Linear Operators and Their Essential Pseudospectra

Aref Jeribi, PhD Department of Mathematics, University of Sfax, Tunisia E-mail: [email protected] Apple Academic Press Inc. Apple Academic Press Inc. 3333 Mistwell Crescent 9 Spinnaker Way Oakville, ON L6L 0A2 Canada Waretown, NJ 08758 USA © 2018 by Apple Academic Press, Inc. Exclusive worldwide distribution by CRC Press, a member of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper International Standard Book Number-13: 978-1-77188-699-4 (Hardcover) International Standard Book Number-13: 978-1-351-04627-5 (eBook) All rights reserved. No part of this work may be reprinted or reproduced or utilized in any form or by any electric, mechanical or other means, now known or hereafter invented, including photocopying and re- cording, or in any information storage or retrieval system, without permission in writing from the publisher or its distributor, except in the case of brief excerpts or quotations for use in reviews or critical articles. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission and sources are indicated. Copyright for individual articles remains with the authors as indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors, editors, and the publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors, editors, and the publisher have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify in any future reprint. Trademark Notice: Registered trademark of products or corporate names are used only for explanation and identification without intent to infringe. Library and Archives Canada Cataloguing in Publication

Jeribi, Aref, author Linear operators and their essential pseudospectra / Aref Jeribi, PhD (Department of Mathematics, University of Sfax, Tunisia).

Includes bibliographical references and index. Issued in print and electronic formats. ISBN 978-1-77188-699-4 (hardcover).--ISBN 978-1-351-04627-5 (PDF)

1. Linear operators. 2. Spectral theory (Mathematics). I. Title.

QA329.2.J47 2018 515'.7246 C2018-900406-1 C2018-900407-X

CIP data on file with S Library of C ongress

Apple Academic Press also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information about Apple Academic Press products, visit our website at www.appleacademicpress.com and the CRC Press website at www.crcpress.com To my mother Sania, my father Ali, my wife Fadoua, my children Adam and Rahma, my brothers Soﬁen and Mohamed Amin, my sister Elhem, and all members of my extended family

Contents

About the Author xiii

Preface xv

1 Introduction 1 1.1 Essential Spectra and Relative Essential Spectra . . . . . 2 1.2 Essential Pseudospectra ...... 8 1.3 Structured Essential Pseudospectra and Relative Struc- tured Essential Pseudospectra ...... 12 1.4 Condition Pseudospectrum ...... 14 1.5 Outline of Contents ...... 15

2 Fundamentals 19 2.1 Operators ...... 19 2.1.1 Linear Operators ...... 19 2.1.2 Bounded Operators ...... 20 2.1.3 Closed and Closable Operators ...... 20 2.1.4 Adjoint Operator ...... 22 2.1.5 Direct Sum ...... 24 2.1.6 Resolvent Set and Spectrum ...... 25 2.1.7 Compact Operators ...... 26 2.1.8 A-Deﬁned, A-Bounded, and A-Compact Operators 27 2.1.9 Weakly Compact and A-Weakly Compact Opera- tors ...... 29 2.1.10 Dunford-Pettis Property ...... 29 2.1.11 Strictly Singular Operators ...... 30 viii Contents

2.1.12 Strictly Cosingular ...... 31 2.1.13 Perturbation Function ...... 32 2.1.14 Measure of Non-Strict-Singularity ...... 33 2.1.15 Semigroup Theory ...... 33 2.2 Fredholm and Semi-Fredholm Operators ...... 34 2.2.1 Deﬁnitions ...... 34 2.2.2 Basics on Bounded Fredholm Operators ...... 37 2.2.3 Basics on Unbounded Fredholm Operators . . . . 43 2.3 Perturbation ...... 48 2.3.1 Fredholm and Semi-Fredholm Perturbations . . . . 48 2.3.2 Semi-Fredholm Perturbations ...... 51 2.3.3 Riesz Operator ...... 52 2.3.4 Some Perturbation Results ...... 53 2.3.5 A-Fredholm Perturbation ...... 54 2.3.6 A-Compact Perturbations ...... 55 2.3.7 The Convergence Compactly ...... 56 2.4 Ascent and Descent Operators ...... 57 2.4.1 Bounded Operators ...... 57 2.4.2 Unbounded Operators ...... 62 2.5 Semi-Browder and Browder Operators ...... 63 2.5.1 Semi-Browder Operators ...... 63 2.5.2 Fredholm Operator with Finite Ascent and Descent 65 2.6 Measure of Noncompactness ...... 67 2.6.1 Measure of Noncompactness of a Bounded Subset 68 2.6.2 Measure of Noncompactness of an Operator . . . . 69 2.6.3 Measure of Non-Strict-Singularity ...... 71 2.6.4 γ-Relatively Bounded ...... 73 2.6.5 Perturbation Result ...... 73 2.7 γ-Diagonally Dominant ...... 75 2.8 Gap Topology ...... 76 2.8.1 Gap Between Two Subsets ...... 76 2.8.2 Gap Between Two Operators ...... 77 Contents ix

2.8.3 Convergence in the Generalized Sense ...... 78 2.9 Quasi-Inverse Operator ...... 79 2.10 Limit Inferior and Superior ...... 83

3 Spectra 85 3.1 Essential Spectra ...... 85 3.1.1 Deﬁnitions ...... 85 3.1.2 Characterization of Essential Spectra ...... 88 3.2 The Left and Right Jeribi Essential Spectra ...... 91 3.3 S-Resolvent Set, S-Spectra, and S-Essential Spectra . . . . 92 3.3.1 The S-Resolvent Set ...... 92 3.3.2 S-Spectra ...... 98 3.3.3 S-Browder’s Resolvent ...... 100 3.3.4 S-Essential Spectra ...... 104 3.4 Invariance of the S-Essential Spectrum ...... 108 3.5 Pseudospectra ...... 113 3.5.1 Pseudospectrum ...... 113 3.5.2 S-Pseudospectrum ...... 115 3.5.3 Ammar-Jeribi Essential Pseudospectrum ...... 119 3.5.4 Essential Pseudospectra ...... 122 3.5.5 Conditional Pseudospectrum ...... 123 3.6 Structured Pseudospectra ...... 124 3.6.1 Structured Pseudospectrum ...... 124 3.6.2 The Structured Essential Pseudospectra ...... 125 3.6.3 The Structured S-Pseudospectra ...... 126 3.6.4 The Structured S-Essential Pseudospectra . . . . . 128

4 Perturbation of Unbounded Linear Operators by γ-Relative Boundedness 129 4.1 Sum of Closable Operators ...... 129 4.1.1 Norm Operators ...... 129 4.1.2 Kuratowski Measure of Noncompactness . . . . . 135 4.2 Block Operator Matrices ...... 137 x Contents

4.2.1 2 × 2 Block Operator Matrices ...... 137 4.2.2 3 × 3 Block Operator Matrices ...... 139

5 Essential Spectra 143 5.1 Characterization of Essential Spectra ...... 143 5.1.1 Characterization of Left and Right Weyl Essential Spectra ...... 143 5.1.2 Characterization of Left and Right Jeribi Essential Spectra ...... 149 5.2 Stability of Essential Approximate Point Spectrum and Essential Defect Spectrum of Linear Operator ...... 156 5.2.1 Stability of Essential Spectra ...... 156 5.2.2 Invariance of Essential Spectra ...... 159 5.3 Convergence ...... 162 5.3.1 Convergence Compactly ...... 162 5.3.2 Convergence in the Generalized Sense ...... 169

6 S-Essential Spectra of Closed Linear Operator on a Banach Space 183 6.1 S-Essential Spectra ...... 183 6.1.1 Characterization of S-Essential Spectra ...... 183 6.1.2 Stability of S-Essential Spectra of Closed Linear Operator ...... 188 6.2 S-Left and S-Right Essential Spectra ...... 195 6.2.1 Stability of S-Left and S-Right Fredholm Spectra . 195 6.2.2 Stability of S-Left and S-Right Browder Spectra . . 201

7 S-Essential Spectrum and Measure of Non-Strict-Singularity 205 7.1 A Characterization of the S-Essential Spectrum ...... 205 7.2 The S-Essential Spectra of 2 × 2 Block Operator Matrices 210

8 S-Pseudospectra and Structured S-Pseudospectra 217 8.1 Study of the S-Pseudospectra ...... 217 8.2 Characterization of the Structured S-Pseudospectra . . . . 223 Contents xi

8.3 Characterization of the Structured S-Essential Pseu- dospectra ...... 231

9 Structured Essential Pseudospectra 239 9.1 On a Characterization of the Structured Wolf, Ammar- Jeribi, and Browder Essential Pseudospectra ...... 239 9.1.1 Structured Ammar-Jeribi, and Browder Essential Pseudospectra ...... 239 9.1.2 A Characterization of the Structured Browder Essential Pseudospectrum ...... 252 9.2 Some Description of the Structured Essential Pseu- dospectra ...... 262 9.2.1 Relationship Between Structured Jeribi and Struc- tured Ammar-Jeribi Essential Pseudospectra . . . . 262 9.2.2 A Characterization of the Structured Ammar- Jeribi Essential Pseudospectrum ...... 264

10 Structured Essential Pseudospectra and Measure of Noncom- pactness 269 10.1 New Description of the Structured Essential Pseudospectra 269 10.1.1 A Characterization of the Structured Ammar- Jeribi Essential Pseudospectrum by Kuratowski Measure of Noncompactness ...... 269 10.1.2 A Characterization of the Structured Browder Essential Pseudospectrum by Means of Measure of Non-Strict-Singularity ...... 274

11 A Characterization of the Essential Pseudospectra 279 11.1 Approximation of ε-Pseudospectrum ...... 279 11.2 A Characterization of Approximation Pseudospectrum . . 283 11.3 Essential Approximation Pseudospectrum ...... 293 11.4 Properties of Essential Pseudospectra ...... 298 11.5 Pseudospectrum of Block Operator Matrices ...... 305 xii Contents

12 Conditional Pseudospectra 315 12.1 Some Properties of Σε (A) ...... 315 12.2 Characterization of Condition Pseudospectrum ...... 322

Bibliography 327

Index 345 About the Author

Aref Jeribi, PhD Professor, Department of Mathematics, University of Sfax, Tunisia

Aref Jeribi, PhD, is Professor in the Department of Mathematics at the University of Sfax, Tunisia. He is the author of the book Spectral Theory and Applications of Linear Operators and Block Operator Matrices (2015) and co-author of the book Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications (CRC Press, 2015). He has published many journal articles in international journals. His areas of interest include spectral theory, matrice operators, transport theory, Gribov operator, Bargman space, ﬁxed point theory, Riesz basis, and linear relations.

