I.3.1 (Gohberg and Krein, 1970), the sequence {V − Vn } tends to zero as n tends to inﬁnity. This proves the required result. 2 Lemma 10.6.2 Let V be a Volterra operator with a maximal resolution of the identity P (t) deﬁned on a segment [a,b]. Then → −1 (I + V dP (t)). (I − V ) = [a,b]

Proof: Due to Lemma 10.6.1, V is the limit in the operator norm of the sequence of operators Vn , deﬁned by (6.1) . Due to Lemma 10.1.2, (I − Vn )−1 =

→ 1≤k≤n

(n)

(I + Vn ∆P (tk )).

Hence the required result follows. 2

10.7

Resolvents of P -Triangular Operators

In this section [a, b] is a ﬁnite real segment, again. Theorem 10.7.1 Let A be a P -triangular operators with a m.r.i. P (.) deﬁned on [a, b], a (compact) nilpotent part V and the diagonal part b φ(t)dP (t), (7.1) D= a

where φ is a scalar function integrable in the Riemann-Stieljes sense with respect to P (.). Then → V dP (t) ) (λ ∈ σ(A)). (7.2) (I − Rλ (A) = Rλ (D) φ(t) −λ [a,b]

160

10. Multiplicative Representations of Resolvents

Proof: By Lemma 7.3.4 V Rλ (D) is a Volterra operator. We invoke Lemma 10.6.3. It asserts that → (I − V Rλ (D)dP (t)). (7.3) (I + V Rλ (D))−1 = [a,b]

But according to (7.1) Rλ (D)dP (t) = Thus, −1

(I + V Rλ (D))

1 dP (t). φ(t) − λ

→

= [a,b]

(I −

V dP (t) ). φ(t) − λ

Hence relation (3.2) yields the required result. 2 Furthermore, from (7.2) it follows that Rλ (A) =

a

b

dP (s) φ(s) − λ

→

[a,b]

(I −

V dP (t) ) φ(t) − λ

for all regular λ. But dP (s)V dP (t) = 0 for t ≤ s. We thus get Corollary 10.7.2 Let the hypothesis of Theorem 10.7.1 hold. Then Rλ (A) =

a

b

dP (s) φ(s) − λ

→

[s,b]

(I −

V dP (t) ) (λ ∈ σ(A)). φ(t) − λ

Let us suppose that A is a normal operator. Then V = 0 and Theorem 10.7.1 yields b dP (s) Rλ (A) = . φ(s) −λ a Thus, Theorem 10.7.1 generalizes the classical representation for the resolvent of a normal operator. Corollary 10.7.3 Let the hypothesis of Theorem 10.7.1 hold. Then → 2i(P (t)AI − Im φ(t))dP (t) Rλ (A) = Rλ (D) ) (λ ∈ σ(A)). (I − φ(t) − λ [a,b] Indeed, since A = D + V , we have AI = VI + DI with DI = (D − D∗ )/2i and VI = (V − V ∗ )/2i. But P (t)V dP (t) = V dP (t), dP (t)V dP (t) = 0 and P (t)V ∗ dP (t) = 0.

10.8. Notes

161

Thus, V dP (t) = 2iP (t)VI dP (t). Moreover, since DI dP (t) = Im φ(t)dP (t), we get V dP (t) = 2i[P (t)AI − Im φ(t)]dP (t). Thus, applying Theorem 10.7.1, we get Corollary 10.7.3. In particular, let A have a purely real spectrum. Then Corollary 10.7.3 implies the representation → b dP (s) 2iAI dP (t) ) Rλ (A) = (I − φ(t) − λ a φ(s) − λ [s,b] for all regular λ. Let AR , VR and DR are the real components of A, V and D, respectively. Repeating the above arguments, by Theorem 10.7.1, we easily obtain the following result. Corollary 10.7.4 Let the hypothesis of Theorem 10.7.1 hold. Then → 2(P (t)AR − Re φ(t))dP (t) ) (λ ∈ σ(A)). Rλ (A) = Rλ (D) (I − φ(t) − λ [a,b]

10.8

Notes

The contents of Sections 10.1-10.3 and 10.5-10.8 is based on the papers (Gil’, 1973) and (Gil’, 1980). The results presented in Sections 10.4 and 10.8 are probably new. For more details about the multiplicative integral see (Gohberg and Krein, 1970), (Brodskii, 1971), (Feintuch and Saeks, 1982).

References [1] Brodskii, M. S. (1971). Triangular and Jordan Representations of Linear Operators, Transl. Math. Monogr., v. 32, Amer. Math. Soc. Providence, R. I. [2] Feintuch, A., Saeks, R. (1982). System Theory. A Hilbert Space Approach. Ac. Press, New York. [3] Gil’, M. I. (1973). On the representation of the resolvent of a nonselfadjoint operator by the integral with respect to a spectral function, Soviet Math. Dokl., 14 : 1214-1217. [4] Gil’, M. I. (1980). On spectral representation for the resolvent of linear operators. Sibirskij Math. Journal, 21: 231. [5] Gohberg, I. C. and Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space, Trans. Mathem. Monogr., v. 24, Amer. Math. Soc., Providence, R. I.

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11. Relatively P -Triangular Operators

This chapter is devoted to operators of the type A = D + W , where D is a normal boundedly invertible operator in a separable Hilbert space H, and W has the following property: V := D−1 W is a Volterra operator in H. If, in addition, A has a maximal resolutions of the identity, then it is called a relatively P -triangular operator. Below we derive estimates for the resolvents of various relatively P -triangular operators and investigate spectrum perturbations of such operators.

11.1

Deﬁnitions and Preliminaries

Let P (.) be a maximal resolution of the identity deﬁned on a real segment [a, b] (see Section 7.2). In the present chapter paper we consider a linear operator A in H of the type A=D+W

(1.1)

where D is a normal boundedly invertible, generally unbounded operator and W is linear operator with the properties H ⊇ Dom (W ) ⊃ Dom (D) = Dom (A). In addition, P (t)W P (t)h = W P (t)h (t ∈ [a, b], h ∈ Dom (W ))

(1.2)

DP (t)h = P (t)Dh (t ∈ [a, b], h ∈ Dom (A)).

(1.3)

and

M.I. Gil’: LNM 1830, pp. 163–172, 2003. c Springer-Verlag Berlin Heidelberg 2003

164

Moreover,

11. Relatively P -Triangular Operators

V := D−1 W is a Volterra operator .

(1.4)

We have P (t)D−1 W P (t)h = D−1 P (t)W P (t)h = D−1 W P (t)h (t ∈ [a, b], h ∈ Dom (A)). So P (t)V P (t) = V P (t) (t ∈ [a, b]). Deﬁnition 11.1.1 Let relations (1.1)-(1.4) hold. Then A is said to be a relatively P -triangular operator (RPTO), D is the diagonal part of A and V is the relatively nilpotent part of A. Everywhere in the present chapter A denotes a relatively P -triangular operator, D denotes its diagonal part and V ∈ Y denotes the relatively nilpotent part of A. Recall that Y is an ideal of linear compact operators in H with a norm |.|Y and the following property: any Volterra operator V ∈ Y satisﬁes the inequalities (1.5) V k ≤ θk |V |kY (k = 1, 2, ...), √ k where constants θk are independent of V and θk → 0 (k → ∞). Under (1.5) put ni(V )−1 JY (V ) = θk |V |kY (θ0 = 1), k=0

where ni (V ) is the ”nilpotency” index (see Deﬁnition 1.4.2). In the sequel one can replace ni (V ) by ∞. Let rl (D) = inf |σ(D)|, again. Since D is invertible, rl (D) > 0. Lemma 11.1.2 Under conditions (1.1)-(1.5), operator A is boundedly invertible. Moreover, A−1 ≤ rl−1 (D)JY (V ) and A−1 D ≤ JY (V ). Proof: Clearly, A = (D + W ) = D(I + V ). So A−1 = (I + V )−1 D−1 . Due to (1.5) (I + V )−1 ≤ JY (V ). Since D is normal, D−1 = rl−1 (D). This proves the required results. 2

11.2. Resolvents of Relatively P -Triangular Operators

11.2

165

Resolvents of Relatively P -Triangular Operators

Denote w(λ, D) ≡ inf | t∈σ(D)

λ − 1| t

and ν(λ) := ni ((D − λI)−1 W ) = ni ((I − D−1 λ)−1 V ) (λ ∈ σ(D)). Under (1.5) put JY (V, m, z) =

m−1

z −1−k θk |V |kY (z > 0).

k=0

Lemma 11.2.1 Under conditions (1.1)-(1.5), let λ ∈ σ(D). Then λ is a regular point of operator A. Moreover, Rλ (A) ≤ rl−1 (D)JY (V, ν(λ), w(λ, D))

(2.1)

Rλ (A)D ≤ JY (V, ν(λ), w(λ, D)).

(2.2)

and Proof:

According to (1.1), A − λI = (D − λI)(I + (D − λI)−1 W ).

Consequently

Rλ (A) = (I + Rλ (D)W )−1 Rλ (D).

Taking into account that (D − λI)−1 = (I − D−1 λ)−1 D−1 , we have Rλ (A) = (I + (I − D−1 λ)−1 D−1 W )−1 (I − D−1 λ)−1 D−1 .

(2.3)

But D is normal. Therefore, (I − D−1 λ)−1 ≤

1 . w(λ, D)

Moreover, due to Lemma 7.3.3, (I − D−1 λ)−1 V is a Volterra operator. So ν(λ)−1

(I + (I − D−1 λ)−1 V )−1 ≤

((I − D−1 λ)−1 V )k ≤

k=0 ν(λ)−1

k=0

θk |((I − D−1 λ)−1 V )k |Y ≤

11. Relatively P -Triangular Operators

166

ν(λ)−1

θk w−k (λ, D)|V |kY = w(λ, D)JY (V, ν(λ), w(λ, D)).

k=0

Thus (2.3) yields Rλ (A) ≤ (I + (I − D−1 λ)−1 V )−1 D−1 (I − D−1 λ)−1 ≤ JY (V, ν(λ), w(λ, D))rl−1 (D). Therefore (2.1) is proved. Moreover, thanks to (2.3) Rλ (A)D ≤ (I + (I − D−1 λ)−1 V )−1 (I − D−1 λ)−1 ≤ JY (V, ν(λ), w(λ, D)). So inequality (2.2) is also proved. 2

11.3

Invertibility of Perturbed RPTO

In the sequel Z is a linear operator in H satisfying the condition m(Z) := D−1 Z < ∞.

(3.1)

Lemma 11.3.1 Under conditions (1.1)-(1.5) and (3.1), let JY (V )m(Z) < 1.

(3.2)

Then the operator A + Z is boundedly invertible. Moreover, the inverse operator satisﬁes the inequality (A + Z)−1 ≤ Proof:

JY (V ) . ρl (D)(1 − JY (V )m(Z))

Due to Lemma 11.1.2 we have A−1 Z = A−1 DD−1 Z ≤ JY (V )m(Z).

But (A + Z)−1 − A−1 = −A−1 Z(A + Z)−1 = −A−1 DD−1 Z(A + Z)−1 . Hence,

(A + Z)−1 ≤ A−1 + A−1 DD−1 Z(A + Z)−1 ≤ A−1 + JY (V )m(Z)(A + Z)−1

and

(A + Z)−1 ≤ A−1 (1 − JY (V )m(Z))−1 .

Lemma 11.1.2 yields now the required result. 2

11.4. Resolvents of Perturbed RPTO

11.4

167

Resolvents of Perturbed RPTO

Recall that JY (V, ν(λ), w(λ, D)) is deﬁned in Section 11.2. Theorem 11.4.1 Under conditions (1.1)-(1.5) and (3.1), for a λ ∈ σ(D), let m(Z)JY (V, ν(λ), w(λ, D)) < 1. (4.1) Then λ is a regular point of operator A + Z. Moreover, Rλ (A + Z) ≤ Proof:

JY (V, ν(λ), w(λ, D)) . ρl (D)(1 − m(Z)JY (V, ν(λ), w(λ, D)))

Due to Lemma 11.2.1, Rλ (A)Z = Rλ (A)DD−1 Z ≤ m(Z)JY (V, ν(λ), w(λ, D)).

But Rλ (A + Z) − Rλ (A) = −Rλ (A)ZRλ (A + Z). Hence, Rλ (A + Z) ≤ Rλ (A) + Rλ (A)ZRλ (A + Z) ≤ Rλ (A) + m(Z)JY (V, ν(λ), w(λ, D))Rλ (A + Z). Due to (4.1) Rλ (A + Z) ≤ Rλ (A)(1 − m(Z)JY (V, ν(λ), w(λ, D))−1 . Now Lemma 11.2.1 yields the required result. 2 Theorem 11.4.1 implies Corollary 11.4.2 Under conditions (1.1)-(1.5) and (3.1), for any µ ∈ σ(A+ Z), there is a µ0 ∈ σ(D), such that, either µ = µ0 , or m(Z)JY (V, ν(µ), |1 − µµ−1 0 |) ≥ 1.

11.5

(4.2)

Relative Spectral Variations

Deﬁnition 11.5.1 Let A and B be linear operators in H. Then the quantity rsvA (B) := sup

inf |1 −

µ∈σ(B) λ∈σ(A)

µ | λ

will be called the relative spectral variation of B with respect to A. In addition, rhd(A, B) := max{rsvA (B), rsvB (A)} is said to be the relative Hausdorﬀ distance between the spectra of A and B.

11. Relatively P -Triangular Operators

168

Put ν˜0 = sup

λ∈σ(D)

ν(λ) = sup

λ∈σ(D)

ni ((D − λI)−1 W ).

In the sequel one can replace ν˜0 by ∞. Theorem 11.5.2 Under conditions (1.1)-(1.5) and (3.1), the equation m(Z)JY (V, ν˜0 , z) = 1

(5.1)

has a unique positive root z0 (Y, V, Z). Moreover, rsvD (A + Z) ≤ z0 (Y, V, Z).

(5.2)

Proof: Comparing equation (5.2) with inequality (4.2), we arrive at the required result. 2 Lemma 8.3.1 and Theorem 11.5.2 imply Corollary 11.5.3 Under conditions (1.1)-(1.5) and (3.1), we have the inequality (5.3) rsvD (A + Z) ≤ δY (A, Z), where δY (A, Z) ≡ 2 max

j=1,2,...

j θj−1 |V |j−1 Y Z.

Due to (5.3), for any µ ∈ σ(A + Z), there is a µ0 ∈ σ(D), such that 1 ≤ |µµ−1 0 | + δY (A, Z). Thus, |µ| ≥ |µ0 |(1 − δY (A, Z)). Let µ = rl (A + Z). Then we get Corollary 11.5.4 Under conditions (1.1)-(1.5) and (3.1), we have the inequality rl (A + Z) ≥ rl (D) max{0, 1 − δY (A, Z)}.

11.6 11.6.1

Operators with von Neumann-Schatten Relatively Nilpotent Parts Invertibility conditions

Let A be a relatively P -triangular operator (RPTO). Throughout this section it is assumed that its relatively nilpotent part V := D−1 W ∈ C2p

(6.1)

11.6. Neumann-Schatten Relatively Nilpotent Parts

169

for a natural p ≥ 1. Again put, ni(V )−1

Jp (V ) =

k=0

where (p) θk =

(p)

k θk N2p (V ),

1 [k/p]!

and [.] means the integer part. Due to Corollary 6.9.4, (p)

j (V ) (j = 1, 2, ...). V j ≤ θj N2p

(6.2)

Then Lemma 11.1.2 implies that under (1.1)-(1.4) and (6.1), A−1 ≤ rl−1 (D)Jp (V ) and A−1 D ≤ Jp (V ).

(6.3)

Moreover, due to Lemma 11.3.1, we get Lemma 11.6.1 Under conditions (1.1)-(1.4), (3.1) and (6.1), let m(Z)Jp (V ) < 1. Then operator A + Z is boundedly invertible. Moreover, (A + Z)−1 ≤

Jp (V ) . (1 − m(Z)Jp (V ))rl (D)

Under (6.1) for a z > 0, put ψp (V, z) = a0

p−1 j N2p (V ) j=0

z j+1

2p b0 N2p (V ) exp [ ], z 2p

where constants a0 , b0 can be taken from the relations c , b0 = c/2 for a c > 1, a0 = c−1 or

a0 = e1/2 , b0 = 1/2.

Lemma 11.6.2 Under conditions (1.1)-(1.4), (3.1) and (6.1), let m(Z)ψp (V, 1) < 1. Then operator A + Z is boundedly invertible. Moreover, (A + Z)−1 ≤

ψp (V, 1) . rl (D) (1 − m(Z)ψp (V, 1))

(6.4)

(6.5)

(6.6)

170

Proof:

11. Relatively P -Triangular Operators

Due to Theorem 6.7.3, (I − V )−1 ≤ ψp (V, 1).

(6.7)

But A−1 = (I + V )−1 D−1 . So −1 D ≤ ψp (V, 1). A−1 ≤ ρ−1 l (D)ψp (V, 1) and A

Now, using the arguments of the proof of Lemma 11.3.1, we arrive at the required result. 2

11.6.2

The norm of the resolvent

Recall that ν(λ) is deﬁned in Section 11.2. Relation (6.2) and Lemma 11.2.1, under (6.1) imply Rλ (A) ≤ rl−1 (D)J˜p (V, ν(λ), w(λ, D)) and Rλ (A)D ≤ J˜p (V, ν(λ), w(λ, D)) (λ ∈ σ(A)), where J˜p (V, m, z) =

m−1 k=0

(p)

k θk N2p (V ) (z > 0). z k+1

Now thanks to Theorem 11.4.1, we get Theorem 11.6.3 Under conditions (1.1)-(1.4), (3.1) and (6.1), let m(Z)J˜p (V, ν(λ), w(λ, D)) < 1. Then λ is a regular point of operator A + Z. Moreover, Rλ (A + Z) ≤ Put

J˜p (V, ν(λ), w(λ, D)) . rl (D) (1 − m(Z)J˜p (V, ν(λ), w(λ, D))

V˜ (λ) = (I − D−1 λ)−1 V.