Preface

This book is intended to provide a fairly comprehensive study of spectral theory of linear operators defined on Banach spaces. The central items of interest include various essential spectra, but we also consider some of their generalizations that have been studied in recent years. As the spectral theory of operators is an important part of functional analysis and has numerous applications in many parts of mathematics and, we require, in this book, some modest prerequisites from functional analysis and operator theory are necessary that the reader can find in the classical texts of functional analysis. We therefore have the hope that this book is accessible to newcomers and graduate students of mathematics with a standard background in analysis. A considerable part of the content of this book corresponds to research activities developed in collaboration with some colleagues as well as with some of my graduate students over the course of several years. This book is considered as an attempt to organize the available material, most of which exists only in the form of research papers scattered throughout the literature. For this reason, it has been a great pleasure for me to organize the material in this book. In recent years, spectral theory has witnessed an explosive development. In this book, we present a survey of results concerning various types of essential spectra and pseudospectra in a unified, axiomatic way, and we also gathered several topics that are new but that relate only to those concepts and methods emanating from other parts of the book. The book covers an important list of topics in spectral theory and will be an excellent choice. It is well written, giving a survey of results xvi Preface concerning various types of spectra in a unified, axiomatic way. The main topics include essential spectra, essential pseudospectra, structured essential pseudospectra, and their relative sets. We do hope that this book will be very useful for researchers, since it represents not only a collection of a previously heterogeneous material but is also an innovation through several extensions. Of course, it is impossible for a single book to cover such a huge field of research. In making personal choices for inclusion of material, we tried to give useful complementary references in this research area, hence prob- ably neglecting some relevant works. We would be very grateful to receive any comments from readers and researchers, providing us with some information concerning some missing references. We would like to thank Salma Charfi for the improvement she has made in the introduction of this book. We are indebted to her. We would also like to thank Aymen Ammar for the improvement he has made throughout this book, and we are very grateful to him. Moreover, we should mention that the thesis work, performed under my direction, by my former students and present colleagues Asrar Elleuch, Mohamemd Zerai Dhahri, Bilel Boukettaya, Kamel Mahfoudhi, and Faten Bouzayeni, obtained results that have helped us in writing this book. Finally, we apologize in case we have forgotten to quote any author who has contributed, directly or indirectly, to this work. —Aref Jeribi, PhD Chapter 1

Introduction

Recently, the interest in description of essential spectra remains high because of the abundance of practical applications that help scientists to deal with information overload. This book is intended to provide an important list of topics in spectral theory of linear operators defined on Banach spaces. Central items of interest include Fredholm operators and various characterizations of essential spectra. In this book, a survey of the state of the art of research related to essential spectra of closed, densely defined, and linear operators subjected to various perturbations is outlined. As important supersets of the essential spectra, the description of essential pseudospectra in this book are interesting objects by themselves, since they carry more information and have a better convergence and approximation properties than essential spectra. We are also interested in giving a much better insight into the essential pseudospectra, by studying some sets called structured essential pseudospectra. Further, a significant amount of research has been done in this book to treat important characterizations of S-essential spectra and S- structured essential pseudospectra. In this book, we also turn our attention to the important concept of condition pseudospectrum which carries more information than spectrum and pseudospectrum, especially, about transient instead of just asymptotic behavior of dynamical systems. 2 Linear Operators and Their Essential Pseudospectra

Now, let us describe its contents in the following sections.

1.1 ESSENTIAL SPECTRA AND RELATIVE ESSENTIAL SPECTRA

The theory of the essential spectra of linear operators in Banach spaces is a modern section of the spectral analysis. It has numerous applications in many parts of mathematics and physics including matrix theory, function theory, complex analysis, differential and integral equations, and control theory. The original definition of the essential spectrum goes back to H. Weyl [158] around 1909, when he defined the essential spectrum for a self-adjoint operator A on a Hilbert space as the set of all points of the spectrum of A that are not isolated eigenvalues of finite algebraic multiplicity and he proved that the addition of a compact operator to A does not affect the essential spectrum. Irrespective of whether A is bounded or not on a Banach space X, there are many ways to define the essential spectrum, most of them are enlargement of the continuous spectrum. Hence, we can find several definitions of the essential spectrum in the literature, which coincide for self-adjoint operators on Hilbert spaces (see, for example, [79, 137]). On the other hand, the concept of essential spectra was introduced and studied by many mathematicians and we can cite the contributions of H. Weyl and his collaborators (see, for instance [1–3, 48, 82, 83, 108, 120, 141, 154, 158]). Further important characterizations concerning essential spectra and their applications to transport operators are established by A. Jeribi and his collaborators see Refs. [1, 4–9, 11–14, 19, 26, 31, 34, 36, 37, 40, 41, 43–47, 51, 53, 59, 60, 84–88, 90, 94, 95, 97–100, 114–116, 123]. Among these essential spectra, the following sets are defined for a closed densely defined linear operator A: Introduction 3

σe1(A) := {λ ∈ C such that λ − A ∈/ Φ+(X)}, σe2(A) := {λ ∈ C such that λ − A ∈/ Φ−(X)}, T σe3(A) := σe1(A) σe2(A), σe4(A) := {λ ∈ C such that λ − A ∈/ Φ(X)}, \ σe5(A) := σ(A + K), K∈K (X) n σe6(A) := C\ λ ∈ C such that λ − A is Fredholm, i(λ − A) = 0, o and all scalars near λ are in ρ(A) , \ σe7(A) := σap(A + K), K∈K (X) \ σe8(A) := σδ (A + K), K∈K (X) \ σewl(A) := σl(A + K), K∈K (X) \ σewr(A) := σr(A + K), K∈K (X) where Φ+(X), Φ−(X), and Φ(X) denote the sets of upper semi-Fredholm, lower semi-Fredholm, and Fredholm operators, respectively; ρ(A), σ(A), and i(A) denote the resolvent set, the spectrum, and the index of A, respectively; K (X) denotes the set of all compact linear operators on X, and n o σap(A) := λ ∈ C such that inf k(λ − A)xk = 0 , kxk=1, x∈D(A) n o σδ (A) := λ ∈ C such that λ − A is not surjective , σl(A) := {λ ∈ C such that λ − A is not left invertible}, σr(A) := {λ ∈ C such that λ − A is not right invertible}.

The sets σe1(·) and σe2(·) are the Gustafson and Weidmann essential spectra, respectively [79]; σe3(·) is the Kato essential spectrum [108]; the subset σe4(·) is the Wolf essential spectrum [160]; σe5(·) is the Schechter essential spectrum [86–88]; and σe6(·) denotes the Browder essential spectrum [48, 96]. The subset σe7(·) was introduced by V. Racoc ˇevic ´ [133] and designates the essential approximate point spectrum, and σe8(·) is 4 Linear Operators and Their Essential Pseudospectra the essential defect spectrum and was introduced by C. Shmoeger [143].

σewl(·) and σewr(·) are the left and the right of Weyl essential spectra, respectively. Considerable attention has also been devoted by A. Jeribi [1,93,113] to give a new characterization of the Schechter essential spectrum named the

Jeribi essential spectrum, σ j(A), of A ∈ C (X) (the set of closed densely deﬁned linear operators on X) deﬁned by

\ σ j(A) = σ(A + K) K∈W ∗(X) where W ∗(X) stands for each one of the sets W (X) (the set of weakly compact operators) or S (X) (the set of strictly singular operators). Recently, an important progress has been made by A. Ammar, B. Boukettaya, and A. Jeribi [20], who were interested in studying the stability problem of the left and right Weyl operator sets; and they defined the left and right Jeribi essential spectra for a closed densely defined operator A as: l \ σ j(A) = σl(A + K) K∈W ∗(X) and r \ σ j (A) = σr(A + K). K∈W ∗(X) l r At the first sight, σewl(A) (resp. σewr(A)) and σ j(A) (resp. σ j (A)) seem l to be not equal. However, the authors has proved that σewl(A) = σ j(A) and r σewr(A) = σ j (A) in L1-spaces or X satisfies the Dunford-Pettis property; l and hence σewl(A) and σewr(A) may be viewed as an extension of σ j(A) r and σ j (A), respectively. On the other hand, the question of stability of the left and the right essential Weyl spectra was well-treated in Ref. [20] by using the concept of polynomially Riesz operators in order to give a refinement on the definition of these essential spectra via this concept and to show that compactness condition can be relaxed in a very general Banach space setting. Introduction 5

Motivated by the notion of measure of noncompactness, A. Ammar, M. Z. Dhahri and A. Jeribi [23] are interested in the description of the essential approximate point spectrum and the essential defect spectrum of a closed densely deﬁned linear operator by means of upper semi-Fredholm and lower semi-Fredholm operators, respectively. In literature, the spectral theory of linear operators have been enriched by S. Goldberg [75], who presented a development of some powerful methods for the study of the convergence of a sequence of linear operators in a Banach space and the investigation of the convergence compactly to zero. Recently, A. Ammar and A. Jeribi are concerned in work [32], not only with the case of bounded linear operators (An)n converging compactly to a bounded operator A, but also with the case of a sequence of closed linear operators (An)n converging in the generalized sense to a closed linear operator A. The notion of generalized convergence was approached as a generalization of convergence in norm for possibly unbounded linear operators, as well as a reliable method for comparing operators. Further, this notion make the authors generalize the results of T. Kato [108], which essentially represents the convergence between their graphs in a certain distance. The obtained results were exploited in Ref. [27] to examine the relationship between the various essential spectrum of An and A, where (An)n converges in the generalized sense to A. In the last years, there have been many studies of the operators pencils, λS − A, λ ∈ C (operator-valued functions of a complex argument) (see, for example, [119, 147]). It is known that many problems of mathematical physics (e.g., quantum theory, transport theory, ···) are reduced to the study of the essential spectra of λS − A. For this, it seems interesting to study the following S-essential spectra: 6 Linear Operators and Their Essential Pseudospectra

σe1,S(A) := {λ ∈ C such that λS − A ∈/ Φ+(X)}, σe2,S(A) := {λ ∈ C such that λS − A ∈/ Φ−(X)}, S σe3,S(A) := {λ ∈ C such that λS − A ∈/ Φ+(X) Φ−(X)}, σe4,S(A) := {λ ∈ C such that λS − A ∈/ Φ(X)}, σe5,S(A) := {λ ∈ C such that λS − A ∈/ W (X)}, σb,S(A) := {λ ∈ C such that λS − A ∈/ B(X)}, σel,S(A) := {λ ∈ C such that λS − A ∈/ Φl(X)}, σer,S(A) := {λ ∈ C such that λS − A ∈/ Φr(X)}, σewl,S(A) := {λ ∈ C such that λS − A ∈/ Wl(X)}, σewr,S(A) := {λ ∈ C such that λS − A ∈/ Wr(X)}, l σbl,S(A) := {λ ∈ C such that λS − A ∈/ B (X)}, r σbr,S(A) := {λ ∈ C such that λS − A ∈/ B (X)}, n o σeap,S(A) := λ ∈ C such that λS − A 6∈ Φ+(X) and i(λS − A) ≤ 0 , n o σeδ,S(A) := λ ∈ C such that λS − A 6∈ Φ−(X) and i(λS − A) ≥ 0 , where B(X) is the set of Riesz-Schauder operators; Φl(X) is the set of left

Fredholm operators; Φr(X) is the set of right Fredholm operators; Wl(X) is the set of left Weyl operators; Wr(X) is the set of right Weyl operators; Bl(X) is the set of left Browder operators; and Br(X) is the set of right Browder operators. They can be ordered as \ σe3,S(A) = σe1,S(A) σe2,S(A) ⊆ σe4,S(A) ⊆ σe5,S(A) ⊆ σb,S(A),

σel,S(A) ⊆ σewl,S(A) ⊆ σbl,S(A),

σer,S(A) ⊆ σewr,S(A) ⊆ σbr,S(A). In Ref. [29], the authors proved that [ σe4,S(A) = σel,S(A) σer,S(A), σe1,S(A) ⊂ σel,S(A) and σe2,S(A) ⊂ σer,S(A).

Note that if S = I, we recover the usual definition of the essential spectra of a bounded linear operator A defined in the first section of this introduction. These relative essential spectra drew the attention of Jeribi and his collaborators in Refs. [1, 10, 20, 29, 101]. Introduction 7

More precisely, the authors A. Ammar, B. Boukettaya, and A. Jeribi pursued the analysis started in Refs. [1, 163, 164] for S-left-right essential spectra and they studied the invariance of σei,S(·)(i = l, r, wl, wr) by some class of perturbations and extended a part of the results obtained in Ref. [98] to a large class of perturbation operators R(X), which contains F (X) for the S-left and S-right spectrum, where R(X) (resp. F (X)) denotes the set of Riesz operators (resp. Fredholm perturbations). They applied their obtained results to describe the S-essential spectra of an integro-differential operator with abstract boundary conditions acting in the Banach space. At the same time, an important progress has been made in order to describe various relative essential spectra and we can cite in this context the contribution of the work of A. Ammar, M. Z. Dhahri and A. Jeribi in Ref. [25], where the authors are inspired by the work [10] and studied some types of S-essential spectra of linear bounded operators on a Banach space X. More precisely, in Ref. [25], we find a detailed treatment of some subsets of S-essential spectra of closed linear operators by means of the measure of non-strict-singularity. When dealing with block operator matrices, we recall that the papers [39, 52, 54–56, 103, 111, 120, 146] are concerned with the study of the I - essential spectra of operators defined by a 2×2 block operator matrix that ! AB L0 = CD acts on the product X ×Y of Banach spaces, where I is the identity operator defined on the product space X ×Y by ! I 0 I = . 0 I Inspired by the works [39,111,120,146], the aim of the authors in Ref. [101] was to generalize the previous work and they considered a bounded operator S formally defined on the product Banach space X ×Y as ! M M S = 1 2 . M3 M4 8 Linear Operators and Their Essential Pseudospectra

A considerable attention has been also devoted in Ref. [24] in order to give the characterization of the S-essential spectra of the 2×2 matrix operator L0 acting on a Banach space are given by using the notion of measure of non-strict-singularity. These results are considered as generalizations of the paper of N. Moalla in Ref. [122], where S-essential spectra of some 2 × 2 operator matrices on X × X are discussed with S = I.

1.2 ESSENTIAL PSEUDOSPECTRA

In 1967, J. M. Varah [155] introduced the first idea of pseudospectra. In 1986, J. H. Wilkinson [159] came up with the modern interpretation of pseudospectrum where he defined it for an arbitrary matrix norm induced by a vector norm. Throughout the 1990s, L. N. Trefethen [135, 150–152] not only initiated the study of pseudospectrum for matrices and operators, but also he talked of approximate eigenvalues and pseudospectrum and used this notion to study interesting problems in mathematical physics. By the same token, several authors worked on this field. For example, we may refer to E. B. Davies [58], A. Harrabi [80], A. Jeribi [1], E. Shar- gorodsky [145], and M. P. H. Wolff [161] who had introduced the term approximation pseudospectrum for linear operators. Pseudospectra are interesting objects by themselves since they carry more information than spectra, especially about transient instead of just asymptotic behavior of dynamical systems. Also, they have better convergence and approximation properties than spectra. The definition of pseudospectra of a closed densely defined linear operator A, for every ε > 0, is given by: [ −1 1 σε (A) := σ(A) λ ∈ C such that k(λ − A) k > , ε Introduction 9 or by [ −1 1 Σε (A) := σ(A) λ ∈ C such that k(λ − A) k ≥ . ε

By convention, we write k(λ − A)−1k = ∞ if λ ∈ σ(A), (spectrum of A).

For ε > 0, it can be shown that σε (A) is a larger set and is never empty. The pseudospectra of A are a family of strictly nested closed sets, which grow to ﬁll the whole complex plane as ε → ∞ (see [80, 151, 152]). From these deﬁnitions, it follows that the pseudospectra associated with various

ε are nested sets. Then, for all 0 < ε1 < ε2, we have

σ(A) ⊂ σε1 (A) ⊂ σε2 (A) and σ(A) ⊂ Σε1 (A) ⊂ Σε2 (A), and that the intersections of all the pseudospectra are the spectrum,

\ \ σε (A) = σ(A) = Σε (A). ε>0 ε>0 In [58], E. B. Davies has deﬁned another equivalent of pseudospectrum. One is in terms of perturbations of the spectrum, that is, for any closed operator A, we have

[ σε (A) := σ(A + D). ||D||<ε

Inspired by the notion of pseudospectra, A. Ammar and A. Jeribi in their works [29], thought to extend these results for essential spectra of densely closed linear operators on a Banach space. They declared the new concept of the essential pseudospectra of densely closed linear operators on a Banach space. Because of the existence of several essential spectra, they were interested in focusing their study on the pseudo-Browder essential spectrum. This set was shown to be characterized in the way one would expect by analogy with essential numerical range. As a consequence, the authors in Ref. [29] located the pseudo-Browder essential spectrum between the essential spectra and the essential numerical range. 10 Linear Operators and Their Essential Pseudospectra

More precisely, for A ∈ C (X) and for every ε > 0, they deﬁned the pseudo- Browder essential spectrum as follows:

[n 1o σe6,ε (A) = σe6(A) λ ∈ C such that kRb(A,λ)k > , ε where Rb(A,λ) is the Browder resolvent of A. The aim of this concept is to study the existence of eigenvalues far from the Browder essential spectrum and also to search the instability of the Browder essential spectrum under every perturbation. Their study of pseudo-Browder essential spectrum enabled them to determine and localize the Browder essential spectrum of a closed, densely deﬁned linear operator on a Banach space. In Refs. [1, 21, 29, 30], A. Jeribi and his collaborators pursued their studies about essential pseudospectra and deﬁned the following sets:

ε σe1,ε (A) := {λ ∈ C such that λ − A ∈/ Φ+(X)}, ε σe2,ε (A) := {λ ∈ C such that λ − A ∈/ Φ−(X)}, ε σe3,ε (A) := {λ ∈ C such that λ − A ∈/ Φ±(X)}, ε σe4,ε (A) := {λ ∈ C such that λ − A ∈/ Φ (X)}, \ σe5,ε (A) := σε (A + K), K∈K (X) S σeap,ε (A) := σe1,ε (A) {λ ∈ C : i(λ − A − D) > 0, ∀ ||D|| < ε}, S σeδ,ε (A) := σe2,ε (A) {λ ∈ C : i(λ − A − D) < 0, ∀ ||D|| < ε},