Thanks to (2.3) Rλ (A)D = (I + V˜ (λ))−1 (I − D−1 λ)−1 . Taking into account (6.7), we get Rλ (A) ≤ ψp (V˜ (λ), 1)w(λ, D). Hence, we arrive at the inequalities Rλ (A) ≤ rl−1 (D)ψp (V, w(λ, D)) and Rλ (A)D ≤ ψp (V, w(λ, D)) (6.8) for all λ ∈ σ(A). Now, taking into account (6.8) and repeating the arguments of the proof of Theorem 11.4.1, we arrive at the following result

11.6. Neumann-Schatten Relatively Nilpotent Parts

171

Corollary 11.6.4 Under conditions (1.1)-(1.4), (3.1) and (6.1), let m(Z)ψp (V, w(λ, D)) < 1. Then λ is a regular point of operator A + Z. Moreover, Rλ (A + Z) ≤

11.6.3

ψp (V, w(λ, D)) . rl (D) (1 − m(Z)ψp (V, w(λ, D))

Localization of the spectrum

Corollary 11.6.4 immediately yield. Corollary 11.6.5 Under conditions (1.1)-(1.4), (3.1) and (6.1), for any µ ∈ σ(A + Z), there is a µ0 ∈ σ(D), such that, either µ = µ0 , or m(Z)ψp (V, |1 −

µ |) ≥ 1. µ0

(6.9)

Theorem 11.6.6 Under conditions (1.1)-(1.4), (3.1) and (6.1), the equation p−1 2p k N2p (V ) N2p (V ) m(Z) exp [(1 + )/2] = 1 (6.10) k+1 2p z z k=0

has a unique positive root zp (V, Z). Moreover, rsvD (A + Z) ≤ zp (V, Z). In particular, the lower spectral radius of A + Z satisﬁes the inequality rl (A + Z) ≥ rl (D) max{0, 1 − zp (V, Z)}. Proof: 2

Comparing (6.9) and (6.10), we have w(D, µ) ≤ zp (V, Z), as claimed.

Substituting the equality z = N2p (V )x, in (6.10) and applying Lemma 8.3.2, we get zp (V, Z) ≤ ∆p (V, Z), where ∆p (V, Z) :=

m(Z)pe N2p (V )[ln (N2p (V )/m(Z)p)]−1/2p

if N2p (V ) ≤ m(Z)pe . if N2p (V ) > m(Z)pe

Thanks to Theorem 11.6.5 we arrive at the following Corollary 11.6.7 Under conditions (1.1)-(1.5), (3.1) and (6.1), rsvD (A + Z) ≤ ∆p (V, Z). In particular, rl (A + Z) ≥ rl (D) max{0, 1 − ∆p (V, Z)}.

172

11. Relatively P -Triangular Operators

11.7

Notes

This chapter is based on the papers (Gil’, 2001) and (Gil’, 2002).

References [1] Gil’, M. I. (2001). On spectral variations under relatively bounded perturbations, Arch. Math., 76, 458-466. [2] Gil’, M. I. (2002). Spectrum localization of inﬁnite matrices, Mathematical Physics, Analysis and Geometry, 4, 379-394 (2002).

12. Relatively Compact Perturbations of Normal Operators

The present chapter is devoted to linear operators of the type A = D + W+ + W− , where D is a normal invertible operator and D−1 W± are Volterra (compact quasinilpotent) operators. Numerous diﬀerential and integro-diﬀerential operators are examples of such operators. We derive estimates for the resolvents and bounds for the spectra.

12.1

Invertibility Conditions

Let H be a separable Hilbert space. In the present chapter we consider a linear operator A in H of the type A = D + W+ + W− ,

(1.1)

where D is a normal bounded invertible, generally unbounded operator and W± are linear operators, such that H ⊇ Dom (W± ) ⊃ Dom (D) = Dom (A). Let P (t) (−∞ ≤ t ≤ ∞) be a maximal resolution of the identity (m.r.i.) (see Section 7.2). It is assumed that P (t)W+ P (t)h = W+ P (t)h (h ∈ Dom (W+ ); t ∈ R),

(1.2a)

P (t)W− P (t)h = P (t)W− h (h ∈ Dom (W− ); t ∈ R)

(1.2b)

M.I. Gil’: LNM 1830, pp. 173–180, 2003. c Springer-Verlag Berlin Heidelberg 2003

174

12. Relatively Compact Perturbations

and P (t)Dh = DP (t)h (h ∈ Dom (D); t ∈ R).

(1.2c)

Again we use a normed ideal Y of linear compact operators in H with a norm |.|Y and having the property: any Volterra operator V ∈ Y satisﬁes the inequalities V k ≤ θk |V |kY (k = 1, 2, ...) where constants θk are independent √ k of V and θk → 0 (k → ∞). It is assumed that V± ≡ D−1 W± are Volterra operators from ideal Y . That is, V±k ≡ (D−1 W± )k ≤ θk |V± |kY (k = 1, 2, ...).

(1.3)

Recall that ni (V ) means the nilpotency index of a quasinilpotent operator V (see Deﬁnition 7.5.2). Put ni(V± )−1

JY (V± ) =

θk |V± |kY (θ0 = 1).

k=0

Lemma 12.1.1 Under condition (1.2), (1.3), let ζ0 (A) ≡ max {

1 1 − V+ , − V− } > 0. JY (V− ) JY (V+ )

(1.4)

Then operator A represented by (1.1) is boundedly invertible. Moreover, the inverse operator satisﬁes the inequality A−1 ≤

1 . rl (D)ζ0 (A)

(1.5)

Recall that rl (.) denotes the lower spectral radius. Proof: Condition (1.4) implies that either V+ JY (V− ) < 1

(1.6)

or V− JY (V+ ) < 1. If (1.6) holds, then replacing in Lemma 11.3.1 A by D + W− and Z by W+ we get the invertibility and estimate A−1 ≤

Exchanging V−

JY (V− ) = rl (D)(1 − V+ JY (V− ))

1 . rl (D)(JY−1 (V− ) − V+ ) and V+ , we have the estimate A−1 ≤

1 . rl (D)(JY−1 (V+ ) − V− )

This and (1.7) imply the required result. 2

(1.7)

12.2. Estimates for Resolvents

12.2

175

Estimates for Resolvents

Under condition (1.3) denote JY (V± , m, z) =

m−1

z −1−k θk |V± |kY (z > 0).

(2.1)

k=0

Lemma 12.2.1 Under conditions (1.2), the operator (D − λI)−1 W± is quasinilpotent for any λ ∈ σ(D). Proof:

Condition (1.2a) implies P (t)V+ P (t) = P (t)D−1 W+ P (t) = D−1 P (t)W+ P (t) = D−1 W+ P (t) = V+ P (t)

and

(I − D−1 λ)−1 P (t) = P (t)(I − D−1 λ)−1 (t ∈ (−∞, ∞)).

Thus, due to Lemma 7.3.3, the operator (I − D−1 λ)−1 V+ is a Volterra one. But (I − D−1 λ)−1 V+ = (D − λI)−1 W+ . Similarly, we can prove that (D − λI)−1 W− is a Volterra operator. 2 Put ν± (λ) ≡ ni((D − λI)−1 W± ) = ni((I − λD−1 )−1 V± ) and w(λ, D) ≡ inf | t∈σ(D)

λ − 1|. t

Now we are in a position to formulate the main result of the chapter Theorem 12.2.2 Under conditions (1.2) and (1.3), for a λ ∈ σ(D), let ζ(A, λ) ≡ max {

1 − V+ , JY (V− , ν− (λ), w(λ, D))

1 − V− } > 0. JY (V+ , ν+ (λ), w(λ, D))

(2.2)

Then λ is a regular point of operator A represented by (1.1). Moreover, Rλ (A) ≤ Proof: holds:

1 . ζ(A, λ)ρl (D)

(2.3)

Condition (1.5) means that at least one of the following inequalities JY (V− , ν− (λ), w(λ, D))V+ < 1

or JY (V+ , ν+ (λ), w(λ, D))V− < 1.

(2.4)

176

12. Relatively Compact Perturbations

If (2.4) holds, then replacing in Lemma 11.4.1 operator A by D + W− and operator Z by W+ we get the regularity of λ and estimate Rλ (A) ≤

JY (V+ , ν+ (λ), w(λ, D)) . ρl (D)(1 − V− JY (V+ , ν+ (λ), w(λ, D))

Rλ (A) ≤

1 . ρl (D)(JY−1 (V+ , ν+ (λ), w(λ, D)) − V− )

Or

Exchanging V+ and V− , we get Rλ (A) ≤

1 . ρl (D)(JY−1 (V− , ν− (λ), w(λ, D)) − V+ )

These inequalities yield the required result. 2

12.3

Bounds for the Spectrum

Recall that rsvD (A) denotes the relative spectral variation of operator A with respect to D. Again put τ (A) := min{V− , V+ }, V+ if V+ ≥ V− , ˜ V := V− if V− > V+ and ν˜0 = sup

λ∈σ(D)

(3.1) (3.2)

ni ((D − λI)−1 V˜ ).

In the sequel one can replace ν˜0 by ∞. Theorem 12.3.1 Under conditions (1.1), (1.2), let V˜ ∈ Y and V˜ k ≤ θk |V˜ |kY (k = 1, 2, ...).

(3.3)

τ (A)JY (V˜ , ν˜0 , z) = 1

(3.4)

Then the equation has a unique positive root zY (A). Moreover, rsvD (A) ≤ zY (A). Proof:

(3.5)

Due to Theorem 12.2.2, τ (A)JY (V˜ , ν˜0 , w(D, µ)) ≥ 1

for any µ ∈ σ(A). Comparing this inequality with (3.4), we have w(D, µ) ≤ zY (A). This inequality proves the theorem. 2 Theorem 12.3.1 and Lemma 8.3.1 give us the following result

12.4. Relatively Neumann-Schatten Parts

177

Corollary 12.3.2 Under conditions (1.1), (1.2) and (3.3), we have rsvD (A) ≤ ∆Y (A),

where ∆Y (A) := 2 max

j

j=1,2,...

θj−1 |V˜ |j−1 Y τ (A).

Due to (3.3), for all µ ∈ σ(A), there is µ0 ∈ σ(D), such that 1 ≤ |µµ−1 0 | + ∆Y (A). Thus |µ| ≥ |µ0 |(1 − ∆Y (A)). Now the previous corollary yields Corollary 12.3.3 Let conditions (1.1), (1.2) and (3.3) hold. Then rl (A) ≥ rl (D) max{0, 1 − ∆Y (A)}. Remark 12.3.4 Everywhere below V± can be replaced by upper bounds.

12.4

Operators with Relatively von Neumann - Schatten Oﬀ-diagonal Parts

12.4.1

Invertibility conditions

In this section it is assumed that V± = D−1 W± are quasinilpotent operators belonging to the Neumann-Schatten ideal C2p with some integer p ≥ 1. That is, (4.1) N2p (V± ) ≡ [T race (V±∗ V± )p ]1/2p < ∞. Put

ni(V )−1

Jp (V± ) =

k=0

where (p) θk =

(p)

k θk N2p (V± ),

1 [k/p]!

and [.] means the integer part. Lemma 12.4.1 Let relations (1.2) and (4.1) hold. In addition, let ζ2p (A) := max{Jp−1 (V− ) − V+ , Jp−1 (V+ ) − V− } > 0.

(4.2)

Then operator A represented by (1.1) is boundedly invertible. Moreover, A−1 ≤

1 . ζ2p (A)rl (D)

(4.3)

178

12. Relatively Compact Perturbations

Proof:

Due to Corollary 6.9.4, (p)

k (V ) (k = 1, 2, ...). V k ≤ θk N2p

(4.4)

for a quasinilpotent operator V ∈ C2p . Now the required result is due to Lemma 12.1.1. 2 Clearly,

p−1 j+pk ∞ N2p (V± ) √ Jp (V± ) ≤ Ip (V± ) := . k! j=0 k=1

According to (4.4) one can replace Jp (V± ) in (4.2) by Ip (V± ). Furthermore, for a z > 0, put ψp (V± , z) =

p−1 Npj (V± ) j=0

z j+1

exp [(1 +

2p N2p (V± ) )/2]. z 2p

(4.5)

Then we have Corollary 12.4.2 Let relations (1.2) and (4.1) hold. In addition, let ζ˜2p (A) ≡ max{

1 1 − V+ , − V− } > 0. ψp (V− , 1) ψp (V+ , 1)

Then operator A represented by (1.1) is boundedly invertible. Moreover, A−1 ≤

1 . ζ˜2p (A)rl (D)

Indeed, taking into account Lemma 11.6.2 and repeating the arguments of the proof of the previous lemma, we arrive at the required result.

12.4.2

Estimates for resolvents

Under (4.1) denote, J˜p (V± , m, z) =

m−1 k=0

(p)

k θk N2p (V± ) (z > 0). k+1 z

Recall that ν± (λ) ≡ ni ((D − λ)−1 V± ) ≤ ∞. Due to Theorem 12.2.2 and inequality (4.4), we get Theorem 12.4.3 Under conditions (1.1), (1.2) and (4.1), for a λ ∈ σ(D), let 1 ζ˜2p (λ, A) ≡ max{ − V+ , ˜ Jp (V− , ν− (λ), w(λ, D))

12.5. Notes

179

1 − V− } > 0. ˜ Jp (V+ , ν+ (λ), w(λ, D)) Then λ is a regular point of operator A, represented by (1.1). Moreover, Rλ (A) ≤

1 . ρl (D)ζ˜2p (λ, A)

As it was shown in Subsection 11.6.2, J˜p (V, ν± , z) can be replaced by ψp (V, z) for an arbitrary Volterra operator V ∈ C2p . Now Theorem 12.4.3 yields Corollary 12.4.4 Under conditions (1.1), (1.2) and (4.1), for a λ ∈ σ(D), let 1 (1) − V+ , ζ2p (λ, A) ≡ max{ ψp (V− , w(λ, D)) 1 − V− } > 0. ψp (V+ , w(λ, D)) Then λ is a regular point of operator A, represented by (1.1). Moreover, Rλ (A) ≤

12.4.3

1 (1) ρl (D)ζ2p (λ, A)

.

(4.6)

Spectrum localization

Let τ (A) and V˜ be deﬁned by (3.1) and (3.2), respectively. Theorem 12.3.1 and Corollary 12.4.4 yield Lemma 12.4.5 Under the conditions (1.1), (1.2) and V˜ ∈ C2p ,

(4.7)

N2p (V˜ ) 1 exp [ (1 + )] = 1, 2 z 2p

(4.8)

the equation τ (A)

p−1 j ˜ N2p (V ) j=0

z j+1

has a unique positive root z2p (A). Moreover, rsvD (A) ≤ z2p (A).

(4.9)

By virtiue of Lemma 8.3.2, we can assert that z2p (A) ≤ φp (A), where peτ (A) if N2p (V˜ ) ≤ τ (A)pe φp (A) := . (4.10) −1/2p ˜ if N2p (V˜ ) > τ (A)pe [ln (N2p (V )/pτ (A))] Now the previous lemma yields Corollary 12.4.6 Under conditions (1.1), (1.2) and (4.7), we have rsvD (A) ≤ φp (A). In particular, rl (A) ≥ rl (D) max{0, 1 − φp (A)}.

180

12. Relatively Compact Perturbations

12.5

Notes

The present chapter is based on the papers (Gil’, 2001) and (Gil’, 2002).

References [1] Gil’, M. I. (2001). On spectral variations under relatively bounded perturbations, Arch. Math., 76, 458-466. [2] Gil’, M. I. (2002). Spectrum localization of inﬁnite matrices, Mathematical Physics, Analysis and Geometry, 4, 379-394 (2002).

13. Inﬁnite Matrices in Hilbert Spaces and Diﬀerential Operators The present chapter is concerned with applications of some results from Chapters 7-12 to integro-diﬀerential and diﬀerential operators, as well as to inﬁnite matrices in a Hilbert space. In particular, we suggest estimates for the spectral radius of an inﬁnite matrix.

13.1 13.1.1

Matrices with Compact oﬀ Diagonals Upper bounds for the spectrum

Let {ek }∞ k=1 be an orthogonal normal basis in a separable Hilbert space H. Let A be a linear operator in H represented by a matrix with the entries ajk = (Aek , ej ) (j, k = 1, 2, ...),

(1.1)

where (., .) is the scalar product in H. Then A = D + V + + V− ,

(1.2)

where V− , V+ and D are the upper triangular, lower triangular, and diagonal parts of A, respectively: (V+ ek , ej ) = ajk for j < k, (V+ ek , ej ) = 0 for all j > k; (V− ek , ej ) = ajk for j > k, (V− ek , ej ) = 0 for all j < k; (Dek , ek ) = akk , (Dek , ej ) = 0 for j = k (j, k = 1, 2, ...). M.I. Gil’: LNM 1830, pp. 181–188, 2003. c Springer-Verlag Berlin Heidelberg 2003

(1.3)

182

13. Inﬁnite Matrices and Diﬀerential Operators

Let {Pk }∞ k=1 be a maximal orthogonal resolution of the identity, where Pk are deﬁned by k (., ej )ej (k = 1, 2, ...). (1.4) Pk = j=1

Simple calculations show that Pk V+ Pk = V+ Pk , Pk V− Pk = Pk V−

(1.5)

Pk Dh = Pk Dh (h ∈ Dom (D)) (k = 1, 2, ...).

(1.6)

and In addition, Dom (D) = {h = (hk ) ∈ H :

∞

|akk hk |2 < ∞; hk = (h, ek ); k = 1, 2, ....}

k=1

if D is unbounded. We restrict ourselves by the conditions N22 (V+ ) =

∞ ∞

|ajk |2 < ∞, N22 (V− ) =

j=1 k=j+1

j−1 ∞

|ajk |2 < ∞.

(1.7)

j=2 k=1

That is, V+ , V− are Hilbert-Schmidt matrices, but the results of Chapters 8 and 9 allow us to investigate considerably more general conditions than (1.7). Without any loss of generality, assume that N2 (V− ) ≤ N2 (V+ ).

(1.8)

The case N2 (V− ) ≥ N2 (V+ ) can be considered absolutely similarly. Put eN2 (V− ) if N2 (V+ ) ≤ eN2 (V− ), ˜ φ1 = . N2 (V+ )[ln (N2 (V+ )/N2 (V− ))]−1/2 if N2 (V+ ) > eN2 (V− ) (1.9) Due to Corollary 9.4.3 and Remark 9.4.4 we get Lemma 13.1.1 Under conditions (1.7), the spectrum of operator A = (ajk )∞ j,k is included in the set {z ∈ C : |akk − z| ≤ φ˜1 , k = 1, 2, ..., }.

(1.10)

In particular, the spectral radius of A satisﬁes the inequality rs (A) ≤ sup |akk | + φ˜1 , k

(1.11)

provided D is bounded: D ≡ supk |akk | < ∞. In addition, α(A) := sup Re σ(A) ≤ sup Re akk + φ˜1 , provided sup Re akk < ∞. So the considered matrix operator is stable (that is, its spectrum is in the open left half plane), if Re akk + φ˜1 < 0 (k = 1, 2, ....).