ε ε where Φ+(X) (resp. Φ−(X)) denotes the set of upper (resp. lower) pseudo ε semi-Fredholm operator, Φ±(X) denotes the set of pseudo semi-Fredholm operator, and Φε (X) denotes the set of pseudo Fredholm operators. Note that if ε tends to 0, we recover the usual definition of the essential spectra of a closed linear operator A defined in the first section of this introduction. In [161], M. P. H. Wolff has given a motivation to study the essential approximation pseudospectrum. In [33], A. Ammar, A. Jeribi and K. Mahfoudhi showed that the notion of essential approximation pseudospectrum can be extended by devoting our studies to the essential approxima- Introduction 11 tion spectrum. For ε > 0 and A ∈ C (X), they define

\ σeap,ε (A) = σap,ε (A + K), K∈K (X) \ Σeap,ε (A) = Σap,ε (A + K), K∈K (X) where

σap,ε (A) := λ ∈ C such that inf k(λ − A)xk < ε , x∈D(A), kxk=1 and Σap,ε (A) := λ ∈ C such that inf k(λ − A)xk ≤ ε . x∈D(A), kxk=1

In their work, the authors measure the sensitivity of the set σap(A) with respect to additive perturbations of A by an operator D ∈ L (X) of a norm less than ε. So, they deﬁne the approximation pseudospectrum of A by

[ σap,ε (A) = σap(A + D), kDk<ε and they give a characterization of the essential approximation pseudospectrum σeap,ε (·) which nicely blends these properties of the essential and the approximation pseudospectra, and accordingly the authors are interested in the following essential approximation spectrum

\ σeap(A) := σap(A + K). K∈K (X)

As a consequence, the authors obtain the following result

[ σeap,ε (A) = σeap(A + D). kDk<ε 12 Linear Operators and Their Essential Pseudospectra

1.3 STRUCTURED ESSENTIAL PSEUDOSPECTRA AND RELATIVE STRUCTURED ESSENTIAL PSEUDOSPECTRA

Structured pseudospectra are very useful tools in control theory and other ﬁelds [81]. In particular, D. Hinrichsen and B. Kelb [81] presented in 1993 a graphical method to determine and visualize the spectral value sets of a matrix A.

We also refer the reader to E. B. Davies [58] who deﬁned the structured pseudospectra, or spectral value sets of a closed densely deﬁned linear operator A on a Banach space X by

[ σ(A,B,C,ε) = σ(A +CDB), kDk<ε where B ∈ L (X,Y) (the set of bounded linear operators from X into Y) and C ∈ L (Z,X). The notion of structured pseudospectra drew the attention of A. Elleuch and A. Jeribi who have recently been concerned in [67,68] with the study of essential spectra of an operator A which is subjected to structured perturbation of the form A A + CDB, where B, C are given bounded operators and D is unknown disturbance operator satisfies kDk < ε for a given ε > 0. More specifically, inspired by [1, 86–89, 91, 112], where the authors proved the invariance of the Schechter essential spectrum of a closed densely defined linear operators, A. Elleuch and A. Jeribi, in [67, 68], extended the obtained results for the case of structured Ammar- Jeribi essential pseudospectrum. For this analysis, they needed to characterize the structured pseudospectra of bounded operator and they defined the following structured essential pseudospectra of a closed, densely defined linear operator A with respect to the perturbation structure (B,C), Introduction 13 where B ∈ L (X,Y), C ∈ L (Z,X) and uncertainty level ε > 0, [ σe4(A,B,C,ε) := σe4(A +CDB), kDk<ε

[ σe5(A,B,C,ε) := σe5(A +CDB), kDk<ε

[ σe6(A,B,C,ε) := σe6(A +CDB). kDk<ε

The subset σe4(A,B,C,ε) is the structured Wolf essential pseudospectrum,

σe5(A,B,C,ε) is the structured Ammar-Jeribi essential pseudospectrum, and σe6(A,B,C,ε) denotes the structured Browder essential pseudospectrum. The main purpose of the authors in Ref. [67] consists in studying the structured essential pseudospectra of the sum of two operators in order to improve and extend the results of Refs. [1, 12, 144] which were done for various essential spectra. In addition, the authors described the behavior of structured Ammar-Jeribi essential pseudospectrum and they presented a fine description of this set by means of compact and Fredholm perturbations. Further, they used in Ref. [68] some criteria in order to prove the stability of the structured Wolf and the structured Ammar-Jeribi essential pseudospectra. They also used the notion of the measure of noncompactness to give a new characterization of the structured Ammar-Jeribi essential pseudospectrum and they presented some properties of the structured Ammar-Jeribi and Browder essential pseudospectra by means of the measure of non-strict-singularity. On the other hand, based on the definition of the S-essential spectra and the structured S-pseudospectra, A. Elleuch and A. Jeribi, in [67, 68], introduced and characterized the structured S-pseudospectrum of a closed, densely defined linear operator on a Banach space X. They also established 14 Linear Operators and Their Essential Pseudospectra a relationship between the structured S-essential pseudospectra and the S- essential spectra of a bounded linear operator under some conditions.

1.4 CONDITION PSEUDOSPECTRUM

There are several generalizations of the concept of the spectrum in literature such as Ransford spectrum [134], pseudospectrum [1, 80, 152], and condition spectrum [109]. It is natural to ask whether there are any results similar for operator in infinite dimensional Banach spaces for these sets. The concept of condition pseudospectrum is very important since it carries more information than spectra and pseudospectra, especially, about transient instead of just asymptotic behavior of dynamical systems. Also, condition pseudospectrums have better convergence and approximation properties than spectra and pseudospectra. For a detailed study of the properties of the condition spectrum in finite dimensional space or in Banach algebras, we may refer to S. H. Kulkarni and D. Sukumar in Ref. [109], M. Karow in [106] and G. K. Kumar and S. H. Lui in Ref. [110]. The aim of the authors A. Ammar, A. Jeribi and K. Mahfoudhi in [35] is to show some properties of condition pseudospectra of a linear operator A in Banach spaces and reveal the relation between their condition pseudospectrum and the pseudospectrum (resp. numerical range). One of the central questions consists in characterizing the condition pseudospectra of all bounded linear operators acting in a Banach space. More precisely, the authors consider, in their work, one more such extension in terms of the condition number. In a complex Banach space, the condition pseudospectrum of a bounded linear operator A in infinite dimensional Banach spaces is given for every 0 < ε < 1 by:

ε [ −1 1 Σ (A) := σ(A) λ ∈ C such that kλ − Akk(λ − A) k > , ε Introduction 15 with the convention that kλ − Akk(λ − A)−1k = ∞, if λ − A is not invertible. The authors summarize some properties and useful results of the condition pseudospectrum and they characterize it in order to improve lots recent results of Refs. [106, 109, 110] in inﬁnite dimensional Banach spaces.

1.5 OUTLINE OF CONTENTS

Going on to a brief indication of the contents of the individual chapters, we mention, first of all that our book consists of 12 chapters. We recall in Chapter 2 some definitions, notations, and basic information on both bounded and unbounded Fredholm operators. We also introduce the concept of semigroup theory, compactly convergence, measure of noncompactness, gap topology, and quasi-inverse operator. Basic concepts and classical results are summarized in the first two sections. In the subse- quent sections we study the approximate point spectrum, which is one of the most important examples of a spectrum in Banach algebras. The approximate point spectrum is closely related with the notions of removable and non-removable ideals. We continue Chapter 3 by giving some fundamentals about characterization of various spectra and essential spectra such as left and right Jeribi essential spectra, S-resolvent set, S-spectra, S-essential spectra, pseudospectra and structured pseudospectra. The aim of Chapter 4 is to investigate the perturbation of unbounded linear operators via the concept of γ-relative boundedness introduced in [92]. In fact, this chapter is a survey of the paper [22] where the authors try to generalize the some results of [61] by using the new concept of the gap and the γ-relative boundedness in order to establish some sufficient conditions imposed to three unbounded linear operators and to obtain the 16 Linear Operators and Their Essential Pseudospectra closedness of their algebraic sum. The obtained results are useful to study some specific properties of both 2 × 2 and 3 × 3 block operator matrices. Chapter 5 addresses the characterization of essential spectra such as left and right Jeribi and Weyl essential spectra. We also study the stability of essential approximate point spectrum and essential defect spectrum of linear operators and we treat the elegant interaction between essential spectra of closed linear operators and the notion of convergence in the generalized sense. In Chapters 6 and 7, we focus on the study of the stability of S-left and S-right essential spectra. Moreover, we give the characterization of the Schechter S-essential spectrum by means of compact and Fredholm perturbations and also via the concept of measure of non-strict-singularity. Finally, illustrate the obtained results to matrix operator. We concentrate ourselves exclusively in Chapter 8 on the characterization of S-pseudospectra, structured S-pseudospectra, and structured S- essential pseudospectra. Inspired by the definition of essential spectra, we define and characterize in Chapter 9 the structured Ammar-Jeribi, Browder and Wolf, essential pseudospectra and in particular, we treat the relationship between structured Jeribi and structured Ammar-Jeribi essential pseudospectra. Chapter 10 focuses on a new description of the structured essential pseudospectra. Not only do we give a characterization of the structured Ammar-Jeribi essential pseudospectrum by Kuratowski measure of noncompactness but also we investigate the structured Browder essential pseudospectrum by means of measure of non-strict-singularity. The aim of Chapter 11 is to investigate the essential approximation pseudospectrum of closed, densely defined linear operators on a Banach space. We refine the notion of the ε-pseudospectrum and we focus on the characterization of the approximation pseudospectrum and the essential approximation pseudospectrum. We also establish some properties of essential pseudospectra. Further, we study the pseudospectrum of block operator matrix. Introduction 17

Finally, in Chapter 12, we summarize some properties and useful results of the condition pseudospectrum and we characterize it in inﬁnite dimensional Banach spaces.

Chapter 2

Fundamentals

The aim of this chapter is to introduce the basic concepts, notations, and elementary results which are used throughout the book.