13.1. Compact Oﬀ-diagonals

13.1.2

183

Lower bounds for the spectrum

Now assume that under (1.7), DI = (D − D∗ )/2i is a Hilbert-Schmidt operator: ∞ |ajj − ajj |2 ]1/2 /2 < ∞. (1.12) N2 (DI ) = [ j=1 ∗

Then uder (1.7) AI = (A − A )/2i is a Hilbert-Schmidt operator: ∞ ∞ N2 (AI ) = [ |ajk − akj |2 ]1/2 /2 < ∞. j=1 k=1

Recall that gI (A) = [2N22 (AI ) − 2 One can replace gI (A) by ˜ H (A) := ∆

√

∞

|Im λk (A)|2 ]1/2 .

k=1

2N2 (AI ). Put if gI (A) ≤ eN2 (V− ) . if gI (A) > eN2 (V− ) (1.13)

eN2 (V− ) gI (A)[ ln (gI (A)/N2 (V− )) ]−1/2

Due to Corollary 9.8.2, we get Lemma 13.1.2 Under conditions (1.7), (1.12), for the matrix A = (ajk )∞ j,k , the following relations are true: ˜ H (A)}, rs (A) ≥ max{0, sup |akk | − ∆ k

(1.14)

˜ H (A) and α(A) ≥ sup Re akk − ∆ ˜ H (A). rl (A) ≤ inf |akk | + ∆ k

k

˜ H (A) ≥ 0. Note that, according So A is unstable, provided, supk Re akk − ∆ to Corollary 9.6.2, in the case N2 (A) ≡ [

∞

|ajk |2 ]1/2 < ∞,

j,k=1

one can replace gI (A) by g(A) = [N22 (A) −

∞

|λk (A)|2 ]1/2 ≤ [N22 (A) − |T race A2 |]1/2 .

k=1

Example 13.1.3 In the complex Hilbert space H = L2 [0, 1], let us consider an operator A deﬁned by

184

13. Inﬁnite Matrices and Diﬀerential Operators

1

(Au)(x) = u(x) + 0

K(x, s)u(s)ds (0 ≤ x ≤ 1),

(1.15)

where K is a scalar Hilbert-Schmidt kernel. Take the orthogonal normal basis ek (x) = e2πikx (0 ≤ x ≤ 1; k = 0, ±1, ±2, ....). Let

∞

K(x, s) =

(1.16)

bjk ek (x)ej (s)

(1.17)

j,k=−∞

be the Fourier expansion of K with the Fourier coeﬃcients bjk . Put ajk = (Aek , ej ) = bjk (j = k), and ajj = (Aej , ej ) = 1 + bjj for j, k = 0, ±1, ±2, ...). Here (., .) is the scalar product in L2 [0, 1]. Assume that 1 + bjj = 0 for any integer j. According to (1.3), under consideration we have ∞ ∞ |bjk |2 < ∞, N 2 (V+ ) = j=−∞ k=j+1

and N 2 (V− ) =

∞

j−1

|bjk |2 < ∞.

j=−∞ k=−∞

Now relations (1.11) and (1.14) give us the bounds max{0,

13.2

sup

k=0,±1,±2,...

˜ H (A)} ≤ rs (A) ≤ |1 + bkk | − ∆

sup

k=0,±1,±2,...

|1 + bkk | + φ˜1 .

Matrices with Relatively Compact Oﬀdiagonals

Under (1.1), assume that ρl (D) ≡

inf

k=1,2,...

|akk | > 0

(2.1)

and the operators D−1 V± are Hilbert-Schmidt ones: N2 (D−1 V± ) = v± , where j−1 ∞ ∞ ∞ |ajk |2 |ajk |2 2 2 v− = < ∞; v = < ∞. (2.2) + 2 |ajj | |ajj |2 j=2 j=1 k=1

k=j+1

Note that the results of Section 12.4 allow us to investigate considerably more general conditions than (2.2). Without any loss of generality assume that v− ≤ v+ .

(2.3)

13.3. A Nonselfadjoint Diﬀerential Operator

The case v− ≥ v+ can be similarly considered . Put if v+ ≤ ev− , ev− . δ(v− , v+ ) := v+ [ln (v+ /v− )]−1/2 if v+ > ev−

185

(2.4)

Due to Lemma 12.4.5 and Corollary 12.4.6, the spectrum of the matrix A = (ajk )∞ j,k , under conditions (2.2), (2.3) is included in the set {z ∈ C : |1 −

z | ≤ δ(v− , v+ ), k = 1, 2, ..., }. akk

(2.5)

In particular, the lower spectral radius of A satisﬁes the inequality rl (A) ≥ max{0, 1 − δ(v− , v+ )} inf |akk |. k

13.3

(2.6)

A Nonselfadjoint Diﬀerential Operator

In space H = L2 [0, 1], let us consider an operator A deﬁned by (Au)(x) = −

1 d2 u(x) w(x) du(x) + l(x)u(x) + 4 dx2 2 dx

(0 < x < 1, u ∈ Dom (A))

(3.1)

with the domain Dom (A) = {h ∈ L2 [0, 1] : h ∈ L2 [0, 1], h(0) = h(1), h (0) = h (1)}. (3.2) Here w(.), l(.) ∈ L2 [0, 1] are scalar functions. So the periodic boundary conditions (3.3) u(0) = u(1), u (0) = u (1) are imposed. With the orthogonal normal basis (1.16), let l=

∞ k=−∞

˜lk ek and w =

∞

w ˜k ek (w ˜k = (w, ek ), ˜lk = (l, ek ))

(3.4)

k=−∞

be the Fourier expansions of l and w, respectively. Omitting simple calculations, we have ˜j−k + ˜lj−k (k = j) (Aek , ej ) = iπk w and

(Aek , ek ) = π 2 k 2 + iπk w ˜0 + ˜l0 (j, k = 0, ±1, ±2, ...).

Here (., .) is the scalar product in L2 [0, 1]. Take Dom (D) = Dom (A) and rewrite operator A as the matrix (ajk )∞ j,k=−∞ with the entries akk = π 2 k 2 + iπk w ˜0 + ˜l0

186

13. Inﬁnite Matrices and Diﬀerential Operators

and

˜j−k + ˜lj−k (j = k; j, k = 0, ±1, ±2, ...). ajk = iπk w

Assume that rl (D) = inf |akk | > 0. k

Then N2 (D 2 v+

−1

=

V+ ) = v± , where ∞

k−1

|(π 2 k 2 + iπk w ˜0 + ˜l0 )−1 (iπk w ˜j−k + ˜lj−k )|2 =

k=−∞ j=−∞ ∞

−1

|(π 2 k 2 + iπk w ˜0 + ˜l0 )−1 (iπk w ˜m + ˜lm )|2 ≤

k=−∞ m=−∞ ∞

2

|π 2 k 2 + iπk w ˜0 + ˜l0 |−2 π|k|2

2

|w ˜m |2 +

m=−∞

k=−∞ ∞

−1

|(π 2 k 2 + iπk w ˜0 + ˜l0 )−1 |2

−1

|˜lm |2 < ∞

m=−∞

k=−∞ 2

since w, l ∈ L . Similarly, 2 v− =

∞

∞

|(π 2 k 2 + iπk w ˜0 + ˜l0 )−1 (iπk w ˜j−k + ˜lj−k )|2 < ∞.

k=−∞ j=k+1

Acccording to (2.5), the spectrum of the operator A deﬁned by (3.1) is included in the set z | ≤ δ(v− , v+ ), k = 0, ±1, ±2, ..., }, {z ∈ C : |1 − 2 2 π k + iπk w ˜0 + ˜l0 where δ(v− , v+ ) is deﬁned by (2.4). In particular, the lower spectral radius of A satisﬁes the inequality rl (A) ≥ min |π 2 k 2 + iπk w ˜0 + ˜l0 | max{0, 1 − δ(v− , v+ )}. k

13.4

Integro-diﬀerential Operators

In space H = L2 [0, 1] let us consider the operator 1 d2 u(x) + w(x)u(x) + K(x, s)u(s)ds (Au)(x) = − 4dx2 0 (u ∈ Dom (A), 0 < x < 1)

(4.1)

13.5. Notes

187

with the domain Dom (A) deﬁned by (3.2). So the periodic boundary conditions (3.3) hold. Here K is a Hilbert-Schmidt kernel and w(.) ∈ L2 [0, 1] is a scalar-valued function. Take the orthonormal basis (1.16). Let (1.7) and (3.4) be the Fourier expansions of K and of w, respectively. Obviously, for all j, k = 0, ±1, ±2, ..., ajk = (Aej , ek ) = w ˜j−k + bjk (j = k) and ˜0 + bkk . akk = (Aek , ek ) = π 2 k 2 + w Assume that rl (D) =

inf

k=0,±1,±2,...

˜0 + bkk | > 0. |π 2 k 2 + w

(4.2)

Then we have N2 (D−1 V± ) = v± with 2 = v+

∞

k−1

|(π 2 k 2 + w ˜0 + bkk )−1 (w ˜j−k + bjk )|2 < ∞,

k=−∞ j=−∞

and 2 v− =

∞

∞

|(π 2 k 2 + w ˜0 + bkk )−1 (w ˜j−k + bjk )|2 < ∞.

k=−∞ j=k+1

According to (2.5), the spectrum of the operator A deﬁned by (4.1) is included in the set z {z ∈ C : |1 − 2 2 | ≤ δ(v− , v+ ), k = 0, ±1, ±2, ..., }, π k +w ˜0 + bkk where δ(v− , v+ ) is deﬁned by (2.4).

13.5

Notes

The results presented in this chapter are based on the paper (Gil’, 2001). In particular, inequality (1.11) is sharper than the well-known estimate rs (A) ≤ sup j

∞

|ajk |,

(5.1)

k=1

cf. (Krasnosel’skij et al, 1989, inequality (16.2)), provided sup j

∞ k=1

|ajk | > sup |akk | + φ˜1 (A). k

For nonnegative matrices the following estimate is well-known, cf. (Krasnosel’skij et al, 1989 inequality (16.15)): rs (A) ≥ r˜∞ (A) ≡

min

j=1,...,∞

∞ k=1

ajk

(5.2)

188

13. Inﬁnite Matrices and Diﬀerential Operators

Our relation (1.14) is sharper than estimate (5.2) in the case |ajk | = ajk (j, k = 1, 2, ...), provided ˜ H (A) > r˜∞ (A). max akk − ∆ k

That is, (1.11) improves estimate (5.1) and (1.14) improves estimate (5.2) for matrices which are ”close” to triangular ones. The results in Section 13.4 supplement the well-known results on diﬀerential operators, cf. (Edmunds and Evans, 1990), (Egorov and Kondratiev, 1996), (Locker, 1999) and references therein.

References [1] Edmunds, D.E. and Evans V.D. (1990). Spectral Theory and Differential Operators. Clarendon Press, Oxford. [2] Egorov, Y. and Kondratiev, V. (1996). Spectral Theory of Elliptic Operators. Birkh¨ auser Verlag, Basel. [3] Gil’, M.I. (2001). Spectrum localization of inﬁnite matrices, Mathematical Physics, Analysis and Geometry, 4, 379-394 [4] Krasnosel’skii, M.A., Lifshits, J. and A. Sobolev (1989). Positive Linear Systems. The Method of Positive Operators, Heldermann Verlag, Berlin. [5] Locker, J. (1999). Spectral Theory of Nonself-Adjoint Two Point Diﬀerential Operators. Amer. Math. Soc, Mathematical Surveys and Monographs, Volume 73, R.I.

14. Integral Operators in Space L2

The present chapter is concerned with integral operators in L2 . In particular, we suggest estimates for the spectral radius of an integral operator.

14.1

Scalar Integral Operators

Consider a scalar integral operator A deﬁned in H = L2 [0, 1] by 1 K(x, s)u(s)ds (u ∈ L2 [0, 1]; x ∈ [0, 1]), (Au)(x) = a(x)u(x) +

(1.1)

0

where a(.) is a real bounded measurable function, K is a real Hilbert-Schmidt kernel. Deﬁne the maximal resolution of the identity P (t) (− ≤ t ≤ 1; > 0) by 0 if − ≤ t < x, (P (t)u)(x) = u(x) if x ≤ t ≤ 1 with x ∈ [0, 1]. Then, the conditions (1.1) and (1.2) from Section 9.1 are valid with 1 (Du)(x) = a(x)u(x), (V+ u)(x) = K(x, s)u(s)ds, x

and (V− u)(x) =

0

x

K(x, s)u(s)ds (u ∈ L2 [0, 1]; x ∈ [0, 1]).

So N22 (V+ ) =

1

0

M.I. Gil’: LNM 1830, pp. 189–197, 2003. c Springer-Verlag Berlin Heidelberg 2003

1

x

K 2 (x, s) ds dx

190

14. Integral Operators

and N22 (V− ) =

0

1

x

K 2 (x, s)ds dx.

0

Without any loss of generality, assume that N2 (V− ) ≤ N2 (V+ ).

(1.2)

The case N2 (V− ) ≥ N2 (V+ ) can be similarly considered. So according to relations (4.1) and (4.2) from Section 9.4, we have τ (A) ≤ N2 (V− ) and V˜ = V+ . Put eN2 (V− ) if N2 (V+ ) ≤ eN2 (V− ) . φ˜1 = N2 (V+ )[ln (N2 (V+ )/ N2 (V− )) ]−1/2 if N2 (V+ ) > eN2 (V− ) (1.3) Due to Corollary 9.4.3 and Remark 9.4.4, the spectrum of operator A is included in the set {z ∈ C : |a(x) − z| ≤ φ˜1 , 0 ≤ x ≤ 1}. Hence, the spectral radius of A satisﬁes the inequality rs (A) ≤ sup |a(x)| + φ˜1 . x∈[0,1]

In particular, if a(x) ≡ 0, then rs (A) ≤ φ˜1 (A).

(1.4)

Let us derive the lower estimates for the spectrum. Clearly, 1 1 N22 (AI ) ≡ N22 ((A − A∗ )/2i) = |K(x, s) − K(s, x)|2 ds dx/4. 0

Recall that gI (A) = [2N22 (AI ) − 2 √

0

∞

|Im λk (A)|2 ]1/2

k=1

and one can replace gI (A) by 2N2 (AI ). Put eN2 (V− ) if gI (A) ≤ eN2 (V− ) ˜ H := ∆ . gI (A)[ln (gI (A)/N2 (V− )) ]−1/2 if gI (A) > eN2 (V− )

(1.5)

Due to Corollary 9.8.2, for the integral operator deﬁned by (1.1), the following relations are true: ˜ H }, rs (A) ≥ max{0, sup |a(x)| − ∆ x∈[0,1]

˜ H and α(A) ≥ sup Re a(x) − ∆ ˜ H. rl (A) ≤ inf |a(x)| + ∆ x

x∈[0,1]

(1.6)

14.2. Relatively Small Kernels

14.2

191

Matrix Integral Operators with Relatively Small Kernels

Let ω ⊆ Rm be a set with a ﬁnite Lebesgue measure, and H ≡ L2 (ω, Cn ) be a Hilbert space of functions deﬁned on ω with values in Cn and equipped with the scalar product (f, h)H = (f (s), h(s))C n ds, ω

where (., .)C n is the scalar product in Cn . Consider in L2 (ω, Cn ) the operator (Ah)(x) = Q(x)h(x) + K(x, s)h(s)ds (h ∈ L2 (ω, Cn )), (2.1) ω

where Q(x), K(x, s) are matrix-valued functions deﬁned on ω and ω × ω, respectively. It is assumed that Q is bounded measurable and K is a HilbertSchmidt kernel. So ˜ + K, ˜ A=Q where ˜ (Qh)(x) = Q(x)h(x)

and ˜ (Kh)(x) =

ω

Besides,

K(x, s)h(s)ds (x ∈ ω).

˜ =[ N2 (K)

ω

ω

K(x, s)2C n ds]1/2 ,

where .C n is the Euclidean norm. Lemma 14.2.1 The spectrum of operator A deﬁned by (2.1) lies in the set ˜ sup (Q(x) − IC n λ)−1 C n ≥ 1}. {λ ∈ C : N2 (K) x∈ω

Proof:

Since, ˜ ˜+K ˜ − λI = (Q ˜ − λI)(I + (Q ˜ − λI)−1 K), A − λI = Q

if

˜ H < 1, ˜ − λI)−1 K (Q

then λ is a regular point. So for any µ ∈ σ(A), ˜ − µI)−1 H K ˜ H ≤ (Q ˜ − µI)−1 H N2 (K). ˜ 1 ≤ (Q But

˜ − µI)−1 H ≤ sup (Q(x) − IC n µ)−1 C n . (Q x∈ω

192

14. Integral Operators

This proves the lemma. 2 Due to Corollary 2.1.2, for a ﬁxed x we have (Q(x) − IC n λ)−1 C n ≤

n−1

√

k=0

g k (Q(x)) . k!ρk+1 (Q(x), λ)

(2.2)

Now Lemma 14.2.1 yields Lemma 14.2.2 Let operator A be deﬁned by (2.1). Then its spectrum lies in the set ˜ {λ ∈ C : N2 (K)

n−1

√

k=0

g k (Q(x)) ≥ 1, x ∈ ω}. k!ρk+1 (Q(x), λ)

Corollary 14.2.3 Let operator A be deﬁned by (2.1). In addition, let ˜ sup N2 (K) x∈ω

n−1

√

k=0

g k (Q(x)) < 1, k!dk+1 (Q(x)) 0

where d0 (Q(x)) =

min |λk (Q(x))|.

k=1,...,n

(2.3)

Then A is boundedly invertible in L2 (ω, Cn ). With a ﬁxed x ∈ ω, consider the algebraic equation ˜ z n = N2 (K)

n−1 k=0

g k (Q(x))z n−k−1 √ . k!

(2.4)

Lemma 14.2.4 Let z0 (x) be the extreme right (unique positive) root of (2.4). Then for any point µ ∈ σ(A) there are x ∈ ω and an eigenvalue λj (Q(x)) of matrix Q(x), such that |µ − λj (Q(x))| ≤ z0 (x).

(2.5)

In particular, rs (A) ≤ sup(rs (Q(x)) + z0 (x)). x

Proof: Due to Lemma 14.2.2, for any point µ ∈ σ(A) there is x ∈ ω, such that the inequality ˜ N2 (K)

n−1 k=0

√

g k (Q(x)) ≥1 k!ρk+1 (Q(x), µ)

is valid. Comparing this with (2.4), we have ρ(Q(x), µ) ≤ z0 (x). This proves the required result. 2

14.3. Perturbations of Convolutions

193

Corollary 14.2.5 Let Q(x) be a normal matrix for all x ∈ ω. Then for any point µ ∈ σ(A) there are x ∈ ω and λj (Q(x))σ(Q(x)), such that ˜ |µ − λj (Q(x))| ≤ N2 (K). ˜ + supx (rs (Q(x))). In particular, rs (A) ≤ N2 (K) ˜ Now Indeed, since Q(x) is normal, we have g(Q(x)) = 0 and z0 (x) = N2 (K). the result is due to the latter theorem. Put n−1 g k (Q(x)) ˜ √ b(x) := N2 (K) . k! k=0 Due to Corollary 1.6.2, z0 (x) ≤ δn (x), where δn (x) = n b(x) if b(x) ≤ 1 and δn (x) = b(x) if b(x) > 1. Now Theorem 14.2.4 implies Theorem 14.2.6 Under condition (2.7), for any point µ ∈ σ(A), there are x ∈ ω and an eigenvalue λj (Q(x)) of Q(x), such that |µ − λj (Q(x))| ≤ δn (x). In particular, rs (A) ≤ supx (rs (Q(x)) + δn (x)).