2.1 OPERATORS

Let X and Y be two Banach spaces. A mapping A which assigns to each element x of a set D(A) ⊂ X a unique element y ∈ Y is called an operator (or transformation), we use the notation A(X −→ Y). The set D(A) on which A acts is called the domain of A.

2.1.1 Linear Operators

The operator A is called linear, if D(A) is a subspace of X, and if A(λx + βy) = λAx + βAy for all scalars λ, β and all elements x, y in D(A). By an operator A from X into Y, we mean a linear operator with a domain D(A) ⊂ X. We denote by N(A) its null space, and R(A) its range. For injective A, the inverse A−1(Y −→ X) is an operator with D(A−1) = R(A) and R(A−1) = D(A). The nullity, α(A), of A is defined as the dimension of N(A) and the defi- ciency, β(A), of A is defined as the codimension of R(A) in Y. 20 Linear Operators and Their Essential Pseudospectra

2.1.2 Bounded Operators

The operator A is called bounded if there is a constant M such that

kAxk ≤ Mkxk, x ∈ X.

The norm of such an operator is deﬁned by kAxk kAk = sup . x6=0 kxk For an introduction to the theory of bounded or unbounded linear operators, we refer to the books of E. B. Davies [58], N. Dunford and J. T. Schwartz [65], I. C. Gohberg, S. Goldberg and M. A. Kaashoek [73], A. Jeribi [1–3] and T. Kato [108].

2.1.3 Closed and Closable Operators

The graph G(A) of a linear operator A on D(A) ⊂ X into Y is the set

{(x,Ax) such that x ∈ D(A)} in the product space X ×Y. Then, A is called a closed linear operator when its graph G(A) constitutes a closed linear subspace of X ×Y. Hence, the notion of a closed linear operator is an extension of the notion of a bounded linear operator. The operator B will be called A-closed if xn → x, Axn → y, Bxn → z for {xn} ⊂ D(A) implies that x ∈ D(B) and Bx = z. A sequence (xn)n ⊂ D(A) will be called A-convergent to x ∈ X if both (xn)n and (Axn)n are Cauchy sequences and xn → x. A linear operator A on D(A) ⊂ X into Y is said to be closable if A has a closed extension. It is equivalent to the condition that the graph G(A) is a submanifold (or subspace) of a closed linear manifold (or space) which is at the same time a graph. It follows that A is closable if, and only if, the closure G(A) of G(A) is a graph. We are thus led to the criterion: A is closable if, and only if, no element of the form (0,x), x 6= 0 is the limit of elements of the form (x,Ax). In other words, A is closable if, and only if, (xn)n ∈ D(A), xn → 0 and Axn → x imply x = 0. When A is closable, there is a closed operator A with G(A) = G(A). A Fundamentals 21 is called the closure of A. It follows immediately that A is the smallest closed extension of A, in the sense that any closed extension of A is also an extension of A. Since x ∈ D(A) is equivalent to (x,Ax) ∈ G(A), x ∈ X belongs to D(A) if, and only if, there exists a sequence (xn)n that is A- convergent to x. In this case, we have

Ax = lim Axn. n→∞

The operator B will be called A-closable if xn → 0, Axn → 0, Bxn → z implies z = 0. By C (X,Y), we denote the set of all closed, densely deﬁned linear operators from X into Y, and by L (X,Y) the Banach space of all bounded linear operators from X into Y. If X = Y, the sets L (X,Y) and C (X,Y) are replaced, respectively, by L (X) and C (X). It is easy to see that a bounded operator deﬁned on the whole Banach space X is closed. The inverse is also true and follows from the closed graph theorem which is the following theorem.

Theorem 2.1.1 If X, Y are Banach spaces, and A is a closed linear operator from X into Y, with D(A) = X, then A ∈ L (X,Y). ♦

Theorem 2.1.2 If A is a one-to-one closed linear operator from X into Y, then R(A) is closed in Y if, and only if, A−1 is a bounded linear operator from Y into X. ♦

Theorem 2.1.3 Let X, Y be Banach spaces, and let A be a closed linear operator from X into Y. Then, a necessary and sufﬁcient condition that R(A) be closed in Y is that

dist(x,N(A)) ≤ CkAxk, x ∈ D(A) holds, where C is a constant and dist(x,N(A)) denotes the distance between x and N(A). ♦

We will recall the Hahn-Banach theorem for normed spaces: 22 Linear Operators and Their Essential Pseudospectra

Theorem 2.1.4 Let X be a normed space and let x0 6= 0 be any element of X. Then, there exists a bounded linear functional f on X such that k f k = 1 and f (x0) = kx0k. ♦

We will recall the Liouville theorem.

Theorem 2.1.5 Let f (z) be an analytic function on the complex plane. If f (z) is bounded, then f (z) is constant. ♦

2.1.4 Adjoint Operator

Let A ∈ L (X,Y). For each y0 ∈ Y ∗ (the adjoint space of Y), the expression y0(Ax) assigns a scalar to each x ∈ X. Thus, it is a functional F(x). Clearly, F is linear. It is also bounded since

|F(x)| = |y0(Ax)| ≤ ky0kkAxk ≤ ky0kkAkkxk.

Thus, there is an x0 ∈ X∗ (the adjoint space of X) such that

y0(Ax) = x0(x), x ∈ X. (2.1)

This functional is unique, for any other functional satisfying (2.1) would have to coincide with x0 on each x ∈ X. Thus, (2.1) can be written in the form y0(Ax) = A∗y0(x). (2.2)

The operator A∗ is called the adjoint of A, depending on the mood one is in. If A, B are bounded operators defined everywhere, it is easily checked that (BA)∗ = A∗B∗. We just follow the definition of adjoint for bounded operator (see (2.2)) for define the adjoint of unbounded operator. Fundamentals 23

By an operator A from X into Y, we mean a linear operator with a domain D(A) ⊂ X. We want

A∗y0(x) = y0(Ax), x ∈ D(A).

Thus, we say that y0 ∈ D(A∗) if there is an x0 ∈ X∗ such that

x0(x) = y0(Ax), x ∈ D(A). (2.3)

Then, we deﬁne A∗y0 to be x0. In order that this deﬁnition make sense, we need x0 to be unique, i.e., that

x0(x) = 0 for all x ∈ D(A) should imply that x0 = 0. Thus, it is true if D(A) is dense in X. To summarize, we can deﬁne A∗ for any linear operator from X into Y provided D(A) is dense in X. We take D(A∗) to be the set of those y0 ∈ Y ∗ for which there is an x0 ∈ X∗ satisfying (2.3). This x0 is unique, and we set

A∗y0 = x0.

If A, B are only densely deﬁned, D(AB) need not be dense, and consequently, (BA)∗ need not exist. If D(BA) is dense, it follows that

D(A∗B∗) ⊂ D((BA)∗) and (BA)∗z0 = A∗B∗z0, for all z0 ∈ D(A∗B∗). Let N be a subset of a normed vector space X.A functional x0 ∈ X∗, is called an annihilator of N if x0(x) = 0, x ∈ N. The set of all annihilators of N is denoted by N⊥. The subspace N⊥ is closed. For any subset R of X∗, we call x ∈ X an annihilator of R if

x0(x) = 0, x0 ∈ R. We denote the set of such annihilators of R by ⊥R. The subspace ⊥R is closed. If M is a closed subspace of X, then

⊥(M⊥) = M. 24 Linear Operators and Their Essential Pseudospectra

0 0 Lemma 2.1.1 (M. Schechter [140, Lemma 4.14, p. 93]) If x1,··· ,xm are ∗ linearly independent vectors in X , then there are vectors x1,··· ,xm in X, such that  0 1 if j = k x j(xk) = δ jk := 1 ≤ j,k ≤ m. (2.4) 0 if j 6= k

Moreover, if x1,··· ,xm are linearly independent vectors in X, then there 0 0 ∗ are vectors x1,··· ,xm in X , such that (2.4) holds. ♦ We will recall a simple lemma due to F. Riesz. Lemma 2.1.2 Let M be a closed subspace of a normed vector space X. If M is not the whole of X, then for each number θ satisfying 0 < θ < 1, there is an element xθ ∈ X, such that kxθ k = 1 and

dist(xθ ,M) := inf kxθ − xk ≥ θ, x∈M where dist(xθ ,M) denotes the distance between xθ and M. ♦

2.1.5 Direct Sum

It is obvious that the sum M + N of two linear subspaces M and N of a vector space X is again a linear subspace. If M TN = {0}, then this sum is called the direct sum of M and N, and will be denoted by M LN. In this case, for every z = x + y in M + N, the components x, y are uniquely determined. A subspace M ⊂ X is said to be complemented, if there exists a closed subspace N ⊂ X, such that X = M LN. If X = M LN, then N is called an algebraic complement of M. A particularly important class of endomorphisms are the so-called projections. If X = M LN and x = y+z, with x ∈ M and y ∈ N, deﬁne P : X −→ M by Px := y. The linear map P projects X onto M along N. Clearly, I − P projects X onto N along M and, we have P(X) = N(I − P) = M, N(P) = (I − P)(X) = N, with P2 = P, i.e., P is an idempotent operator. Suppose now that X is a Banach space. If X = M LN and the projection P is continuous, then M Fundamentals 25 is said to be complemented and N is said to be a topological complement of M. Note that each complemented subspace is closed, but the converse is not true, for instance c0, the Banach space of all sequences which converge to 0, is a not complemented closed subspace of l∞, where l∞ denotes the Banach space of all bounded sequences, see [126].

Lemma 2.1.3 (V.Muller¨ [124, Lemma 2, p. 156]) Let T ∈ L (X,Y) and let F ⊂ Y be a ﬁnite-dimensional subspace. Suppose that R(T) + F is closed. Then, R(T) is closed. In particular, if codimR(T) < ∞, then R(T) is closed. ♦

T Proof. Let F0 = R(T) F and choose a subspace F1 such that M F = F0 F1.

Let M S : X|N(T) F1 −→ Y be the operator deﬁned by

S(x + N(T))M f = Tx + f .

Then, R(S) = R(T) + F and S is one-to-one. Hence, S is bounded below, and consequently, the space M R(T) = S X|N(T) {0} is closed. Q.E.D.

2.1.6 Resolvent Set and Spectrum

Definition 2.1.1 Let A be a closable linear operator in a Banach space X. The resolvent set and the spectrum of A are, respectively, defined as − ρ(A) = {λ ∈ C such that λ − A is injective and (λ − A) 1 ∈ L (X)}, σ(A) = C\ρ(A) 26 Linear Operators and Their Essential Pseudospectra and the point spectrum, continuous, and the residual spectrum are defined as σp(A) = {λ ∈ C such that λ − A is not injective}, σc(A) = {λ ∈ C such that λ − A is injective, R(λ − A) = X, and R(λ − A) 6= X}, σR(A) = {λ ∈ C such that λ − A is injective, R(λ − A) 6= X}. ♦

Remark 2.1.1 Note that, if ρ(A) 6= /0, then A is closed. In fact, if λ ∈ ρ(A), then (λ −A)−1 is closed, which is also valid for λ −A. Then, according to the closed graph theorem (see Theorem 2.1.1), we deduce that

ρ(A) = {λ ∈ C such that λ − A is bijective} and hence, S S σ(A) = σp(A) σc(A) σR(A). ♦

Theorem 2.1.6 Let A, B be elements of L (X). Then, σ(AB)\{0} = σ(BA)\{0}. ♦

2.1.7 Compact Operators

Deﬁnition 2.1.2 Let X and Y be two Banach spaces. A bounded linear operator K from X into Y is said to be compact (or completely continuous) if it transforms every bounded set of X in a relatively compact set of Y. In a similar way, K is said to be compact if for any bounded sequence (xn)n of X, the sequence (Kxn)n has a convergent sequence in Y. ♦

We denote by K (X,Y) the set of all compact linear operators from X into Y. If X = Y, then the set K (X,X) is replaced by K (X).

Theorem 2.1.7 Let K ∈ L (X,Y). Then,

(i) If (Kn)n is compact and converge in norm to K, then K is compact. (ii) K∗ is compact if, and only if, K is compact. (iii) K (X,Y) is a two-sided ideal of L (X,Y). ♦ Fundamentals 27

Lemma 2.1.4 For any n elements a1,··· ,an of a complex Banach space X, the set E, of their linear combinations, forms a subspace of X of dimension ≤ n. ♦

Deﬁnition 2.1.3 A bounded operator A on X is said to be ﬁnite rank if dimR(A) < ∞. ♦

Lemma 2.1.5 Operators of ﬁnite rank are compact. ♦

2.1.8 A-Deﬁned, A-Bounded, and A-Compact Operators

Let A ∈ C (X,Y). The graph norm of A is deﬁned by

kxkA = kxk + kAxk, x ∈ D(A). (2.5)

From the closedness of A, it follows that D(A) endowed with the norm k·kA is a Banach space. Let us denote by XA, the space D(A) equipped with the norm k·kA. Clearly, the operator A satisﬁes

kAxk ≤ kxkA and consequently,

A ∈ L (XA,X). Let J : X −→ Y be a linear operator on X. If D(A) ⊂ D(J), then J will be called A-deﬁned. If J is A-deﬁned, we will denote by Jbits restriction to D(A). Moreover, if Jb∈ L (XA,Y), we say that J is A-bounded. We can easily check that, if J is closed (or closable), then J is A-bounded. Thus, we have the obvious relation  α(Ab) = α(A), β(Bb) = β(B), R(Ab) = R(A),  α(Ab+ Bb) = α(A + B), (2.6)   β(Ab+ Jb) = β(A + J) and R(Ab+ Jb) = R(A + J), where α(A) := dimN(A) and β(A) := codimR(A).

Deﬁnition 2.1.4 Let X and Y be two Banach spaces, A ∈ C (X,Y) and let J be an A-deﬁned linear operator on X. We say that J is A-compact if

Jb∈ K (XA,Y). ♦ 28 Linear Operators and Their Essential Pseudospectra

We can deﬁne the A-compact by the sequence.

Deﬁnition 2.1.5 Let X, Y and Z be Banach spaces, and let A and S be two linear operators from X into Y and from X into Z, respectively. S is called relatively compact with respect to A (or A-compact), if D(A) ⊂ D(S) and for every bounded sequence (xn)n ∈ D(A) such that (Axn)n ⊂ Y is bounded, the sequence (Sxn)n ⊂ Z contains a convergent subsequence. ♦

Let AK (X,Y) denotes the set of A-compact on X. If X = Y we write AK (X) for AK (X,X). We can deﬁne the iterates A2, A3,··· of A. If n > 1, D(An) is the set n o x ∈ X such that x,Ax,··· ,An−1x ∈ D(A) , and Anx = A(An−1x).

Deﬁnition 2.1.6 Let X be a Banach space. We consider the following linear operators A : D(A) ⊂ X −→ X and K : D(K) ⊂ X −→ X. We say that K commutes with A if (i) D(A) ⊂ D(K), (ii) Kx ∈ D(A) whenever x ∈ D(A), and (iii) KAx = AKx for x ∈ D(A2). ♦

Deﬁnition 2.1.7 Let X, Y and Z be Banach spaces, and let A and S be two linear operators from X into Y and from X into Z, respectively. S is called relatively bounded with respect to A (or A-bounded), if D(A) ⊂ D(S) and there exist two constants aS ≥ 0, and bS ≥ 0, such that

kSxk ≤ aSkxk + bSkAxk, x ∈ D(A). (2.7)

The inﬁmum δ of all bS that (2.7) holds for some aS ≥ 0 is called relative bound of S with respect to A (or A-bounded of S). ♦ Fundamentals 29

Clearly, a compact operator is always A-compact, and an A-compact operator is always A-bounded.

Deﬁnition 2.1.8 An operator B is said to be A-pseudo-compact if

kxnk + kAxnk + kBxnk ≤ c, for all {xn} ∈ D(A) implies that {Bxn} has a convergent subsequence, where c is a constant. ♦

2.1.9 Weakly Compact and A-Weakly Compact Operators

Deﬁnition 2.1.9 An operator A ∈ L (X,Y) is said to be weakly compact, if A(B) is relatively weakly compact in Y, for every bounded subset B ⊂ X. ♦

The family of weakly compact operators from X into Y is denoted by W (X,Y). If X = Y, then the family of weakly compact operators on X, W (X) := W (X,X), is a closed two-sided ideal of L (X) containing K (X) (cf. [65, 75]).

Deﬁnition 2.1.10 Let X and Y be two Banach spaces and let S and A be two linear operators from X into Y. The operator S is said to be relatively weakly compact with respect to A (or A-weakly compact) if

D(A) ⊂ D(S) and, for every bounded sequence (xn)n ∈ D(A) such that (Axn)n is bounded, the sequence (Sxn)n contains a weakly convergent subsequence. ♦

2.1.10 Dunford-Pettis Property

Deﬁnition 2.1.11 Let X be a Banach space. We say that X possesses the Dunford-Pettis property (for short, property DP) if, for each Banach space Y, every weakly compact operator A : X −→ Y takes weakly compact sets in X into norm compact sets of Y. ♦ 30 Linear Operators and Their Essential Pseudospectra

It is well known that if X has the property DP, then

W (X)W (X) ⊂ K (X). (2.8)

Indeed, Let A1 and A2 ∈ W (X). If U is a bounded subset of X, then

A1(U) is relatively weakly compact. Accordingly, since X has the DP property, then A2(A1(U)) is a relatively compact subset of X. That is,

A2A1 ∈ K (X). It is well known that any L1-space has the property DP [64]. Also, if Ω is a compact Hausdorff space, then C(Ω) has the property DP [78]. For further examples we refer to [62]. Actually, if X is a separable Hilbert space, then K (X) is the unique proper closed two-sided ideal of

L (X). This result hold true for the space lp, 1 ≤ p ≤ ∞ and c0.

2.1.11 Strictly Singular Operators

Deﬁnition 2.1.12 Let X and Y be two Banach spaces. An operator A ∈ L (X,Y) is called strictly singular if, for every inﬁnite-dimensional subspace M, the restriction of A to M is not a homeomorphism. ♦

Let S (X,Y) denotes the set of strictly singular operators from X into Y. The concept of strictly singular operators was introduced by T. Kato in his pioneering paper [108] as a generalization of the notion of compact operators. For a detailed study of the properties of strictly singular operators, we may refer to [75,108]. For our own use, let us recall the following four facts. The set S (X,Y) is a closed subspace of L (X,Y). If X = Y, then S (X) := S (X,X) is a closed two-sided ideal of L (X) containing K (X). If X is a Hilbert space, then

K (X) = S (X) and the class of weakly compact operators on L1-spaces (resp. C(K)- spaces with K being a compact Haussdorff space) is, nothing else but, the family of strictly singular operators on L1-spaces (resp. C(K)-spaces)

(see [128, Theorem 1]). In fact, if X = L1(Ω,Σ, µ), where (Ω,Σ, µ) is a Fundamentals 31 positive measure space or X = C(K) with K is a compact Hausdorff space, then W (X) = S (X). (2.9) For the detailed study of the basic properties of the classes S (X), we refer the reader to [127, Theorem 1].

Lemma 2.1.6 (A. Pelczynski [127]) Let (Ω,Σ, µ) be a positive measure space and let p ≥ 1. If X is isomorphic to one of the spaces Lp(Ω,Σ,dµ), then S (X)S (X) ⊂ K (X). ♦

Deﬁnition 2.1.13 Let X and Y be two Banach spaces, A ∈ C (X,Y) and let J be an A-deﬁned linear operator on X. We say that J is A-strictly singular if Jb∈ S (XA,Y). ♦

Let AS (X,Y) denotes the set of A-strictly singular from X into Y. If X = Y we write AS (X) for AS (X,X).