14.3

Perturbations of Matrix Convolutions

Consider in H = L2 ([−π, π], Cn ) the convolution operator π (Ch)(x) = Q0 h(x) + K0 (x − s)h(s)ds (h ∈ L2 ([−π, π], Cn )),

(3.1)

−π

where Q0 is a constant matrix, K0 is a matrix-valued function deﬁned on [−π, π] with K0 C n ∈ L2 [−π, π], having the Fourier expansion K0 (x) =

∞

Dk eikx

k=−∞

with the matrix Fourier coeﬃcients π 1 Dk = √ K0 (s)e−iks ds. 2π −π

194

14. Integral Operators

Put Bk = Q0 + Dk . We have

Ceikx = Bk eikx .

(3.2)

Let djk be an eigenvector of Bk , corresponding to an eigenvalue λj (Bk ) (j = 1, ...n). Then π ikx ikx K0 (x − s)djk eiks ds = Ce djk = e Q0 djk + −π

eikx Bk djk = eikx λj (Bk )djk . Since the set

{eikx }k=∞ k=−∞

is a basis in L2 [−π, π] we have the following result Lemma 14.3.1 The spectrum of operator (3.1) consists of the points λj (Bk ) (k = 0, ±1, ±2, ... ; j = 1, ...n). Let Pk be orthogonal projectors deﬁned by π ikx 1 h(s)e−iks ds. (Pk h)(x) = e 2π −π Since

∞

Pk = IH ,

k=−∞

it can be directly checked by (3.2) that the equality C=

∞

Bk Pk

k=−∞

holds. Hence, the relation −1

(C − IH λ)

=

∞

(Bk − IC n λ)−1 Pk

k=−∞

is valid for any regular λ. Therefore, (C − IH λ)−1 H ≤ Using Corollary 2.1.2, we get

sup

k=0, ±1,...

(Bk − IC n λ)−1 C n .

14.3. Perturbations of Convolutions

195

Lemma 14.3.2 The resolvent of convolution C deﬁned by (3.1) satisﬁes the inequality n−1 g k (Bl ) √ (C − λI)−1 H ≤ sup . k!ρk+1 (Bl , λ) l=0, ±1,... k=0 Consider now the operator π (Ah)(x) ≡ Q0 h(x) + K0 (x − s)h(s)ds + (Zh)(x) (−π ≤ x ≤ π). (3.3) −π

where Z is a bounded operator in L2 ([−π, π], Cn ). We easily have by the previous lemma that the inequalities ZH (C − λI)−1 H ≤ ZH

n−1

sup

l=0, ±1,...

√

k=0

g k (Bl ) <1 k!ρk+1 (Bl , λ)

imply that λ is a regular point. Hence we arrive at Lemma 14.3.3 The spectrum of operator A deﬁned by (3.3) lie in the set {λ ∈ C : ZH

sup

n−1

l=0, ±1,...

k=0

√

g k (Bl ) ≥ 1}. k!ρk+1 (Bl , λ)

In other words, for any µ ∈ σ(A), there are l = 0, ±1, ±2, ... and j = 1, ..., n, such that ZH

n−1 k=0

√

g k (Bl ) ≥ 1. k!|µ − λj (Bl )|k+1

Corollary 14.3.4 Operator A deﬁned by (3.3) is invertible provided that ZH

n−1 k=0

√

g k (Bk ) ≤ c0 < 1 (c0 = const) k!|λj (Bl )|k+1

for all l = 0, ±1, ±2, ... and j = 1, ..., n. Let zl be the extreme right (unique positive) root of the equation z n = ZH

n−1 k=0

z n−1−k g k (Bl ) √ . k!

(3.4)

Since the function in the right part of (3.4) monotonically increases as z > 0 increases, Lemma 14.3.4 implies

196

14. Integral Operators

Theorem 14.3.5 For any point µ of the spectrum of operator (3.3), there are indexes l = 0, ±1, ±2, ... and j = 1, ..., n, such that |µ − λj (Bl )| ≤ zl ,

(3.5)

where zl is the extreme right (unique positive) root of the algebraic equation (3.4). In particular, rs (A) ≤

max rs (Bl ) + zl .

l=0,±1,...

If all the matrices Bl are normal, then g(Bl ) ≡ 0, zl = ZH , and (3.5) takes the form |µ − λj (Bl )| ≤ ZH . Assume that bl := ZH

n−1 k=0

g k (Bl ) √ ≤ 1 (l = 0, ±1, ±2, ...). k!

Then due to Lemma 1.6.1 zl ≤

n

(3.6)

bl .

Now Theorem 14.3.5 implies Corollary 14.3.6 Let A be deﬁned by (3.3) and condition (3.6) hold. Then for any µ ∈ σ(A) there are l = 0, ±1, ±2, ... and j = 1, ..., n, such that |µ − λj (Bl )| ≤ In particular, rs (A) ≤

14.4

sup

l=0,±1,±2,...

n bl .

n bl + rs (Bl ).

Notes

Inequality (1.4) improves the well-known estimate rs (A) ≤ δ˜0 (A) ≡ vrai sup x

0

1

|K(x, s)|ds,

cf. (Krasnosel’skii et al., 1989, Section 16.6) for operators which are ”close” to Volterra ones. The material in this chapter is taken from the papers (Gil’, 2000), (Gil’, 2003).

14.4. Notes

197

References [1] Gil’, M.I. (2000). Invertibility conditions and bounds for spectra of matrix integral operators, Monatshefte f¨ ur mathematik, 129, 15-24. [2] Gil’, M.I. (2003). Inner bounds for spectra of linear operators, Proceedings of the American Mathematical Society , (to appear). [3] Krasnosel’skii, M. A., J. Lifshits, and A. Sobolev (1989). Positive Linear Systems. The Method of Positive Operators, Heldermann Verlag, Berlin. [4] Pietsch, A. (1987). Eigenvalues and s-Numbers, Cambridge University Press, Cambridge.

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15. Operator Matrices

In the present chapter we consider the invertibility and spectrum of matrices, whose entries are unbounded, in general, operators. In particular, under some restrictions, we improve the Gershgorin-type bounds. Applications to matrix diﬀerential operators are also discussed.

15.1

Invertibility Conditions

Let H be an orthogonal sum of Hilbert spaces Ek (k = 1, ..., n < ∞) with norms .Ek : H ≡ E1 ⊕ E2 ⊕ ... ⊕ En . Consider in H the operator matrix  A11 A12  A21 A22 A=  . ... An1 An2

 . . . A1n . . . A2n  , . .  . . . Ann

(1.1)

where Ajk are linear operators acting from Ek to Ej . In the present chapter, invertibility conditions and bounds for the spectrum of operator (1.1) are investigated under the assumption that we have an information about the spectra of diagonal operators. Let h = (hk ∈ Ek )nk=1 be an element of H. Everywhere in the present chapter the norm in H is deﬁned by the relation h ≡ hH = [

n

k=1

and I = IH is the unit operator in H. M.I. Gil’: LNM 1830, pp. 199–213, 2003. c Springer-Verlag Berlin Heidelberg 2003

hk 2Ek ]1/2

(1.2)

200

15. Operator Matrices

Denote by V, W and D the upper triangular, lower triangular, and diagonal parts of A, respectively. That is,   0 A12 . . . A1n  0 0 . . . A2n  , V =  . ... . .  0 0 ... 0 

0  A21  W = . An1

0 0 ... An2

... ... . ...

0 0 . An,n−1

 0 0    0

and D = diag [A11 , A22 , ..., Ann ]. Recall that, for a linear operator A, Dom (A) means the domain, σ(A) is the spectrum, λk (A) (k = 1, 2, ...) are the eigenvalues with their multiplicities, ρ(A, λ) is the distance between the spectrum of A and a λ ∈ C. Theorem 15.1.1 Let the diagonal operator D be invertible and the operators VA ≡ D−1 V, WA ≡ D−1 W be bounded .

(1.3)

In addition, let the condition

n−1 j,k=1

(−1)k+j VAk WAj < 1

(1.4)

hold. Then operator A deﬁned by (1.1) is invertible. Proof:

We have

A = D + V + W = D(I + VA + WA ) = D[(I + VA )(I + WA ) − VA WA ]. Simple calculations show that VAn = WAn = 0.

(1.5)

So VA and WA are nilpotent operators and, consequently, the operators, I + VA and I + WA are invertible. Thus, A = D(I + VA )[I − (I + VA )−1 VA WA (I + WA )−1 ](I + WA ). Therefore, the condition (I + VA )−1 VA WA (I + WA )−1 < 1

15.1. Invertibility Conditions

201

provides the invertibility of A. But according to (1.5), (I + VA )−1 VA =

n−1

(−1)k−1 VAk , WA (I + WA )−1 =

k=1

n−1

(−1)k−1 WAk .

k=1

Hence, the required result follows. 2 Corollary 15.1.2 Let operator matrix A deﬁned by (1.1) be an upper triangular one (W=0), D an invertible operator and D−1 V a bounded one. Then A is invertible. Similarly, let operator matrix A be a lower triangular one (V=0) and D−1 W a bounded operator. Then A is invertible. Corollary 15.1.3 Let the diagonal operator D be invertible and the conditions (1.3) and n−2 VA WA VA k WA j < 1 (1.6) j,k=0

hold. Then operator (1.1) is invertible. In particular, let VA , WA = 1. Then (1.6) can be written in the form VA WA

(1 − VA n−1 )(1 − WA n−1 ) < 1. (1 − VA )(1 − WA )

(1.7)

Indeed, taking into account that n−2 k=0

n−2

1 − VA n−1 1 − WA n−1 , VA = WA k = 1 − VA 1 − WA k

k=0

and using Theorem 15.1.1, we arrive at the required result. 2 We need also the following Lemma 15.1.4 Let ajk ≡ Ajk Ek →Ej < ∞ (j, k = 1, ..., n). Then the norm of operator A deﬁned by (1.1) is subject to the relation A ≤ ˜ aC n , where a ˜ is the linear operator in the Euclidean space Cn , deﬁned by the matrix with the entries ajk and .C n is the Euclidean norm.

202

15. Operator Matrices

The proof is a simple application of relation (1.2) and it is left to the reader. The latter corollary implies A2 ≤

n

Ajk 2Ek →Ej .

(1.8)

j,k=1

Consider the case n = 2:

A=

A11 A21

A12 A22

.

(1.9)

Clearly, VA =

0 A−1 11 A12 0 0

Hence,

and WA =

−1 A−1 11 A12 A22 A21 0

VA W A =

0

A−1 22 A21 0 0

0 0

.

.

Thus, due to Theorem 15.1.1, if −1 A−1 11 A12 A22 A21 < 1,

then operator (1.9) is invertible.

15.2

Bounds for the Spectrum

Theorem 15.2.1 For any regular point λ of D, let ˜ (λ) := (D − IH λ)−1 W be bounded operators . V˜ (λ) := (D − IH λ)−1 V and W (2.1) Then the spectrum of operator A deﬁned by (1.1) lies in the union of the sets σ(D) and n−2

˜ (λ) {λ ∈ C : V˜ (λ)W

˜ (λ)j ≥ 1}. V˜ (λ)k W

j,k=0

Indeed, if for some λ ∈ σ(A), ˜ (λ) V˜ (λ)W

n−2

˜ (λ)j < 1, V˜ (λ)k W

(2.2)

j,k=0

then due to Corollary 15.1.3, A − λI is invertible. This proves the required result. 2

15.2. Bounds for the Spectrum

203

Corollary 15.2.2 Let operator matrix (1.1) be an upper triangular one, and V˜ (λ) be bounded for all regular λ of D. Then σ(A) = ∪nk=1 σ(Akk ) = σ(D).

(2.3)

˜ (λ) be bounded for all regular Similarly, let (1.1) be lower triangular and W λ of D. Then (2.3) holds. ˜ (λ) = 0. Now the result is due to Indeed, let A be upper triangular. Then W Theorem 15.2.1. The lower triangular case can be similarly considered. This result shows that Theorem 15.2.1 is exact. Lemma 15.2.3 Let W and V be bounded operators and the condition (D − IH λ)−1 ≤ Φ(ρ−1 (D, λ)) (λ ∈ / σ(D))

(2.4)

hold, where Φ(y) is a continuous increasing function of y ≥ 0 with the properties Φ(0) = 0 and Φ(∞) = ∞. In addition, let z0 be the unique positive root of the scalar equation n−1

Φk+j (y)V j W k = 1.

(2.5)

j,k=1

Then the spectral variation of operator A deﬁned by (1.1) with respect to D satisﬁes the inequality 1 svD (A) ≤ . z0 Proof:

Due to (2.4) ˜ (λ) ≤ W Φ(ρ−1 (D, λ)). V˜ (λ) ≤ V Φ(ρ−1 (D, λ)), W

For any λ ∈ σ(A) and λ ∈ / σ(D), Theorem 15.2.1 implies n−1

Φk+j (ρ−1 (D, λ))V k W j ≥ 1.

j,k=1

Taking into account that Φ is increasing and comparing the latter inequality with (2.5), we have ρ−1 (D, λ) ≥ z0 for any λ ∈ σ(A). This proves the required result. 2 For instance, let n = 2. Then (2.5) takes the form Φ2 (y)V W = 1.

(2.6)

204

15. Operator Matrices

Hence it follows that

1 ), z0 = Ψ( V W

where Ψ is the function inverse to Φ: Φ(Ψ(y)) = y. Thus, in the case n = 2, svD (A) ≤

15.3

1

Ψ( √

. 1 ) V W

Operator Matrices with Normal Entries

Assume that H is an orthogonal sum of the same Hilbert spaces Ek ≡ E (k = 1, ..., n) with norm .E . Consider in H the operator matrix deﬁned by (1.1), assuming that (3.1) Ajj = Sj (j = 1, ..., n), where Sj are normal, unbounded in general operators in Ej , and Ajk = φk (Sj ) (j = k; j, k = 1, ..., n),

(3.2)

where φk (s) are scalar-valued measurable functions of s ∈ C. In addition, assume that (3.3) αjk ≡ sup |φk (t)t−1 | < ∞. t∈σ(Sj )

Then A−1 jj Ajk are bounded normal operators with the norms A−1 jj Ajk E = αjk . Moreover, due to Lemma 15.1.4, with the notation n

vA := max j

we have

αjk and wA := max j

k=j+1

j−1

αjk

k=1

D−1 V ≤ vA and D−1 W ≤ wA .

Now Corollary 15.1.3 implies Lemma 15.3.1 Let the conditions (3.1)-(3.3) and n−1 j,k=1

k j vA wA < 1

(3.4)

hold. Then operator (1.1) is invertible. In particular, let vA , wA = 1. Then (3.4) can be written in the form vA wA

n−1 n−1 (1 − vA )(1 − wA ) < 1. (1 − vA )(1 − wA )

(3.5)

15.4. Matrices with Bounded oﬀ Diagonals

205

Furthermore, under (3.1), (3.2) assume that for all regular λ of D, the relations n sup |φk (t)(λ − t)−1 | < ∞ and v˜(λ) := max j

k=j+1 t∈σ(Sj )

w(λ) ˜ := max j

j−1

sup |φk (t)(λ − t)−1 | < ∞.

k=1 t∈σ(Sj )

(3.6)

are fulﬁlled. Due to Lemma 15.1.4 with the notation from (2.1), we have ˜ (λ) ≤ w(λ) W ˜ and V˜ (λ) ≤ v˜(λ). Now Theorem 15.2.1 gives. Lemma 15.3.2 Under conditions (3.1), (3.2) and (3.6), the spectrum of operator A deﬁned by (1.1) lies in the union of the sets σ(D) and {λ ∈ C :

n−1

v˜k (λ)w ˜ j (λ) ≥ 1}.

j,k=1

15.4

Operator Matrices with Bounded oﬀ Diagonal Entries

Again assume that H is an orthogonal sum of the same Hilbert spaces Ek ≡ E (k = 1, ..., n). In addition, Ajk (j = k) are arbitrary bounded operators: v0 ≡ V < ∞, w0 ≡ W < ∞.

(4.1)

Lemma 15.4.1 Let the conditions (3.1),(4.1) and n−1

ρ−k−j (D, 0)v0k w0j < 1

j,k=1

hold. Then operator (1.1) is invertible. Proof:

Since D is normal, (D − λIH )−1 = ρ−1 (D, λ).

(4.2)

Therefore D−1 = ρ−1 (D, 0). Due to (4.1) D−1 V ≤ D−1 V ≤ ρ−1 (D, 0)v0 . Similarly, D−1 W ≤ ρ−1 (D, 0)w0 . Now Corollary 15.1.3 implies the required result. 2

206

15. Operator Matrices

Lemma 15.4.2 Under conditions (3.1) and (4.1), let z1 be the unique nonnegative root of the algebraic equation n−2

z k+j v0n−j−1 w0n−k−1 = z 2(n−1) .

(4.3)

j,k=0

Then the spectral variation of A with respect to D satisﬁes the inequality svD (A) ≤ z1 . Proof:

With the substitution y = 1/z equation (4.3) takes the form n−1

y k+j v0j w0k = 1.

(4.4)

j,k=1

So under consideration equation (2.5) can be rewritten as (4.4). Due to Lemma 15.2.3 we arrive at the result. 2 Due to Lemma 1.6.1, if p(A) :=

n−1

v0j w0k < 1,

(4.5)

j,k=1 2(n−1)

then z1

≤ p(A). So under (4.5), the previous lemma gives the inequality svD (A) ≤

2(n−1)

p(A).

(4.6)

Let us improve Lemma 15.4.1 in the case, when Ajk (j = k) are HilbertSchmidt operators. Recall that N2 (.) denotes the Hilbert-Schmidt norm. Clearly, N22 (Ajk ), N22 (V ) = 1≤j
and

N22 (W ) =

N22 (Ajk ).

1≤k<j≤n

Lemma 15.4.3 Under condition (3.1), let V and W be Hilbert-Schmidt operators and the inequality n−1

(k!j!)−1/2 N2k (V )N2j (W )ρ−k−j (D, 0) < 1

j,k=1

hold. Then operator A deﬁned by (1.1) is invertible.

(4.7)

15.5. Hilbert-Schmidt Diagonals

207

Proof: Due to (4.7) D−1 is bounded, D−1 V and D−1 W are Hilbert- Schmidt operators. In addition, according to (1.5), they are nilpotent. Due to Corollary 6.9.2, (D−1 V )k ≤ N2k (D−1 V )(k!)−1/2 , (D−1 W )k ≤ N2k (D−1 W )(k!)−1/2 (4.8) for any natural k < n. But N2 (D−1 V ) ≤ D−1 N2 (V ) = ρ−1 (D, 0)N2 (V ). Similarly,

N2 (D−1 W ) ≤ ρ−1 (D, 0)N2 (W ).

Now Theorem 15.1.1 implies the required result. 2

15.5

Operator Matrices with Hilbert-Schmidt Diagonal Operators

Again, let H be an orthogonal sum of Hilbert spaces Ek , k = 1, ..., n. In addition, Ek are separable, condition (4.1) holds and the diagonal operators have the form Ajj = IE + Kj where Kj (j = 1, ..., n) are Hilbert-Schmidt operators . (5.1) Due to Theorem 6.4.1, for any Hilbert-Schmidt operator K in H, we can write (K −λI)

−1

∞

≤ G(K, ρ(λ, K)) ≡

k=0

where G(K, y) ≡

√

g k (K) for all regular λ, (5.2) k!ρk+1 (K, λ)

∞ g k (K) √ (y > 0) k!y k+1 k=0

and g(K) = (N22 (K) −

∞

|λk (K)|2 )1/2 (K ∈ C2 ).

k=1 ∗

If K is a normal operator: KK = K ∗ K, then g(K) = 0. The following relations are true: g 2 (K) ≤ N22 (K) − |T race(K 2 )| and g 2 (K) ≤

1 2 ∗ N (K − K) (K ∈ C2 ) 2 2

208

15. Operator Matrices

(see Section 6.3). So due to (5.1) and (5.2) (Kj − λI)−1 Ej ≤ G(Kj , ρ(Kj , λ))

(5.3)

and therefore with λ = −1 we get D−1 = max (IE + Kj )−1 Ej ≤ b0 (D), j

(5.4)

where b0 (D) = max G(Kj , ρ(Kj , −1)) = max j

j

∞

√

k=0

g k (Kj ) . k!ρk+1 (Kj , −1)

Lemma 15.5.1 Let the conditions (4.1), (5.1) and n−1

bk+j (D)v0k w0j < 1 0

(5.5)

j,k=1

be fulﬁlled. Then operator A deﬁned by (1.1) is invertible. Proof:

Due to (4.1) and (5.4) D−1 V ≤ D−1 v0 ≤ b0 (D)v0 .

Similarly,

D−1 W ≤ b0 (D)w0 .

Now Theorem 15.1.1 implies the required result. 2 Thanks to Theorem 15.1.1, we also get the following result. Lemma 15.5.2 Under conditions (4.1) and (5.1) the spectrum of operator A deﬁned by (1.1), lies in the set ∪nj=1 Ωj (λ), where Ωj (λ) = {λ ∈ C :

n−1

Gs+k (Kj , ρ(Kj , λ − 1))v0k w0s ≥ 1}.

s,k=1

In other words, for any µ ∈ σ(A), there are an integer j and a λ(Kj ) ∈ σ(Kj ), such that

n−1 s,k=1

Gs+k (Kj , |λ(Kj ) + 1 − µ|)v0k w0s ≥ 1.

15.6. Example

Put h(y, D) =

209

∞ g0k y k+1 √ , k! k=0

where g0 = maxj g(Kj ). Let us consider the scalar equation n−1

hs+l (y, D)v0l w0s = 1.

(5.6)

s,l=1

Thanks to (5.3), one can write (D − λIH )−1 = max (IE + Kj − λIE )−1 ≤ h(ρ−1 (D, λ), D). j

Now Lemma 15.2.3 yields Theorem 15.5.3 Let x0 be the extreme right (unique positive) root of equation (5.6). Then the spectral variation of operator A deﬁned by (1.1) with respect to D satisﬁes the inequality svD (A) ≤ x−1 0 .

15.6

Example

Let H = L2 ([0, π], Cn ) be the Hilbert space of functions deﬁned on [0, π] with values in a Euclidean space Cn and the scalar product π (h(x), w(x))C n dx, (h, w)H = 0

where (., .)C n is the scalar product in Cn . Consider the operator A deﬁned by the expression Au(x) = −

d du(x) d0 (x) + B0 (x)u(x) (u ∈ Dom (A), 0 < x < π) (6.1) dx dx

on the domain Dom (A) = {u ∈ H, u ∈ H, u(0) = u(π) = 0 },

(6.2)

with continuous real n × n-matrices d0 (x) = diag [a1 (x), ..., an (x)], B0 (x) = (bjk (x))nj,k=1 , where functions aj (x) are diﬀerentiable and positive: d˜j ≡ min aj (x) > 0. x

(6.3)

210

15. Operator Matrices

Take E = L2 ([0, π], C1 ) and deﬁne operators Ajk by Ajj v(x) = −

d dv(x) + bjj (x)v(x) aj (x) dx dx

(v ∈ Dom (Ajj ), 0 < x < π), where Dom (Ajj ) = {v ∈ E : v ∈ E, v(0) = v(π) = 0 } and Ajk v(x) = bjk (x)v(x) (v ∈ E; 0 < x < π, j = k). Assume that βj ≡ inf bjj (x) + d˜j > 0 (j = 1, ..., n). x

(6.4)

Omitting simple calculations, we have (Ajj v(x), v)E = (aj v , v )E + (bjj v, v)E ≥ d˜j (v , v )E + (bjj v, v)E ≥ d˜j (v, v)E + (bjj v, v)E = βj (v, v)E . Consequently, −1 A−1 (j = 1, ..., n). jj E ≤ βj

Clearly, Ajk E ≤ max |bjk (x)|. x

So −1 A−1 jj Ajk E ≤ βj max |bjk (x)|. x

With the notation n−1

v˜A ≡ (

n

j=1 k=j+1

and

βj−2 max |bjk (x)|2 )1/2 . x

j−1 n βj−2 max |bjk (x)|2 )1/2 w ˜A ≡ ( j=2 k=1

x

we have D−1 V ≤ v˜A , and D−1 W ≤ w ˜A . Now Corollary 15.1.3 yields

15.6. Example

211

Proposition 15.6.1 Let the conditions (6.3), (6.4) and n−1 j,k=1

k j v˜A w ˜A < 1

(6.5)

hold. Then operator A deﬁned by (6.1), (6.2) is invertible. In particular, let v˜A , w ˜A = 1. Then (6.5) can be written in the form v˜A w ˜A

n−1 n−1 (1 − v˜A )(1 − w ˜A ) < 1. (1 − v˜A )(1 − w ˜A )

In the case n = 2 one can write v˜A = β1−1 max |b12 (x)|, w ˜A = β2−1 max |b21 (x)|. x

x

Inequality (6.5) takes the form max |b12 (x)| max |b21 (x)| < β1 β2 . x

x

To investigate the spectrum of operator (6.1) assume for simplicity that aj ≡ const > 0, bjj ≡ const (j = 1, 2, ...).

(6.6)

Then it is simple to check that the eigenvalues of Ajj are λk (Ajj ) = aj k 2 + bjj (k = 1, 2, ...). Denote

n−1

vb ≡ (

n

max |bjk (x)|2 )1/2 x

j=1 k=j+1

and

j−1 n wb ≡ ( max |bjk (x)|2 )1/2 . j=1 k=2

x

Clearly, V ≤ vb , W ≤ wb . Now Lemma 15.4.2 yields Proposition 15.6.2 Let z2 be the unique non-negative root of the algebraic equation n−2 z k+j vbn−j−1 wbn−k−1 = z 2(n−1) . (6.7) j,k=0

Then the spectral variation of A with respect to D satisﬁes the inequality svD (A) ≤ z2 .

212

15. Operator Matrices

In other words for any µ ∈ σ(A), there are natural k = 1, 2, ... and j ≤ n, such that |µ − aj k 2 − bjj | ≤ z2 . In particular, if n = 2, then √ z2 = vb wb = [max |b12 (x)| max |b21 (x)|]1/2 . x

x

Certainly, instead of the ordinary diﬀerential operator, in (6.1) we can consider an elliptic one.

15.7

Notes

The spectrum of operator matrices and related problems were investigated in many works cf. (Kovarik, 1975, 1977 and 1980), (Kovarik and Sherif, 1985), (Gaur and Kovarik, 1991), (Stampli, 1964), (Davis, 1958) and references given therein. In particular, in the paper (Kovarik, 1975), the Gershgorin type bounds for spectra of operator matrices with bounded operator entries are derived. They generalize the well-known results for block-matrices (Varga, 1965), (Levinger and Varga, 1966). But the Gershgorin-type bounds give good results in the cases when the diagonal operators are dominant. Theorem 15.1.1 improves the Gershgorin type bounds for operator matrices, which are close to triangular ones. Moreover, we also consider unbounded operators. Proposition 15.6.2 on the bounds for the spectrum of a matrix diﬀerential operator supplements the well known results on diﬀerential operators, cf. (Edmunds and Evans, 1990), (Egorov and Kondratiev, 1996) and references therein. The material in this chapter is taken from the paper (Gil’, 2001).

References [1] Davis, C. C. (1958). Separation of two subspaces. Acta Sci. Math. (Szeged) 19, 172-187. [2] Edmunds, D.E. and Evans W.D. (1990). Spectral Theory and Diﬀerential Operators. Clarendon Press, Oxford. [3] Egorov, Y and Kondratiev, V. (1996). Spectral Theory of Elliptic Operators. Birkh¨ auser Verlag, Basel. [4] Gaur A.K. and Kovarik, Z. V. (1991). Norms, states and numerical ranges on direct sums, Analysis 11, 155-164. [5] Gil’, M. I. (2001). Invertibility conditions and bounds for spectra of operator matrices. Acta Sci. Math, 67/1, 353-368

15.7. Notes

213

[6] Kato, T. (1966). Perturbation Theory for Linear Operators, SpringerVerlag. New York. [7] Kovarik, Z. V. (1975). Spectrum localization in Banach spaces II, Linear Algebra and Appl. 12, 223-229 . [8] Kovarik, Z. V. (1977). Similarity and interpolation between projectors, Acta Sci. Math. (Szeged), 39, 341-351 [9] Kovarik, Z. V. (1980). Manifolds of frames of projectors, Linear Algebra and Appl. 31, 151-158. [10] Kovarik, Z. V. and Sherif, N. (1985). Perturbation of invariant subspaces, Linear Algebra and Appl. 64, 93-113. [11] Levinger, B.W. and Varga, R.S. (1966). Minimal Gershgorin sets II, Paciﬁc. J. Math., 17, 199-210. [12] Stampli, J. (1964), Sums of projectors, Duke Math. J., 31, 455-461. [13] Varga, R.S. (1965). Minimal Gershgorin sets, Paciﬁc. J. Math., 15, 719729.

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16. Hille - Tamarkin Integral Operators

In the present chapter, the Hille-Tamarkin integral operators on space Lp [0, 1] are considered. Invertibility conditions, estimates for the norm of the inverse operators and positive invertibility conditions are established. In addition, bounds for the spectral radius are suggested. Applications to nonselfadjoint diﬀerential operators and integro-diﬀerential ones are also discussed.

16.1

Invertibility Conditions

Recall that Lp ≡ Lp [0, 1] (1 < p < ∞) is the space of scalar-valued functions deﬁned on [0, 1] and equipped with the norm 1 |h(s)|p ds]1/p . |h|Lp = [ 0

˜ is a linear operator in Lp deﬁned by Everywhere below K 1 ˜ K(x, s)h(s)ds (h ∈ Lp , x ∈ [0, 1]), (Kh)(x) =

(1.1)

0

where K(x, s) is a scalar kernel deﬁned on [0, 1]2 and having the property 1 1 |K(x, s)|q ds]p/q dx]1/p < ∞ (p−1 + q −1 = 1). (1.2) Mp (K) ≡ [ [ 0

0

˜ is a Hille-Tamarkin operator (Pietsch, 1987, p. 245). Deﬁne the That is, K Volterra operators x (V− h)(x) = K(x, s)h(s)ds (1.3) 0

M.I. Gil’: LNM 1830, pp. 215–226, 2003. c Springer-Verlag Berlin Heidelberg 2003

216

16. Hille - Tamarkin Integral Operators

and

(V+ h)(x) =

Set

1

K(x, s)h(s)ds.

(1.4)

x

1 t Mp (V− ) ≡ [ ( |K(t, s)|q ds)p/q dt]1/p , 0

0

1 Mp (V+ ) ≡ [ ( 0

and Jp±

1

t

|K(t, s)|q ds)p/q dt]1/p

∞ Mpk (V± ) √ . ≡ p k! k=0

Now we are in a position to formulate the main result of the chapter. Theorem 16.1.1 Let the conditions (1.2) and Jp+ Jp− < Jp+ + Jp−

(1.5)

˜ is boundedly invertible in Lp and the inverse hold. Then operator I − K operator satisﬁes the inequality ˜ −1 |Lp ≤ |(I − K)

Jp+

Jp− Jp+

+ Jp− − Jp+ Jp−

.

(1.6)

The proof of this theorem is presented in the next two sections. Note that condition (1.5) is equivalent to the following one: θ(K) ≡ (Jp+ − 1)(Jp− − 1) < 1.

(1.7)

Besides (1.6) takes the form ˜ −1 |Lp ≤ |(I − K)

Jp− Jp+ . 1 − θ(K)

(1.8)

Due to H¨ older’s inequality, for arbitrary a > 1 ∞ Mpk (V± ) √ ≤ p k! k=0

[

∞

k=0

a−qk ]1/q [

∞ apk Mpkp (V± )

k=0

Take a = 21/p . Then where

k!

p

]1/p = (1 − a−q )−1/q ea p

Jp± ≤ mp e2Mp (V± )/p mp = (1 − 2−q/p )−1/q .

Mpp (V± )/p

.

(1.9)

16.2. Preliminaries

217

Since, Mpp (K) = Mpp (V− ) + Mpp (V+ ), we have

(1.10)

p

Jp− Jp+ ≤ m2p e2Mp (K)/p . Now relation (1.9) and Theorem 16.1.1 imply Corollary 16.1.2 Let the conditions (1.2) and p

p

p

mp e2Mp (K)/p < e2Mp (V− )/p + e2Mp (V+ )/p ˜ is boundedly invertible in Lp and the inverse hold. Then operator I − K operator satisﬁes the inequality p

˜ −1 |Lp ≤ |(I − K)

16.2

mp e2Mp (K)/p . p p p e2Mp (V− )/p + e2Mp (V+ )/p − mp e2Mp (K)/p

Preliminaries

Let X be a Banach space with a norm .. Recall that a linear operator V˜ in X is called a quasinilpotent one if n V˜ n = 0. lim n→∞

For a quasinilpotent operator V˜ in X, put j(V˜ ) ≡

∞

V˜ k .

k=0

Lemma 16.2.1 Let A be a bounded linear operator in X of the form A = I + V + W,

(2.1)

where operators V and W are quasinilpotent. If, in addition, the condition θA ≡

∞

(−1)k+j V k W j < 1

(2.2)

j,k=1

is fulﬁlled, then operator A is boundedly invertible and the inverse operator satisﬁes the inequality j(V )j(W ) . A−1 ≤ 1 − θA

218

16. Hille - Tamarkin Integral Operators

Proof:

We have A = I + V + W = (I + V )(I + W ) − V W.

(2.3)

Since W and V are quasinilpotent, the operators, I + V and I + W are invertible: (I + V )−1 =

∞

(−1)k V k , (I + W )−1 =

k=0

∞

(−1)k W k .

(2.4)

k=0

Thus, A = I + V + W = (I + V )[I − (I + V )−1 V W (I + W )−1 ](I + W ) =

where

(I + V )(I − BA )(I + W )

(2.5)

BA = (I + V )−1 V W (I + W )−1 .

(2.6)

But according to (2.4) −1

V (I + V )

=

∞

k−1

(−1)

k

−1

V , (I + W )

k=1

=

∞

(−1)k−1 W k .

(2.7)

k=1

So BA =

∞

(−1)k+j V k W j .

(2.8)

j,k=1

If (2.2) holds, then BA < 1 and (I − BA )−1 ≤ (1 − θA )−1 . So due to (2.5) I + V + W is invertible. Moreover, A−1 = (I + W )−1 (I − BA )−1 (I + V )−1 .

(2.9)

But (2.4) implies (I + W )−1 ≤ j(W ), (I + V )−1 ≤ j(V ). Now the required inequality for A−1 follows from (2.9). 2 Furthermore, take into account that by (2.7) V (I + V )−1 ≤

∞

V k ≤ j(V ) − 1.

(2.10)

k=1

Similarly,

W (I + W )−1 ≤ j(W ) − 1.

(2.11)

16.3. Powers of Volterra Operators

219

Thus θA ≤ (j(W ) − 1)(j(V ) − 1). So condition (2.2) is provided by the inequality (j(W ) − 1)(j(V ) − 1) < 1. The latter inequality is equivalent to the following one: j(W )j(V ) < j(W ) + j(V )

(2.12)

Lemma 16.2.1 yields Corollary 16.2.2 Let V, W be quasinilpotent and condition (2.12) be fulﬁlled. Then operator A deﬁned by (2.1) is boundedly invertible and the inverse operator satisﬁes the inequality A−1 ≤

j(V )j(W ) . j(W ) + j(V ) − j(W )j(V )

˜ Under condition (1.2), operators Let us turn now to integral operator K. V± are quasinilpotent due to the well-known Theorem V.6.2 (Zabreiko, et al., 1968, p. 153). Now Corollary 16.2.2 yields. Corollary 16.2.3 Let the conditions (1.2) and j(V+ )j(V− ) < j(V+ ) + j(V− ) ˜ is boundedly invertible and the inverse operator be fulﬁlled. Then I − K satisﬁes the inequality ˜ −1 |Lp ≤ |(I − K)

16.3

j(V− )j(V+ ) . j(V− ) + j(V+ ) − j(V− )j(V+ )

Powers of Volterra Operators

Lemma 16.3.1 Under condition (1.2), operator V− deﬁned by (1.3) satisﬁes the inequality Mpk (V− ) |V−k |Lp ≤ √ (k = 1, 2, ...). (3.1) p k! Proof:

Employing H¨ older’s inequality, we have 1 t |V− h|pLp = | K(t, s)h(s)ds|p dt ≤ 0

0

1

0

t t [ |K(t, s)|q ds]p/q |h(s1 )|p ds1 dt. 0

0

220

16. Hille - Tamarkin Integral Operators

Setting

t |K(t, s)|q ds]p/q , w(t) = [

(3.2)

0

one can rewrite the latter relation in the form 1 s1 p |h(s2 )|p ds2 ds1 . |V− h|Lp ≤ w(s1 ) 0

0

Using this inequality, we obtain 1 |V−k h|pLp ≤ w(s1 )

s1

0

0

|V−k−1 h(s2 )|p ds2 ds1 .

Once more apply H¨ older’s inequality : 1 s1 w(s1 ) w(s2 ) |V−k h|pLp ≤ 0

0

s2

0

|V−k−2 h(s3 )|p ds3 ds2 ds1 .

Repeating these arguments, we arrive at the relation 1 s1 sk k p w(s1 ) w(s2 ) . . . |h(sk+1 )|p dsk+1 . . . ds2 ds1 . |V− h|Lp ≤ 0

0

0

Taking

|h|pLp we get |V−k |pLp ≤

0

1

0

w(s1 )

µ ˜

w(s1 ) . . .

z1

w(s2 ) . . .