Remark 2.1.2 If J is A-deﬁned and strictly singular, then J is A-strictly singular. ♦

2.1.12 Strictly Cosingular

Let X be a Banach space. If N is a closed subspace of X, we denote X by πN the quotient map X −→ X/N. The codimension of N, codim(N), is deﬁned to be the dimension of the vector space X/N.

Deﬁnition 2.1.14 Let X and Y be two Banach spaces. An operator F ∈ L (X,Y) is called strictly cosingular from X into Y if there exists no closed Y subspace N of Y with codim(N) = ∞ such that πN F : X −→ Y/N is surjective. ♦

Let SC (X,Y) denotes the set of strictly cosingular operators from X into Y. This class of operators was introduced by A. Pelczynski [128]. If X = Y, the family of strictly cosingular operators on X, SC (X) := SC (X,X), is a closed two-sided ideal of L (X) (see [156]). 32 Linear Operators and Their Essential Pseudospectra

Deﬁnition 2.1.15 Let X and Y be two Banach spaces, A ∈ C (X,Y) and let J be an A-deﬁned linear operator on X. We say that J is A-strictly cosingular if Jb∈ SC (XA,Y). ♦

Let ASC (X,Y) denotes the set of A-strictly cosingular from X into Y. If X = Y, we write ASC (X) for ASC (X,X).

Remark 2.1.3 If J is A-deﬁned and strictly cosingular, then J is A-strictly cosingular. ♦

2.1.13 Perturbation Function

The family of inﬁnite dimensional subspaces of X is denoted by I (X). For a linear subspace M of X, we denote by iM the canonical injection of

M into X and by AiM the restriction of A to M with the usual convention

AiM = Ai(M TD(A)).

Deﬁnition 2.1.16 Let A : D(A) ⊂ X −→ Y be a linear operator. The minimum modulus of A is deﬁned as follows n o γ1(A) = sup α ≥ 0 such that α dist(x,N(A)) ≤ kAxk ∀ x ∈ D(A) , where dist(x,N(A)) denotes the distance between x and N(A). ♦

Deﬁnition 2.1.17 Let X, Y be normed spaces and let A : D(A) ⊂ X −→ Y be a linear operator. The surjection modulus q(A) is deﬁned by

q(A) = sup{α ≥ 0 such that A(BD(A)) ⊃ BY }, where BD(A) (resp. BY ) denotes the closed unit ball in D(A) (resp. Y), that is, the set of all x ∈ D(A) (resp. x ∈ Y) satisfying kxk ≤ 1. ♦

Deﬁnition 2.1.18 We deﬁne a perturbation function to be a function ψ assigning, to each pair of normed spaces X, Y and a linear operator A : D(A) ⊂ X −→ Y, a number ψ(A) ∈ [0,∞], verifying the following properties: Fundamentals 33

(i) ψ(λA) = |λ|ψ(A), for all λ ∈ C. (ii) ψ(A + S) = ψ(A), where S is a precompact operator. (iii) ψ(A) ≤ kAk whenever A ∈ L (X,Y). Otherwise kAk = ∞. (iv) q(A) ≤ ψ(A) whenever dimD(A∗) = ∞. (v) ψ(AiM) ≤ ψ(A),M ∈ I (D(A)), where I (D(A)) denotes the family of inﬁnite dimensional subspaces of D(A). ♦

2.1.14 Measure of Non-Strict-Singularity

Deﬁnition 2.1.19 Let X, Y be two normed spaces and ψ be a perturbation function. For a linear operator A : D(A) ⊂ X −→Y, we deﬁne the functions

Γψ (·) and ∆ψ (·) in the following way:

If dimD(A) < ∞, then Γψ (A) = 0. However, if dimD(A) = ∞, then n o Γψ (A) = inf ψ(AiM) : M ∈ I (D(A)) (2.10) and

n o ∆ψ (A) = sup Γψ (AiM) : M ∈ I (D(A)) . (2.11)

∆ψ (·) is called the measure of non-strict-singularity. ♦

Theorem 2.1.8 (T. Alvarez [18, Theorem 2.7]) Let T ∈ L (X,Y). Then, T is strictly singular operator if, and only if, ∆ψ (T) = 0. ♦

2.1.15 Semigroup Theory

Deﬁnition 2.1.20 Let X be a Banach space. A strongly continuous semigroup is an operator-valued function T(t) from R+ into L (X) that satis- ﬁes the following properties

T(t + s) = T(t)T(s) for t, s ≥ 0 T(0) = I + kT(t)z0 − z0k → 0 as t → 0 for all z0 ∈ X.

We shall subsequently use the standard abbreviation C0-semigroup for a strongly continuous semigroup. ♦ 34 Linear Operators and Their Essential Pseudospectra

Definition 2.1.21 The infinitesimal generator A of a C0-semigroup on a Banach space X is defined by 1 Az = lim (T(t) − I)z t→0+ t whenever the limit exists; the domain of A, D(A), being the set of elements in X for which the limit exists. ♦

2.2 FREDHOLM AND SEMI-FREDHOLM OPERATORS 2.2.1 Deﬁnitions

Let X and Y be two Banach spaces. By an operator A from X into Y we mean a linear operator with domain D(A) ⊂ X and range R(A) ⊂ Y. The operator A is called left invertible if A is injective and R(A) is closed and a complemented subspaces of X. The operator A is called right invertible if A is onto and N(A) is a complemented subspaces of X. An operator A ∈ C (X,Y) is said to be a Fredholm operator from X into Y if (i) α(A) := dimN(A) is finite, (ii) R(A) is closed in Y, and (iii) β(A) := codimR(A) is finite. The set of upper semi-Fredholm operators is defined by n o Φ+(X,Y) := A ∈ C (X,Y) such that α(A) < ∞ and R(A) is closed in Y .

The set of lower semi-Fredholm operators from X into Y is deﬁned by n o Φ−(X,Y) := A ∈ C (X,Y) such that β(A) < ∞ and R(T) is closed in Y .

The set of semi-Fredholm operators from X into Y is deﬁned by [ Φ±(X,Y) := Φ+(X,Y) Φ−(X,Y).

The set of Fredholm operators from X into Y is deﬁned by \ Φ(X,Y) := Φ+(X,Y) Φ−(X,Y). Fundamentals 35

The set of left invertible and right invertible operators are denoted by Gl(X) and Gr(X), respectively. Note that A is invertible if A is left and right invertible. The set of left Fredholm operators is deﬁned by n Φl(X,Y) = A ∈ C (X,Y) such that R(A) is closed, o complemented subspaces and α(A) < ∞ and the set of right Fredholm operators is deﬁned by n Φr(X,Y) = A ∈ C (X,Y) such that R(A) is closed, o N(A) is complemented subspaces and β(A) < ∞ . Thus, we have the following inclusions

Φ(X,Y) ⊆ Φl(X,Y) ⊆ Φ+(X,Y) and

Φ(X,Y) ⊆ Φr(X,Y) ⊆ Φ−(X,Y).

Let A ∈ C (X,Y) and S ∈ L (X,Y). A complex number λ is in l r Φ+A,S, Φ−A,S, ΦA,S, ΦA,S, Φ±A,S or ΦA,S, if λS − A is in Φ+(X,Y), Φ−(X,Y), Φl(X,Y), Φr(X,Y), Φ±(X,Y) or Φ(X,Y), respectively. Let b b b T Φ (X,Y), Φ+(X,Y), and Φ−(X,Y) denote the sets Φ(X,Y) L (X,Y), T T Φ+(X,Y) L (X,Y), and Φ−(X,Y) L (X,Y), respectively. If X =

Y, then Φ+(X,Y), Φ−(X,Y), Φ±(X,Y), Φ(X,Y), Φl(X,Y), Φr(X,Y), b b b Φ (X,Y), Φ+(X,Y) and Φ−(X,Y) are replaced by Φ+(X), Φ−(X), b b b Φ±(X), Φ(X), Φl(X), Φr(X), Φ (X), Φ+(X) and Φ−(X), respectively. A number complex λ is in Φ+A, Φ−A, ΦlA, ΦrA or ΦA if λ − A is in Φ+(X),

Φ−(X), Φl(X), Φr(X) or Φ(X), respectively.

Theorem 2.2.1 ΦA and ρ(A) are open sets of C. Hence, σ(A) is a closed set. ♦

If A is a semi-Fredholm operator, then the index of A is deﬁned by

i(A) := α(A) − β(A). 36 Linear Operators and Their Essential Pseudospectra

Clearly, if A ∈ Φ(X,Y), then

i(A) < ∞.

If A ∈ Φl(X,Y)\Φ(X,Y), then

i(A) = −∞ and, if A ∈ Φr(X,Y)\Φ(X,Y), then

i(A) = +∞.

The set of left Weyl operators is deﬁned by n o Wl(X) = A ∈ Φl(X) such that i(A) ≤ 0 and, the set of right Weyl operators is deﬁned by n o Wr(X) = A ∈ Φr(X) such that i(A) ≥ 0 .

The set of Weyl operators is deﬁned by \ n o Wl(X) Wr(X) = A ∈ Φ(X) such that i(A) = 0 .

l r Let A ∈ C (X) and S ∈ L (X). A number complex λ is in WA,S or WA,S if λS − A is in Wl(X) or Wr(X), respectively.