0

sk−1

0

dsk . . . ds2 ds1 .

sk−1

w(sk )dsk . . . ds1 =

0

zk−1

... 0

|h(s)|p ds = 1,

s1

0

It is simple to see that 1 0

1

=

dzk dzk−1 . . . dz1 =

0

where zk = zk (sk ) ≡ and

sk

w(s)ds 0

1

µ ˜=

w(s)ds. 0

Thus (3.3) gives |V−k |pLp

1

≤

(

0

w(s)ds)k . k!

µ ˜k , k!

(3.3)

16.4. Spectral Radius

221

But according to (3.2)

1

µ ˜= 0

w(s)ds = Mpp (V− ).

Therefore, |V−k |pLp ≤

M pk (V− ) . k!

As claimed. 2 Similarly, the inequality |V+k |Lp ≤

Mpk (V+ ) √ p k!

(3.4)

can be proved. The assertion of Theorem 16.1.1 follows from Corollary 16.2.3 and relations (3.1), (3.4).

16.4

Spectral Radius of a Hille - Tamarkin Operator

Set Jp± (z) ≡

∞ z k Mpk (V± ) √ (z ≥ 0). p k! k=0

So Jp± = Jp± (1). Clearly, ˜ (λ = 0). ˜ = λ(I − λ−1 K) λI − K Consequently, if Jp+ (|λ|−1 )Jp− (|λ|−1 ) < Jp+ (|λ|−1 ) + Jp− (|λ|−1 ), ˜ is boundedly invertible. We thus get then due to Theorem 16.1.1, λI − K Lemma 16.4.1 Under condition (1.2), any point λ = 0 of the spectrum ˜ of operator K ˜ satisﬁes the inequality σ(K) Jp+ (|λ|−1 )Jp− (|λ|−1 ) ≥ Jp+ (|λ|−1 ) + Jp− (|λ|−1 ).

(4.1)

˜ = sup |σ(K)| ˜ be the spectral radius of K. ˜ Then (4.1) yields Let rs (K) ˜ p− (rs−1 (K)) ˜ ≥ Jp+ (rs−1 (K)) ˜ + Jp− (rs−1 (K)). ˜ Jp+ (rs−1 (K))J

(4.2)

Note that according to (1.9) and (4.2) we have mp exp [

2(Mpp (V− ) + Mpp (V+ )) 2Mpp (V− ) 2Mpp (V+ ) ] ≥ exp [ ] + exp [ ]. ˜ ˜ ˜ rsp (K)p rsp (K)p rsp (K)p

222

16. Hille - Tamarkin Integral Operators

Theorem 16.4.2 Under condition (1.2), let V− = 0 and V+ = 0. Then the equation (4.3) Jp+ (z)Jp− (z) = Jp+ (z) + Jp− (z) ˜ ≤ z −1 (K) has a unique positive zero z(K). Moreover, the inequality rs (K) is valid. Proof:

Equation (4.3) is equivalent to the following one: (Jp+ (z) − 1)(Jp− (z) − 1) = 1.

(4.4)

Clearly, this equation has a unique positive root. In addition, (4.2) is equivalent to the relation ˜ − 1)(Jp− (rs−1 (K)) ˜ − 1) ≥ 1. (Jp+ (rs−1 (K)) Hence the result follows, since the left part of equation (4.4) monotonically increases. 2 Rewrite (4.4) as ∞ ∞ z k Mpk (V− ) z j Mpj (V+ ) √ √ =1 p p j! k! j=1 k=1 Or

∞

bk z k = 1

k=2

with bk =

k−1 j=1

Mpk−j (V− )Mpj (V+ ) (k = 2, 3, ...). p j!(k − j)!

Due to Lemma 8.3.1, with the notation δ(K) = 2 max

j=2,3,...

j bj ,

we get z(K) ≥ δ −1 (K). Now Theorem 16.4.2 yields ˜ ≤ δ(K) is Corollary 16.4.3 Under condition (1.2), the inequality rs (K) true. Theorem 16.4.2 and Corollary 16.4.3 are exact: if either V− → 0, or V+ → 0, then z(K) → ∞, δ(K) → 0.

16.5

Nonnegative Invertibility

We will say that h ∈ Lp is nonnegative if h(t) is nonnegative for almost all t ∈ [0, 1]; a linear operator A in Lp is nonnegative if Ah is nonnegative for each nonnegative h ∈ Lp .

16.6. Applications

223

Theorem 16.5.1 Let the conditions (1.2), (1.5) and K(t, s) ≥ 0 (0 ≤ t, s ≤ 1)

(5.1)

˜ is boundedly invertible and the inverse operator hold. Then operator I − K is nonnegative. Moreover, ˜ −1 ≥ I. (I − K) Proof:

(5.2)

˜ W = V− and V = V+ implies Relation (2.9) with A = I − K, ˜ −1 = (I − V+ )−1 (I − BK )−1 (I − V− )−1 (I − K)

where

(5.3)

BK = (I − V+ )−1 V+ V− (I − V− )−1 .

Moreover, by (5.1) we have V± ≥ 0. So due to (2.4), (I − V± )−1 ≥ 0 and BK ≥ 0. Relations (2.7) and (2.8) according to (2.10) and (2.11) imply |BK |Lp ≤ (Jp (V− ) − 1)(Jp (V+ ) − 1), since j(V± ) ≤ Jp (V± ). But (1.5) is equivalent to (1.7). We thus get |BK |Lp < 1. Consequently, ∞ k (I − BK )−1 = BK ≥ 0. k=0

˜ −1 ≥ 0. Since I − K ˜ ≤ I, we have Now (5.3) implies the inequality (I − K) inequality (5.2). 2

16.6

Applications

16.6.1

A nonselfadjoint diﬀerential operator

Consider a diﬀerential operator A deﬁned by (Ah)(x) = −

dh(x) d2 h(x) + w(x)h(x) (0 < x < 1, h ∈ Dom (A)) + g(x) dx2 dx (6.1)

on the domain Dom (A) = {h ∈ Lp : h ∈ Lp + boundary conditions }.

(6.2)

In addition, the coeﬃcients g, w ∈ Lp and are complex, in general.

(6.3)

224

16. Hille - Tamarkin Integral Operators

Let an operator S be deﬁned on Dom (A) by (Sh)(x) = −h (x), h ∈ Dom (A). It is ssumed that S has the Green function G(t, s). So that, 1 (S −1 h)(x) ≡ G(x, s)h(s)ds ∈ Dom (A) 0

for any h ∈ Lp . Besides, the derivative of the Green function in x satisﬁes the inequality 1 |Gx (x, s)|q ds < ∞. (6.4) vrai supx 0

˜ Thus, A = (I − K)S, where d ˜ + w(x)) (Kh)(x) = −(g(x) dx

1

1

G(x, s)h(s)ds = 0

K(x, s)h(s)ds 0

with K(x, s) = −g(x)Gx (x, s) − w(x)G(x, s). We have

1

[

0

1

0

0

|g(x)| [

1

p

0

|g(x)Gx (x, s)|q ds]p/q dx = |Gx (x, s)|q ds]p/q dx < ∞.

0

Similarly,

1

1

0

1

[ 0

(6.5)

1

|w(x)G(x, s)|q ds]p/q dx =

|w(x)|p [

1

0

|G(x, s)|q ds]p/q dx < ∞.

Thus, condition (1.2) holds. Take into account that by H¨ older’s inequality 1 1 | G(x, s)h(s)ds|p dx]1/p ≤ bp (S)|h|Lp |S −1 h|Lp = [ 0

where

0

1 bp (S) = [ ( 0

Since

0

1

|G(x, s)|q ds)p/q dx]1/p .

˜ −1 , A−1 = S −1 (I − K)

Theorem 16.1.1 immediately implies the following result: Proposition 16.6.1 Under (6.3)-(6.5), let condition (1.5) hold. Then operator A deﬁned by (6.1), (6.2) is boundedly invertible in Lp . In addition, |A−1 |Lp ≤

bp (S)Jp− Jp+

Jp+ + Jp− − Jp+ Jp−

.

16.7. Notes

16.6.2

225

An integro-diﬀerential operator

On domain (6.2), let us consider the operator d2 u(x) + (Eu)(x) = − dx2

1

0

K0 (x, s)u(s)ds (u ∈ Dom (A), 0 < x < 1), (6.6)

where K0 is a kernel with the property

1

0

0

1

|K0 (x, s)|p ds dx < ∞.

(6.7)

Let S be the same as in the previous subsection. Then we can write E = ˜ ˜ is deﬁned by (1.1) with (I − K)S where K K(x, s) = −

0

1

K0 (x, x1 )G(x1 , s)dx1

(6.8)

˜ is invertible, then E is invertible as well. By H¨ So if I − K older’s inequality

1

[

0

0

1

0

0

1

1

|

0

1

K0 (x, x1 )G(x1 , s)dx1 |q ds]p/q dx ≤

|K0 (x, x1 )| dx1 dx [ p

1

0

|G(x1 , s)|q dx1 ds]p/q .

That is, condition (1.2) holds. Since ˜ −1 , E −1 = S −1 (I − K) Theorems 16.1.1 and 16.5.1 yield Proposition 16.6.2 Under (6.4), (6.7) and (6.8), let condition (1.5) hold. Then operator E deﬁned by (6.6), (6.2) is boundedly invertible in Lp and |E −1 |Lp ≤

bp (S)Jp− Jp+

Jp+ + Jp− − Jp+ Jp−

.

If, in addition, G ≥ 0 and K0 ≤ 0, then E −1 is positive. Moreover, (E −1 h)(x) ≥ (S −1 h)(x) = for any nonnegative h ∈ Lp .

1

G(x, s)h(s)ds 0

226

16. Hille - Tamarkin Integral Operators

16.7

Notes

A lot of papers and books are devoted to the spectrum of Hille-Tamarkin integral operators. Mainly, the distributions of the eigenvalues are considered, cf. (Diestel et al., 1995), (K¨ onig, 1986), (Pietsch, 1987) and references therein. Theorem 16.4.2 and Corollary 16.4.3 improve the well-known estimate 1 ˜ ≤ sup |K(x, s)|ds rs (K) x

0

(Krasnosel’skii et al, 1989, Theorem 16.2) for operators, which are close to Volterra ones. The results of Section 16.6 supplement the well-known results on the spectra of diﬀerential operators, cf. (Edmunds and Evans, 1990), (Egorov and Kondratiev, 1996), (Locker, 1999) and references therein. The material in this chapter is taken from the paper (Gil’, 2002).

References [1] Diestel, D., Jarchow, H, Tonge, A. (1995), Absolutely Summing Operators, Cambridge University Press, Cambridge. [2] Edmunds, D.E. and Evans W.D. (1990). Spectral Theory and Diﬀerential Operators. Clarendon Press, Oxford. [3] Egorov, Y. and Kondratiev, V. (1996). Spectral Theory of Elliptic Operators. Birkh¨ auser Verlag, Basel. [4] Gil’, M.I. (2002). Invertibility and positive invertibility of Hille-Tamarkin integral operators, Acta Math. Hungarica, 95 (1-2) 39-53. [5] K¨ onig, H. (1986). Eigenvalue Distribution of Compact Operators, Birkh¨ auser Verlag, Basel- Boston-Stuttgart. [6] Locker, J. (1999). Spectral Theory of Non-Self-Adjoint Two Point Diﬀerential Operators., Amer. Math. Soc., Mathematical Surveys and Monographs, Volume 73. [7] Krasnosel’skii, M. A., J. Lifshits, and A. Sobolev (1989). Positive Linear Systems. The Method of Positive Operators, Heldermann Verlag, Berlin. [8] Pietsch, A. (1987). Eigenvalues and s-Numbers, Cambridge University Press, Cambridge. [9] Zabreiko, P.P., Koshelev A.I., Krasnosel’skii, M. A., Mikhlin, S.G., Rakovshik, L.S. and B.Ya. Stetzenko (1968). Integral Equations, Nauka, Moscow. In Russian

17. Integral Operators in Space L∞

In the present chapter integral operators in space L∞ [0, 1] are considered. Invertibility conditions, estimates for the norm of the inverse operators and positive invertibility conditions are established. In addition, bounds for the spectral radius are suggested. Applications to nonselfadjoint diﬀerential operators and integro-diﬀerential ones are also discussed.

17.1

Invertibility Conditions

Recall that L∞ ≡ L∞ [0, 1] is the space of scalar-valued functions deﬁned on [0, 1] and equipped with the norm |h|L∞ = ess sup |h(x)| (h ∈ L∞ ). x∈[0,1]

˜ is a linear operator in L∞ deﬁned by Everywhere in this chapter K 1 ˜ (Kh)(x) = K(x, s)h(s)ds (h ∈ L∞ , x ∈ [0, 1]),

(1.1)

0

where K(x, s) is a scalar kernel deﬁned on [0, 1]2 and having the property 1 ess sup |K(x, s)|ds < ∞. (1.2) 0

x∈[0,1]

Deﬁne the Volterra operators (V− h)(x) = M.I. Gil’: LNM 1830, pp. 227–234, 2003. c Springer-Verlag Berlin Heidelberg 2003

x

K(x, s)h(s)ds 0

(1.3)

17. Integral Operators in L∞

228

and

(V+ h)(x) =

1

K(x, s)h(s)ds.

(1.4)

x

Set w− (s) ≡ ess

sup

|K(x, s)|,

sup

|K(x, s)|

0≤s≤x≤1

w+ (s) ≡ ess

0≤x≤s≤1

and

M∞ (V± ) ≡

0

1

w± (s)ds.

Now we are in a position to formulate the main result of the chapter. Theorem 17.1.1 Let the conditions (1.2) and eM∞ (V− )+M∞ (V+ ) < eM∞ (V+ ) + eM∞ (V− )

(1.5)

˜ is boundedly invertible in L∞ and the inverse hold. Then operator I − K operator satisﬁes the inequality ˜ −1 |L∞ ≤ |(I − K)

eM∞ (V+ )

eM∞ (V− )+M∞ (V+ ) . + eM∞ (V− ) − eM∞ (V− )+M∞ (V+ )

(1.6)

The proof of this theorem is presented in the next section. Note that condition (1.5) is equivalent to the following one: θ(K) ≡ (eM∞ (V+ ) − 1)(eM∞ (V− ) − 1) < 1.

(1.7)

Besides (1.6) takes the form M∞ (V− )+M∞ (V+ ) ˜ −1 |L∞ ≤ e . |(I − K) 1 − θ(K)

17.2

(1.8)

Proof of Theorem 17.1.1

Under condition (1.2), operators V± are quasinilpotent due to the well-known Theorem V.6.2 (Zabreiko, et al., 1968, p. 153). Now Corollary 16.2.2 yields. Lemma 17.2.1 With the notation j(V± ) ≡

∞

|V±k |L∞ ,

k=0

let the conditions (1.2) and j(V+ )j(V− ) < j(V+ ) + j(V− )

17.2. Proof of Theorem 17.1.1

229

˜ is boundedly invertible in L∞ and the inverse operator be fulﬁlled. Then I − K satisﬁes the inequality j(V− )j(V+ ) . j(V− ) + j(V+ ) − j(V− )j(V+ )

˜ −1 |L∞ ≤ |(I − K)

Lemma 17.2.2 Under condition (1.2), operator V− deﬁned by (1.3) satisﬁes the inequality M k (V− ) (k = 1, 2, ...). (2.1) |V−k |L∞ ≤ ∞ k! Proof:

We have

|V− h|L∞ = ess supx∈[0,1] |

x

0

K(x, s)h(s)ds| ≤

0

1

w− (s)|h(s)|ds.

Repeating these arguments, we arrive at the relation |V−k h|L∞ ≤

1

w− (s1 )

0

s1

w− (s2 ) . . .

0

sk

0

|h(sk )|dsk . . . ds2 ds1 .

Taking |h|L∞ = 1, we get |V−k |L∞

≤

1

w− (s1 )

0

s1

w− (s2 ) . . .

0

0

sk−1

dsk . . . ds2 ds1 .

It is simple to see that

w− (s1 ) . . .

0

1

µ ˜

z1

sk−1

w− (sk )dsk . . . ds1 =

0

zk−1

... 0

0

dzk dzk−1 . . . dz1 =

0

where zj = zk (sj ) ≡ and

µ ˜k , k!

sj

w− (s)ds (j = 1, ..., k)

0

µ ˜= 0

1

w− (s)ds.

Thus (2.2) gives |V−k |L∞ As claimed. 2

1 ( 0 w− (s)ds)k M k (V− ) = ∞ . ≤ k! k!

(2.2)

17. Integral Operators in L∞

230

Similarly, the inequality |V+k |L∞ ≤

k (V+ ) M∞ (k = 1, 2, ...) k!

(2.3)

can be proved. Relations (2.1) and (2.3) imply |(I − V± )−1 |L∞ ≤ j(V± ) ≤ eM∞ (V± ) .

(2.4)

The assertion of Theorem 17.1.1 follows from Lemma 17.2.1 and relations (2.4).

17.3

The Spectral Radius

Clearly,

˜ (λ = 0). ˜ = λ(I − λ−1 K) λI − K

Consequently, if e(M∞ (V− )+M∞ (V+ ))|λ|

−1

< e|λ|

−1

M∞ (V+ )

+ e|λ|

−1

M∞ (V− )

,

˜ is boundedly invertible. We thus get then due to Theorem 17.1.1, λI − K Lemma 17.3.1 Under condition (1.2), any point λ = 0 of the spectrum ˜ of operator K ˜ satisﬁes the inequality σ(K) e(M∞ (V− )+M∞ (V+ ))|λ|

−1

≥ e|λ|

−1

M∞ (V+ )

+ e|λ|

−1

M∞ (V− )

.

(3.1)

˜ be the spectral radius of K. ˜ Then (3.1) yields Let rs (K) −1

ers

˜ (K)(M ∞ (V− )+M∞ (V+ ))

−1

≥ ers

˜ (K)M ∞ (V+ )

−1

+ ers

˜ (K)M ∞ (V− )

.

(3.2)

˜ = 0. Clearly, if V+ = 0 or ( and ) V− = 0, then rs (K) Theorem 17.3.2 Under condition (1.2), let V+ = 0, V− = 0. Then the equation e(M∞ (V− )+M∞ (V+ ))z = ezM∞ (V+ ) + ezM∞ (V− ) (z ≥ 0)

(3.3)

˜ ≤ z −1 (K) has a unique positive zero z(K). Moreover, the inequality rs (K) is valid. Proof:

Equation (3.3) is equivalent to the following one: (eM∞ (V+ )z − 1)(ezM∞ (V− ) − 1) = 1.

In addition, (3.2) is equivalent to the relation −1

(ers

˜ (K)M ∞ (V+ )

−1

− 1)(ers

˜ (K)M ∞ (V− )

− 1) ≥ 1.

(3.4)

17.4. Nonnegative Invertibility

231

Hence, the result follows, since the left part of equation (3.4) monotonically increases. 2 From (3.3) it follows that e(M∞ (V− )+M∞ (V+ ))z ≥ 2 and

ez(M∞ (V+ )−M∞ (V− )) = eM∞ (V+ )z − 1 ≥ exp [ln 2 M∞ (V+ )(M∞ (V− ) + M∞ (V+ ))−1 ] − 1.

Thus with the notation δ∞ (K) = we have

M∞ (V+ ) − M∞ (V− ) + )ln 2 ln [ exp ( M∞M(V∞−(V )+M∞ (V+ ) ) − 1]

(3.5)

−1 (K), z(K) ≥ δ∞

(3.6)

M∞ (V+ ) < M∞ (V− ).

(3.7)

provided Clearly, in (3.5) we can exchang the places of V− and V+ . Now Theorem 17.3.2 yields ˜ ≤ Corollary 17.3.3 Under conditions (1.2) and (3.7), the inequality rs (K) δ∞ (K) is true.

17.4

Nonnegative Invertibility

We will say that h ∈ L∞ is nonnegative if h(t) is nonnegative for almost all t ∈ [0, 1]; a linear operator A in L∞ is nonnegative if Ah is nonnegative for each nonnegative h ∈ L∞ . Recall that I is the identity operator. Theorem 17.4.1 Let the conditions (1.2), (1.5) and K(t, s) ≥ 0 (0 ≤ t, s ≤ 1)

(4.1)

˜ is boundedly invertible and the inverse operator hold. Then operator I − K is nonnegative. Moreover, ˜ −1 ≥ I. (I − K)

(4.2)

˜ W = V− and Proof: Relation (2.9) from Section 16.2 with A = I − K, V = V+ implies ˜ −1 = (I − V+ )−1 (I − BK )−1 (I − V− )−1 (I − K)

(4.3)

17. Integral Operators in L∞

232

where

BK = (I − V+ )−1 V+ V− (I − V− )−1 .

Moreover, by (4.1) we have V± ≥ 0. So (I − V± )−1 ≥ 0 and BK ≥ 0. Relations (2.4) give us the inequalities |(I − V± )−1 V± |L∞ ≤ eM∞ (V± ) − 1. Consequently, |BK |L∞ ≤ (eM∞ (V+ ) − 1)(eM∞ (V− ) − 1). But (1.5) is equivalent to (1.7). We thus get |BK |L∞ < 1. Consequently, (I − BK )−1 =

∞

k BK ≥ 0.

k=0

˜ ≤ I, ˜ −1 ≥ 0. In addition, since I − K Now (4.3) implies the inequality (I − K) we have inequality (4.2). 2

17.5

Applications

17.5.1

A nonselfadjoint diﬀerential operator

Consider a diﬀerential operator A deﬁned by (Ah)(x) = −

d2 h(x) dh(x) + m(x)h(x) (0 < x < 1, h ∈ Dom (A) ) + g(x) 2 dx dx (5.1)

on the domain Dom (A) = {h ∈ L∞ , h ∈ L∞ + some boundary conditions }

(5.2)

In addition, the coeﬃcients g, w ∈ L∞ and are complex, in general. Let an operator S be deﬁned on Dom (A) by (Sh)(x) = −h (x), h ∈ Dom (A). It is assumed that S has the Green function G(t, s). So that, (S −1 h)(x) ≡

0

1

G(x, s)h(s)ds ∈ Dom (A)

(5.3)

17.5. Applications

for any h ∈ L∞ , and the derivative of G in x satisﬁes the condition 1 sup |Gx (x, s)|ds < ∞. 0

x

Put

b∞ (S) :=

1

0

233

(5.4)

sup |G(x, s)|ds. x

We have ˜ A = (I − K)S, where ˜ (Kh)(x) = −(g(x)

d + m(x)) dx

1

1

G(x, s)h(s)ds = 0

K(x, s)h(s)ds 0

with K(x, s) = −g(x)Gx (x, s) − m(x)G(x, s).

(5.5)

According to (5.3) and (5.4), condition (1.2) holds. Take into account that |S −1 h|L∞ ≤ b∞ (S)|h|L∞ . Since

˜ −1 , A−1 = S −1 (I − K)

Theorem 17.1.1 immediately implies the following result: Proposition 17.5.1 Under (5.3)-(5.5), let condition (1.5) hold. Then operator A deﬁned by (5.1), (5.2) is boundedly invertible in L∞ . In addition, |A−1 |L∞ ≤

17.5.2

b∞ (S)eM∞ (V− )+M∞ (V+ ) . eM∞ (V+ ) + eM∞ (V− ) − eM∞ (V− )+M∞ (V+ )

An integro-diﬀerential operator

On domain (5.2), let us consider the operator 1 d2 u(x) (Eu)(x) = − + K0 (x, s)u(s)ds (u ∈ Dom (A), 0 < x < 1), (5.6) dx2 0 where K0 is a kernel with the property 1 ess supx |K0 (x, s)| ds < ∞.

(5.7)

0

Let S and G be the same as in the previous subsection. Then we can write ˜ ˜ is deﬁned by (1.1) with E = (I − K)S where K 1 K(x, s) = − K0 (x, x1 )G(x1 , s)dx1 (5.8) 0

234

17. Integral Operators in L∞

˜ is invertible, then E is invertible as well. Clearly, under (5.4) and So if I − K (5.7), condition (1.2) holds. Since ˜ −1 , E −1 = S −1 (I − K) Theorems 17.1.1 and 17.4.1 yield Proposition 17.5.2 Under (5.4), (5.7) and (5.8), let condition (1.5) hold. Then operator E deﬁned by (5.6), (5.2) is boundedly invertible in L∞ and |E −1 |L∞ ≤

b∞ (S)eM∞ (V− )+M∞ (V+ ) . + eM∞ (V− ) − eM∞ (V− )+M∞ (V+ )

eM∞ (V+ )

If, in addition, G ≥ 0 and K0 ≤ 0, then E −1 is positive. Moreover, 1 (E −1 h)(x) ≥ (S −1 h)(x) = G(x, s)h(s)ds 0

∞

for any nonnegative h ∈ L .

17.6

Notes

The present chapter is based on the paper (Gil’, 2001). About well-known results on the spectrum of integral operators on L∞ , see, for instance, the books (Diestel et al., 1995), (K¨onig, 1986), (Krasnosel’skii et al., 1989), (Pietsch, 1987) and references therein.

References [1] Diestel, D., Jarchow, H, Tonge, A. (1995), Absolutely Summing Operators, Cambridge University Press, Cambridge. [2] Gil’, M.I. (2001). Invertibility and positive invertibility conditions of integral operators in L∞ , J. of Integral Equations and Appl. 13 , 1-14. [3] K¨ onig, H. (1986). Eigenvalue Distribution of Compact Operators, Birkh¨ auser Verlag, Basel- Boston-Stuttgart. [4] Krasnosel’skii, M. A., J. Lifshits, and A. Sobolev (1989). Positive Linear Systems. The Method of Positive Operators, Heldermann Verlag, Berlin. [5] Pietsch, A. (1987). Eigenvalues and s-Numbers, Cambridge University Press, Cambridge. [6] Zabreiko, P.P., A.I. Koshelev, M. A. Krasnosel’skii, S.G. Mikhlin, L.S. Rakovshik, B.Ya. Stetzenko (1968). Integral Equations, Nauka, Moscow. In Russian

18. Hille - Tamarkin Matrices

In the present chapter we investigate inﬁnite matrices, whose oﬀ diagonal parts are the Hille-Tamarkin matrices. Invertibility conditions and estimates for the norm of the inverse matrices are established. In addition, bounds for the spectrum are suggested. In particular, estimates for the spectral radius are derived.

18.1

Invertibility Conditions

Everywhere in this chapter A = (ajk )∞ j,k=1 is an inﬁnite matrix with the entries ajk (j, k = 1, 2, ...). Besides, V+ , V− and D denote the strictly upper triangular, strictly lower triangular, and diagonal parts of A, respectively: 

0 a12  0 0 V+ =   0 0 . .

a13 a23 0 .

a14 a24 a34 .





...   ...   , V− =    ...  ...

0 a21 a31 a41 .

and D = diag [a11 , a22 , a33 , ...]. M.I. Gil’: LNM 1830, pp. 235–241, 2003. c Springer-Verlag Berlin Heidelberg 2003

0 0 a32 a42 .

0 0 0 a43 .

0 0 0 0 ...

 ... ...   ...   ...  . (1.1)

236

18. Hille - Tamarkin Matrices

Throughout this chapter it is assumed that V− and V+ are the Hille-Tamarkin matrices. That is, for some ﬁnite p > 1, ∞

[

∞

|ajk |q ]p/q ]1/p < ∞

(1.2)

j=1 k=1, k=j

with

1 1 + = 1. p q

As usually lp (1 < p < ∞) is the Banach space of number sequences equipped with the norm ∞ |hk |p ]1/p (h = (hk ) ∈ lp ). |h|lp = [ k=1

So under (1.2), A represents a linear operator in lp which is also denoted by A. Clearly, Dom (A) = Dom (D) = {h = (hk ) ∈ lp :

∞

|akk hk |p < ∞}.

k=1

Assume that d0 ≡ inf |akk | > 0 k

(1.3)

and introduce the notations ∞ ∞ q p/q 1/p Mp+ (A) ≡ ( [ |a−1 ) , jj ajk | ] j=1 k=j+1

Mp− (A)

j−1 ∞ q p/q 1/p =( [ |a−1 ) , jj ajk | ] j=2 k=1

and Jp± (A) =

∞ (Mp± (A))k √ . p k! k=0

Now we are in a position to formulate the main result of the chapter. Theorem 18.1.1 Let the conditions (1.2), (1.3) and Jp− (A)Jp+ (A) < Jp+ (A) + Jp− (A)

(1.4)

hold. Then A is boundedly invertible in lp and the inverse operator satisﬁes the inequality |A−1 |lp ≤

(Jp+ (A)

Jp− (A)Jp+ (A)

+ Jp− (A) − Jp− (A)Jp+ (A))d0

The proof of this theorem is presented in the next section.

(1.5)

18.2. Proof of Theorem 18.1.1

18.2

237

Proof of Theorem 18.1.1

Lemma 18.2.1 Under condition (1.2), for the strictly upper and lower triangular matrices V+ and V− , the inequalities (vp± )m |V±m |lp ≤ √ (m = 1, 2, ...) p m! are valid, where

∞ ∞ vp+ = ( [ |ajk |q ]p/q )1/p j=1 k=j+1

and

j−1 ∞ vp− = ( [ |ajk |q ]p/q )1/p . j=2 k=1

This result follows from Lemma 3.2.1 when n → ∞, since 1 . γn,m,p ≤ √ p m! So operators V± are quasinilpotent. The latter lemma yields Corollary 18.2.2 Under conditions (1.2), (1.3), the inequalities |(D−1 V± )m |lp ≤

(Mp± (A))m √ (m = 1, 2, ...) p m!

are valid. Proof of Theorem 18.1.1: We have A = V+ + V− + D = D(D−1 V+ + D−1 V− + I) Clearly,

|D−1 |lp = d−1 0 .

From Lemma 18.2.2, it follows that j(D−1 V± ) ≤ Jp± (A) Now Corollary 16.2.2 and condition (1.4) yield the invertibility of the operator D−1 V+ + D−1 V− + I and the estimate (1.5). 2

238

18. Hille - Tamarkin Matrices

18.3

Localization of the Spectrum

Let σ(A) be the spectrum of A. For a λ ∈ C, assume that ρ(D, λ) ≡ inf |λ − amm | > 0, m

and put

∞ ∞ Mp+ (A, λ) = ( [ |(ajj − λ)−1 ajk |q ]p/q )1/p , j=1 k=j+1 j−1 ∞ Mp− (A, λ) = ( [ |(ajj − λ)−1 ajk |q ]p/q )1/p , j=2 k=1

and Jp± (A, λ) = Clearly

∞ (Mp± (A, λ))k √ . p k! k=0

Mp± (A, 0) = Mp± (A), Jp± (A, 0) = Jp± (A).

Lemma 18.3.1 Under condition (1.2), for any µ ∈ σ(A) we have either µ = ajj for some natural j, or Jp− (A, µ)Jp+ (A, µ) ≥ Jp+ (A, µ) + Jp− (A, µ). Proof:

(3.1)

Assume that Jp− (A, µ)Jp+ (A, µ) < Jp+ (A, µ) + Jp− (A, µ)

for some µ ∈ σ(A). Then due to Theorem 18.1.1, A − µI is invertible. This contradiction proves the required result. 2 Recall that vp± are deﬁned in Section 18.2 and denote, Fp± (z) =

∞ (vp± )k √ (z > 0) z k p k! k=0

(3.2)

Lemma 18.3.2 Under condition (1.2), for any µ ∈ σ(A), either there is an integer m, such that, µ = amm , or Fp− (ρ(D, µ))Fp+ (ρ(D, µ)) ≥ Fp− (ρ(D, µ)) + Fp+ (ρ(D, µ)). Proof:

(3.3)

Let µ = akk for all natural k. Then Mp± (A, µ) ≤ ρ−1 (D, µ)vp± .

Hence,

Jp± (A, µ) ≤ Fp± (ρ(D, µ)).

(3.4)

18.3. Localization of the Spectrum

239

In addition, (3.1) is equivalent to the relation (Jp− (A, µ) − 1)(Jp+ (A, µ) − 1) ≥ 1. Now (3.4) implies (3.3). 2 Theorem 18.3.3 Under condition (1.2), let V+ = 0, V− = 0. Then the equation (3.5) Fp− (z)Fp+ (z) = Fp− (z) + Fp+ (z) has a unique positive root ζ(A). Moreover, ρ(D, µ) ≤ ζ(A) for any µ ∈ σ(A). In other words, σ(A) lies in the closure of the union of the discs {λ ∈ C : |λ − akk | ≤ ζ(A)} (k = 1, 2, ...). Proof:

Equation (3.5) is equivalent to the following one: (Fp− (z) − 1)(Fp+ (z) − 1) = 1.

(3.6)

The left part of this equation monotonically decreases as z > 0 increases; so it has a unique positive root ζ(A). In addition, (3.3) is equivalent to the relation (3.7) (Fp− (ρ(D, µ)) − 1)(Fp+ (ρ(D, µ)) − 1) ≥ 1. Hence the result follows. 2 Rewrite (3.5) as

Or

∞ k=2

∞ ∞ (vp− )k (vp+ )j √ √ = 1. p z k k! j=1 z j p j! k=1

Bk z k = 1 with Bk =

k−1 j=1

(vp+ )k−j (vp− )j (k = 2, 3, ...). p j!(k − j)!

Due to the Lemma 8.3.1, with the notation δp (A) ≡ 2 sup

j=2,3,...

j

Bj ,

we get ζ(A) ≤ δp (A). Now Theorem 18.3.3 yields Corollary 18.3.4 Under condition (1.2), let V+ = 0, V− = 0. Then for any µ ∈ σ(A), the inequality ρ(µ, D) ≤ δ(A) is true. In other words, σ(A) lies in the closure of the union of the sets {λ ∈ C : |λ − akk | ≤ δp (A)} (k = 1, 2, ...).

240

18. Hille - Tamarkin Matrices

Note that Theorem 18.3.3 is exact: if A is triangular: either V− = 0, or V+ = 0, then we due to that lemma σ(A) is the closure of the set {akk , k = 1, 2, ...}. Moreover, Theorem 18.3.3 and Corollary 18.3.4 imply rs (A) ≤

sup |akk | + ζ(A) ≤

k=1,2,...

sup |akk | + δp (A),

k=1,2,...

(3.8)

provided D is bounded. Furthermore, let the condition ∞

sup

j=1,2,...

|ajk | < ∞

(3.9)

k=1

hold. Then the well-known estimate rs (A) ≤

sup

∞

j=1,2,...

|ajk |

(3.10)

k=1

is valid, see (Krasnosel’skii et al. 1989, Theorem 16.2). Under condition (3.9), relations (3.8) improve (3.10), provided ζ(A) <

sup

j=1,2,...

or δp (A) <

sup

j=1,2,...

∞

|ajk |

k=1,k=j

∞

|ajk |.

k=1,k=j

In conclusion, note that Theorem 18.1.1 is exact: if A is upper or lower triangular, then A is invertible, provided D is invertible.

18.4

Notes

The present chapter is based on the paper (Gil’, 2002). About other results on the spectrum of Hille-Tamarkin matrices see, for instance, the books (Diestel et al., 1995), (K¨ onig, 1986), (Pietsch, 1987), and references therein. Note that Hille-Tamarkin matrices arise, in particular, in recent investigations of discrete Volterra equations, see (Kolmanovskii et al, 2000), (Gil’ and Medina, 2002), (Medina and Gil’, 2003).

18.4. Notes

241

References [1] Diestel, D., Jarchow, H, Tonge, A. (1995), Absolutely Summing Operators, Cambridge University Press, Cambridge. [2] Gil’, M.I. (2002), Invertibility and spectrum of Hille-Tamarkin matrices, Mathematische Nachrichten, 244, 1-11 [3] Gil’, M.I. and Medina, R. (2002). Boundedness of solutions of matrix nonlinear Volterra diﬀerence equations. Discrete Dynamics in Nature and Society, 7, No 1, 19-22 [4] Kolmanovskii, V.B., A.D. Myshkis and J.P. Richard (2000). Estimate of solutions for some Volterra diﬀerence equations, Nonlinear Analysis, TMA, 40, 345-363. [5] K¨ onig, H. (1986). Eigenvalue Distribution of Compact Operators, Birkh¨ auser Verlag, Basel- Boston-Stuttgart. [6] Krasnosel’skii, M. A., J. Lifshits, and A. Sobolev (1989). Positive Linear Systems. The Method of Positive Operators, Heldermann Verlag, Berlin. [7] Medina, R. and Gil’, M.I. (2003). Multidimensional Volterra diﬀerence equations. In the book: New Progress in Diﬀerence Equations, Eds. S. Elaydi, G. Ladas and B. Aulbach, Taylor and Francis, London and New York, p. 499-504 [8] Pietsch, A. (1987). Eigenvalues and s-Numbers, Cambridge University Press, Cambridge.

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19. Zeros of Entire Functions

The present chapter is devoted to applications of our abstract results to the theory of ﬁnite order entire functions. We consider the following problem: if the Taylor coeﬃcients of two entire functions are close, how close are their zeros? In addition, we establish bounds for sums of the absolute values of the zeros in the terms of the coeﬃcients of its Taylor series. These bounds supplement the Hadamard theorem.

19.1

Perturbations of Zeros

Consider the entire function f (λ) =

∞

ck λk (λ ∈ C; c0 = 1)

k=0

with complex, in general, coeﬃcients ck , k = 1, 2, ... . Put Mf (r) := max |f (z)| (r > 0). |z|=r

Recall that ρ (f ) := limr→∞

ln ln Mf (r) ln r

is the order of f . Moreover, the relation ρ (f ) = limn→∞ is true, cf. (Levin, 1996, p. 6). M.I. Gil’: LNM 1830, pp. 243–252, 2003. c Springer-Verlag Berlin Heidelberg 2003

n ln n ln (1/|cn |)

244

19. Zeros of Entire Functions

Everywhere in the present chapter it is assumed that the set {zk (f )}∞ k=1 of all the zeros of f taken with their multiplicities is inﬁnite. Note that if f has a ﬁnite number m of the zeros, we can put zk−1 (f ) = 0 for k = m, m + 1, ... and aplly our arguments below. Here and below zk−1 (f ) means zk1(f ) . Rewrite function f in the form ∞ ak λk (a0 = 1) (k!)γ

f (λ) =

(1.1a)

k=0

with a positive γ, and consider the function ∞ bk λ k (b0 = 1). (k!)γ

h(λ) =

(1.1b)

k=0

Assume that

∞

|ak |2 < ∞,

k=0

∞

|bk |2 < ∞.

(1.2)

k=0

Relations (1.1) and (1.2), imply that functions f and h have orders no more than 1/γ. Deﬁnition 19.1.1 The quantity zvf (h) = max min |zk−1 (f ) − zj−1 (h)| j

k

will be called the variation of zeros of function h with respect to function f . For a natural p > 1/2γ, put wp (f ) := 2 [

∞

|ak |2 ]1/2 + 2 [ζ(2γp) − 1]1/2p ,

(1.3)

k=1

where ζ is the Riemann Zeta function, and ψ(f, y) :=

p−1 k w (f ) k=0

y k+1

Finally, denote q := [

1 wp2p (f ) exp [ + ] (y > 0). 2 2y 2p

∞

k=1

|ak − bk |2 ]1/2 .

(1.4)

19.1. Perturbations of Zeros

245

Theorem 19.1.2 Let conditions (1.1) and (1.2) be fulﬁlled. In addition, let r(q, f ) be the unique positive (simple) root of the equation qψ(f, y) = 1. Then zvf (h) ≤ r(q, f ). That is, for any zero z(h) of h there is a zero z(f ) of f , such that |z(h) − z(f )| ≤ r(q, f )|z(h)z(f )|. (1.5) The proof of Theorem 19.1.2 is presented in the next section. Substitute in (1.5) the equality y = xwp (f ) and apply Lemma 8.3.2. Then we have r(q, f ) ≤ δ(q, f ), where

δ(q, f ) :=

epq wp (f ) [ln (wp (f )/qp)]−1/2p

(1.6)

if wp (f ) ≤ epq, . if wp (f ) > epq

Theorem 19.1.2 and inequality (1.6) yield Corollary 19.1.3 Let conditions (1.1) and (1.2) be fulﬁlled. Then zvf (h) ≤ δ(q, f ). That is, for any zero z(h) of h, there is a zero z(f ) of f , such that |z(h) − z(f )| ≤ δ(q, f )|z(h)z(f )|.

(1.7)

Relations (1.5) and (1.7) imply the inequalities |z(f )| − |z(h)| ≤ r(q, f )|z(h)||z(f )| ≤ δ(q, f )|z(h)||z(f )|. Hence, |z(h)| ≥ (r(q, f )|z(f )| + 1)−1 |z(f )| ≥ (δ(q, f )|z(f )| + 1)−1 |z(f )|. This inequality yields the following result Corollary 19.1.4 Under conditions (1.1) and (1.2), for a positive number R0 , let f have no zeros in the disc {z ∈ C : |z| ≤ R0 }. Then h has no zeros in the disc {z ∈ C : |z| ≤ R1 } with R1 =

R0 R0 or R1 = . δ(q, f )R0 + 1 r(q, f )R0 + 1

Let us assume that under (1.1), there is a constant d0 ∈ (0, 1), such that limk→∞ k |ak | < 1/d0 (1.8) and limk→∞

k |bk | < 1/d0

246

19. Zeros of Entire Functions

and consider the functions f˜(λ) =

∞ ak (d0 λ)k k=0

and ˜ h(λ) =

∞ bk (d0 λ)k k=0

(1.9)

(k!)γ

(k!)γ

.

˜ ˜ ≡ h(d0 λ). So functions f˜(λ) and h(λ) That is, f˜(λ) ≡ f (d0 λ) and h(λ) satisfy conditions (1.2). Moreover, wp (f˜) = 2[

∞

2 1/2 d2k + 2[ζ(2γp) − 1]1/2p . 0 |ak | ]

k=1

˜ Thus, we can apply Theorem 19.1.2 and its corollaries to functions f˜(λ), h(λ) and take into account that ˜ = zk (h). d0 zk (f˜) = zk (f ), d0 zk (h)

19.2

(1.10)

Proof of Theorem 19.1.2

For a ﬁnite integer n, consider the polynomials F (λ) =

n ak λn−k k=0

(k!)γ

and Q(λ) =

n bk λn−k k=0

(k!)γ

(a0 = b0 = 1).

(2.1)

Put wp (F ) := 2 [

n

k=1

|ak |2 ]1/2 +2 [

n

1/k 2γp ]1/2p and q(F, Q) := [

k=2

n

|ak −bk |2 ]1/2 .

k=1

In addition, {zk (F )}nk=1 and {zk (Q)}nk=1 are the sets of all the zeros of F and Q, respectively taken with their multiplicities. Deﬁne ψ(F, y) according to (1.4). Lemma 19.2.1 For any zero z(Q) of Q, there is a zero z(F ) of F , such that |z(F ) − z(Q)| ≤ r(Q, F ), where r(Q, F ) be the unique positive (simple) root of the equation q(F, Q)ψ(F, y) = 1.

(2.2)

19.2. Proofs of Theorem 19.1.2

247

Proof: In a Euclidean space Cn with the Euclidean norm ., introduce operators An and Bn by virtue of the n × n-matrices   −a1 −a2 ... −an−1 −an  1/2γ 0 ... 0 0    γ  ... 0 0  1/3 An =  0   . . ... . .  0 0 0 ... 1/nγ 

and

  Bn =   

−b1 1/2γ 0 . 0

−b2 0 1/3γ . 0

... −bn−1 ... 0 ... 0 ... . ... 1/nγ

−bn 0 0 . 0

   .  

It is simple to see that F (λ) = det (λI − An ) and Q(λ) = det (λI − Bn ), where I is the unit matrix. So λk (An ) = zk (F ), λk (Bn ) = zk (Q) (k = 1, 2, ..., n),

(2.3)

where λk (.), k = 1, ..., n are the eigenvalues with their multiplicities. Clearly, An − Bn = q(F, Q). Due to Theorem 8.5.4, for any λj (Bn ) there is an λi (An ), such that |λj (Bn ) − λi (An )| ≤ yp (An , Bn ),

(2.4)

where yp (An , Bn ) is the unique positive (simple) root of the equation q(F, Q)

p−1 (2N2p (An ))k k=0

y k+1

exp [(1 +

(2N2p (An ))2p )/2] = 1, y 2p

where N2p (A) := [T race(AA∗ )p ]1/2p is the Neumann-Schatten norm and the asterisk means the adjointness. But An = M + C, where   −a1 −a2 ... −an−1 −an  0 0 ... 0 0   M =  . . ... . .  0 0 ... 0 0 and

   C=  

0 1/2γ 0 . 0

0 0 1/3γ . 0

... 0 ... 0 ... 0 ... . ... 1/nγ

0 0 0 . 0

   .  

248

19. Zeros of Entire Functions

Therefore, with c=

n

|ak |2 ,

k=1

we have



c  0 ∗ MM =   . 0 

and

  CC =    ∗

0 0 0 . 0

0 1/22γ 0 . 0

0 ... 0 0 ... 0 . ... . 0 ... 0 ... ... ... ... ...

 0 0   .  0 

0 0 0 0 0 0 . . 0 1/n2γ

  .  

Hence, N2p (An ) ≤ N2p (M ) + N2p (C) =

√

c+[

n

1/k 2γp ]1/2p .

k=2

Consequently yp (An , Bn ) ≤ r(Q, F ). Therefore (2.3) and (2.4) imply (2.2), as claimed. 2 Proof of Theorem 19.1.2: Consider the polynomials fn (λ) =

n n ak λk bk λ k and h (λ) = . n (k!)γ (k!)γ

k=0

(2.5)

k=0

Clearly, λn fn (1/λ) = F (λ) and hn (1/λ)λn = Q(λ). So zk (F ) = 1/zk (fn ); zk (Q) = 1/zk (hn ).

(2.6)

Take into account that the roots continuously depend on coeﬃcients, we have the required result, letting in the previous lemma n → ∞. 2

19.3

Bounds for Sums of Zeros

Again consider an entire function f of the form (1.1a) and assume that the condition ∞ θf := [ |ak |2 ]1/2 < ∞ (3.1) k=1

holds. Let the zeros of f be numerated in the increasing way: |zk (f )| ≤ |zk+1 (f )| (k = 1, 2, ...).

(3.2)

19.3. Bounds for Sums of Zeros

249

Theorem 19.3.1 Let f be an entire function of the form (1.1a). Then under conditions (3.1) and (3.2), the inequalities j

−1

|zk (f )|

≤ θf +

j

(k + 1)−γ (j = 1, 2, ...)

k=1

k=1

are valid. The proof of this theorem is presented in this section below. Note that under condition (1.8) we can omit condition (3.1) due to (1.9) and (1.10). To prove Theorem 19.3.1, again consider the polynomial F (λ) deﬁned in (2.1) with the zeros ordered in the following way: |zk (F )| ≥ |zk+1 (F )| (k = 1, ..., n − 1). Set θ(F ) := [

n

|ak |2 ]1/2 .

k=1

Lemma 19.3.2 The zeros of F satisfy the inequalities j

|zk (F )| ≤ θ(F ) +

k=1

and

j

(k + 1)−γ (j = 1, ..., n − 1)

k=1 n

|zk (F )| ≤ θ(F ) +

k=1

Proof:

n−1

(k + 1)−γ .

k=1

Take into account that according to (2.3), j

|λk (An )| ≤

j

sk (An ) (j = 1, ..., n),

(3.3)

k=1

k=1

where sk (An ), k = 1, 2, ... are the singular numbers of An ordered in the decreasing way (Marcus and Minc, 1964, Section II.4.2). But An = M + C, where M and C are introduced in Section 19.2. We can write s1 (M ) = θ(F ), sk (M ) = 0 (k = 2, ..., n). In addition, sk (C) = 1/(k + 1)γ (k = 1, ..., n − 1), sn (C) = 0. Take into account that j k=1

sk (An ) =

j k=1

sk (M + C) ≤

j k=1

sk (M ) +

j k=1

sk (C),

250

19. Zeros of Entire Functions

cf. (Gohberg and Krein, 1969, Lemma II.4.2). So j

sk (An ) ≤ θ(F ) +

k=1

and

j

(k + 1)−γ (j = 1, ..., n − 1)

k=1

n

sk (An ) ≤ θ(F ) +

k=1

n−1

(k + 1)−γ .

k=1

Now (2.3) and (3.3) yield the required result. 2 Proof of Theorem 19.3.1: Again consider the polynomial fn (z) deﬁned as in (2.5). Now Lemma 19.3.2 and (2.6) yield the inequalities j

|zk (fn )|−1 ≤ θf +

k=1

j

(k + 1)−γ (j = 1, ..., n − 1).

(3.4)

k=1

But the zeros of entire functions continuously depend on its coeﬃcients. So for any j = 1, 2, ..., j

−1

|zk (fn )|

→

k=1

j

|zk (f )|−1

k=1

as n → ∞. Now (3.4) implies the required result. 2

19.4

Applications of Theorem 19.3.1

Put τ1 = θf + 2−γ and τk = (k + 1)−γ (k = 2, 3, ...). Corollary 19.4.1 Let φ(t) (0 ≤ t < ∞) be a convex scalar-valued function, such that φ(0) = 0. Then under conditions (1.1a), (3.1) and (3.2), the inequalities j j φ(|zk (f )|−1 ) ≤ φ(τk ) (j = 1, 2, ...) k=1

k=1

are valid. In particular, for any r ≥ 2, j k=1

|zk (f )|−r ≤

j k=1

τkr = (θf + 2−γ )r +

j k=2

(k + 1)−rγ (j = 2, 3, ...). (4.1)

19.4. Applications of Theorem 19.3.1

251

Indeed, this result is due to the well-known Lemma II.3.4 (Gohberg and Krein, 1969) and Theorem 19.3.1. Furthermore, assume that rγ > 1, r ≥ 2.

(4.2)

Then the series ∞ ∞ τkr = (θf + 2−γ )r + (k + 1)−rγ = (θf + 2−γ )r + ζ(γr) − 1 − 2−rγ k=1

k=2

converges. Here ζ(.) is the Riemann Zeta function, again. Now relation (4.1) yields Corollary 19.4.2 Under the conditions (1.1a), (3.1) and (4.2), the inequality ∞ |zk (f )|−r ≤ (θf + 2−γ )r + ζ(γr) − 1 − 2−γr (4.3) k=1

is valid. In particular, if γ > 1, then due to (3.4) ∞

|zk (f )|−1 ≤ θf + ζ(γ) − 1.

(4.4)

k=1

Consider now a positive scalar-valued function Φ(t1 , t2 , ..., tj ) with an integer j, deﬁned on the domain 0 ≤ tj ≤ tj−1 ≤ t2 ≤ t1 < ∞ and satisfying ∂Φ ∂Φ ∂Φ > > ... > > 0 for t1 > t2 > ... > tj . ∂t1 ∂t2 ∂tj

(4.5)

Corollary 19.4.3 Under conditions (1.1a), (3.1), (3.2) and (4.5), Φ(|z1 (f )|−1 , |z2 (f )|−1 , ..., |zj (f )|−1 ) ≤ Φ(τ1 , τ2 , ..., τj ). Indeed, this result is due to Theorem 19.3.1 and the well-known Lemma II.3.5 (Gohberg and Krein, 1969). In particular, let {dk }∞ k=1 be a decreasing sequence of non-negative numbers. Take j Φ(t1 , t2 , ..., tj ) = d k tk . k=1

Then Corollary 19.4.3 yields j k=1

dk |zk (f )|−1 ≤

j

τk d k = d 1 θ f +

k=1

(j = 2, 3, ...).

j k=1

dk (k + 1)−γ

252

19. Zeros of Entire Functions

19.5

Notes

The variation of the zeros of general analytic functions under perturbations was investigated, in particular, by P. Rosenbloom (1969). He established the perturbation result that provides the existence of a zero of a perturbed function in a given domain. In the present chapter a new approach to the problem is proposed. The material in the present chapter is taken from the papers (Gil’, 2000a, 2000b, 2000c and 2001). Corollary 19.4.2 supplements the classical Hadamard theorem (Levin, 1996, p. 18), since it not only asserts the convergence of the series of the zeros, but also gives us the estimate for the sums of the zeros.

References [1] Gil’, M.I. (2000a). Inequalities for imaginary parts of zeros of entire functions. Results in Mathematics, 37, 331-334 [2] Gil’, M.I. (2000b). Perturbations of zeros of a class of entire functions, Complex Variables, 42, 97-106 [3] Gil’, M.I. (2000c). Approximations of zeros of entire functions by zeros of polynomials. J. of Approximation Theory, 106, 66-76 [4] Gil’, M.I. (2001). Inequalities for zeros of entire functions, Journal of Inequalities, 6 463-471. [5] Gohberg, I. C. and Krein, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. of Math. Monographs, v. 18, Amer. Math. Soc., R.I. [6] Levin, B. Ya. (1996). Lectures on Entire Functions, Trans. of Math. Monographs, v. 150. Amer. Math. Soc., R. I. [7] Marcus, M. and Minc, H. (1964). A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston. [8] Rosenbloom, P.C. (1969). Perturbation of zeros of analytic functions. I. Journal of Approximation Theory, 2, 111-126.

List of Main Symbols A operator norm of an operator A (., .) scalar product |A| matrix whose elements are absolute values of A inverse to A A−1 conjugate to A A∗ AI = (A − A∗ )/2i AR = (A + A∗ )/2 Trace class C1 Hilbert-Schmidt ideal C2 the set of all compact operator C∞ Neumann-Schatten ideal Cp complex Euclidean space Cn det (A) determinant of A det2 (A) generalized determinant of A Dom (A) domain of A g(A) gI (A) H separable Hilbert space identity operator (in a space H) I = IH m.r.i. -maximal orthogonal resolution of identity ni(A) nilpotency index of A Np (A) Neumann-Schatten norm of A Hilbert-Schmidt (Frobenius) norm of A N (A) = N2 (A) real Euclidean space Rn resolvent of A Rλ (A) relative spectral variation of B with respect to A rsvA (B) RPTO spectral radius of A rs (A) lower spectral radius of A rl (A) s-number (singular number) of A sj (A) spectral variation of B with respect to A svA (B) T r A = T race A trace of A w(λ, A) α(A) = sup Re σ(A) βp , β˜p γn,k eigenvalue of A λk (A) σ(A) spectrum of A (p) θk = √ 1 where [x] is the integer part of x [k/p]!

ρ(A, λ) distance between a point lambda and the spectrum of A

93 11, 83 106

98 102

167 164

165 108 12

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Index Carleman inequality 33, 93 diagonal part of compact operator 82 matrix 8 noncompact operator 99 estimate for norm of function of Hilbert-Schmidt operator 91 matrix 21 quasi-Hermitian operator 110, 113 quasiunitary operator 119 estimate for norm of powers ﬁnite nilpotent matrices 18, 37 Hilbert-Schmidt Volterra operator 92 Neumann-Schatten Volterra operator 93 estimate for norm of resolvent of Hilbert-Schmidt operator 83 matrix 11 Neumann-Schatten operator 88 operators with Hilbert-Schmidt power 86 quasi-Hermitian operator 106 quasiunitary operator 118 Euclidean norm 1

Hille-Tamarkin integral operator 215 Hille-Tamarkin matrix 235 Lidskij’s theorem 79 matrix-valued function 4 maximal resolution of identity (m.r.i) 98 multiplicative representation for resolvent of ﬁnite dimensional operators 24, 27 π-triangular matrice 69 operator in Hilbert space 151160 multiplicity of eigenvalue of matrix 3 of operator in a Hilbert space 77 Neumann-Schatten ideal 78 nilpotency index 102 nilpotent part of compact operator 82 ﬁnite matrix 8 noncompact operator 99 normal matrix 2 normal operator 76

Frobenius norm of matrix 2 Hausdorﬀ distance between spectra 49, 129 Hilbert-Schmidt ideal 78 Hilbert-Schmidt operator 78 Hilbert-Schmidt norm 2, 78

positive selfadjoint operator 76 P -triangular operator 98 quasi-Hermitian operator 98 quasinilpotent operator 77 quasinormal operator 98

256

Index

quasiunitary operator 116 resolution of identity 98 maximal (m.r.i.) 98 Schur’s basis 7, 81 singular numbers of compact operators 78 spectral variation 49, 129 relative nilpotent part 164 relative P -triangular operator 164 relative spectral variation 167 root vectors 77 Trace class 78 trace of matrix 3 linear operator 78 triangular representation of compact operator 82 matrix 7 quasi-Hermitian operator 104 quasiunitary operator 116 regular function of operator 115 unitary operator 76 Volterra operator 77