Theorem 2.2.2 Assume that T ∗ exists. Then, R(T) is closed if, and only if, R(T ∗) is closed. In this case we have R(T)⊥ = N(T ∗),N(T)⊥ = R(T ∗), α(T ∗) = β(T), and β(T ∗) = α(T). ♦

Deﬁnition 2.2.1 Let ε > 0 and A ∈ C (X). (i) A is called an upper (resp. lower) pseudo semi-Fredholm operators if A + D is an upper (resp. lower) semi-Fredholm operator for all D ∈ L (X) such that ||D|| < ε. (ii) A is called a pseudo semi-Fredholm operator if A + D is a semi- Fredholm operator for all D ∈ L (X) such that ||D|| < ε. (iii) A is called a pseudo Fredholm operator if A + D is a Fredholm operator for all D ∈ L (X) such that ||D|| < ε. ♦ Fundamentals 37

We denote by Φε (X) the set of pseudo Fredholm operators, by ε ε Φ±(X) the set of pseudo semi-Fredholm operator and by Φ+(X)(resp. ε Φ−(X)) the set of upper (resp. lower) pseudo semi-Fredholm operator. ε ε ε ε A complex number λ is in Φ±A, Φ+A, Φ−A or ΦA if λ − A is in ε ε ε ε Φ±(X), Φ+(X), Φ−(X) or Φ (X), respectively.

2.2.2 Basics on Bounded Fredholm Operators

Theorem 2.2.3 (M. Schechter [137, Theorem 5.4, p. 103]) Let X and Y be two Banach spaces, and let A ∈ Φb(X,Y). Then, there is a closed subspace

X0 of X, such that M X = X0 N(A) and, a subspace Y0 of Y of dimension β(A) such that

M Y = R(A) Y0.

Moreover, there is an operator A0 ∈ L (Y,X), such that N(A0) = Y0,

R(A0) = X0,

A0A = I on X0, and

AA0 = I on R(A). In addition,

A0A = I − F1 on X, and

AA0 = I − F2 on Y, where F1 ∈ L (X) with R(F1) = N(A) and F2 ∈ L (Y) with R(F2) = Y0. ♦ 38 Linear Operators and Their Essential Pseudospectra

Theorem 2.2.4 (M. Schechter [137, Theorem 5.5, p. 105]) Let A ∈

L (X,Y). Suppose that there exist A1,A2 ∈ L (Y,X),F1 ∈ K (X) and

F2 ∈ K (Y) such that

A1A = I − F1 on X and

AA2 = I − F2 on Y. Then, A ∈ Φb(X,Y). ♦

Theorem 2.2.5 (M. Schechter [139]) Let A ∈ L (X,Y) and B ∈ L (Y,Z), where X, Y, and Z are Banach spaces. If A and B are Fredholm operators, upper semi-Fredholm operators, lower semi-Fredholm operators, then BA is a Fredholm operator, an upper semi-Fredholm operator, a lower semi- Fredholm operator, respectively, and i(BA) = i(B) + i(A). ♦

Theorem 2.2.6 (M. Schechter [137, Theorem 5.13, p. 110]) Let X, Y, and Z be three Banach spaces. Assume that A ∈ L (X,Y) and B ∈ L (Y,Z) are such that BA ∈ Φb(X,Z). Then, A ∈ Φb(X,Y) if, and only if, B ∈ Φb(Y,Z). ♦

Theorem 2.2.7 (M. Schechter [137, Theorem 5.14, p. 111]) Let X, Y, and Z be three Banach spaces. Assume that A ∈ L (X,Y) and B ∈ L (Y,Z) are such that BA ∈ Φb(X,Z). If α(B) < ∞, then A ∈ Φb(X,Y) and B ∈ Φb(Y,Z). ♦

Theorem 2.2.8 (M. Schechter [137, Theorem 7.3, p. 157]) Let X and Y be Banach spaces. If A ∈ Φb(X,Y) and B ∈ Φb(Y,Z), then BA ∈ Φb(X,Z) and i(BA) = i(B) + i(A). ♦

Lemma 2.2.1 (A. Lebow and M. Schechter [117, Lemma 4.3]) Let X and b Y be Banach spaces. An operator A ∈ L (X,Y) is in Φ+(X,Y) if, and only if, α(A − K) < ∞ for all K ∈ K (X,Y). ♦ Fundamentals 39

Lemma 2.2.2 (A. Lebow and M. Schechter [117, Lemma 4.5]) Let X and b b b Y be Banach spaces. If A ∈ Φ+(X,Y) (resp. Φ−(X,Y), Φ (X,Y)), then b b b there exist an η > 0 such that B ∈ Φ+(X,Y) (resp. Φ−(X,Y), Φ (X,Y)) with i(B) = i(A), for all B ∈ L (X,Y) satisfying kB − Ak < η. ♦

Theorem 2.2.9 (V. Muller¨ [125, Theorem 5, p. 156]) Let X, Y, and Z be three Banach spaces, A ∈ L (X,Y) and B ∈ L (Y,Z). Then,

b b b (i) if A ∈ Φ−(X,Y) and B ∈ Φ−(Y,Z), then BA ∈ Φ−(X,Z), b b b (ii) if A ∈ Φ+(X,Y) and B ∈ Φ+(Y,Z), then BA ∈ Φ+(X,Z), and (iii) if A ∈ Φb(X,Y) and B ∈ Φb(Y,Z), then BA ∈ Φb(X,Z). ♦

Theorem 2.2.10 (V. Muller¨ [125, Theorem 6, p. 157]) Let X, Y, and Z be three Banach spaces, A ∈ L (X,Y) and B ∈ L (Y,Z). Then, b b (i) if BA ∈ Φ+(X,Z), then A ∈ Φ+(X,Y), b b (ii) if BA ∈ Φ−(X,Z), then B ∈ Φ−(Y,Z), and b b b (iii) if BA ∈ Φ (X,Z), then B ∈ Φ−(Y,Z) and A ∈ Φ+(X,Y). ♦

Lemma 2.2.3 Assume that A ∈ L (X) and there exist operators B0,B1 ∈ b b L (X) such that B0A and AB1 are in Φ (X). Then, A ∈ Φ (X). ♦

Proof. By referring to Theorem 2.2.3, there are operators A0, A1 ∈ L (X) such that A0B0A − I and AB1A1 − I are in K (X). The result follows from Theorem 2.2.4. Q.E.D.

Theorem 2.2.11 (V. Muller¨ [125, Theorem 7]) For a bounded operator A on a Banach space X, the following assertions are equivalent: (i) A is a Fredholm operator having index 0, (ii) there exist K ∈ K (X) and an invertible operator S ∈ L (X) such that A = S + K is invertible. ♦

Theorem 2.2.12 (P. Aiena [16, Corollary 1.52]) Suppose that T ∈ L (X). Then, 40 Linear Operators and Their Essential Pseudospectra

(i) T ∈ Φr(X) if, and only if, the class rest Tb = T + K (X) is right invertible in the Calkin algebra L (X)/K (X). (ii) T ∈ Φl(X) if, and only if, the class rest Tb = T + K (X) is left invertible in L (X)/K (X). (iii) T ∈ Φb(X) if, and only if, the class rest Tb = T + K (X) is invertible in L (X)/K (X). ♦

Let P(X) denote the set

∗ r P(X) := {F ∈ L (X) such that there exists r ∈ N satisfying F ∈ K (X)}.

Lemma 2.2.4 (A. Jeribi [84], see also [113]) Let A ∈ P(X) and set A = I − F. Then, (i) dim[N(A)] < ∞, (ii) R(A) is closed, and (iii) codim[R(A)] < ∞. ♦

∗ r Proof. (i) Since F ∈ P(X), there exists r ∈ N such that F ∈ K (X). Let x ∈ N(A), then Frx = x i.e., x ∈ N(I − Fr) and, therefore

N(A) ⊂ N(I − Fr).

On the other hand, the identity I restricted to the kernel of I − Fr is equal to Fr and, consequently compact. Hence, N(I − Fr) is ﬁnite dimensional and, therefore dim[N(A)] < ∞.

(ii) Since A commutes with I, Newton’s binomial formula gives

r r r k k k F = (I − A) = I + ∑(−1) Cr A . (2.12) k=1 Let E be a closed complement for N(A), so that

X = N(A)ME. Fundamentals 41

Thus, we obtain two linear continuous maps

A|E : E −→ X and

F|E : E −→ X, the restrictions of A and F to E. It is clear that the kernel of A|E is {0}. In order to conclude, it sufﬁces to show that

A|E (E) = A(E) = A(X) is closed. For this, it sufﬁces to show that the map

−1 (A|E ) : A(E) −→ E

−1 is continuous. By linearity, it even sufﬁces to prove that (A|E ) is continuous at 0. Suppose that this is not the case. Then, we can ﬁnd a sequence

(xn)n in E such that

Axn → 0, but (xn)n does not converge to 0. Selecting a suitable subsequence, we can assume without loss of generality that

kxnk ≥ η > 0 for all n. Then,

1 1 < kxnk η for all n and, consequently A( xn ) also converges to 0. Furthermore, xn kxnk kxnk has norm 1, and hence some subsequence of Fr( xn ) converges. It follows kxnk from (2.12) that xn has a converging subsequence to an element z in E kxnk verifying kzk = 1 and 42 Linear Operators and Their Essential Pseudospectra

Fr(z) = z. On the other hand,

Fr = F − FA − F2A − ··· − Fr−1A and so, we get Fr(z) = F(z). Hence, we infer that F(z) − z = 0, which implies that z ∈ N(A). This contradicts the fact that \ E N(A) = {0}

(because kzk = 1) and completes the proof of (ii). (iii) If A(X) does not have ﬁnite codimension, we can ﬁnd a sequence of closed subspaces

A(X) = M0 ⊂ M1 ⊂ M2 ⊂ ··· ⊂ Mn ⊂ ··· such that each Mn is closed and of codimension 1 in Mn+1 just by adding one-dimensional spaces to A(X) inductively. By Riesz’s lemma

(see Lemma 2.1.2), we can ﬁnd, in each Mn, an element xn such that kxnk = 1 and,

kxn − yk ≥ 1 − ε for all y in Mn−1 with 0 < ε < 1. Then, by using (2.12), together with the fact that X ⊃ R(A) ⊃ R(A2)··· ⊃ R(An) ⊃ ··· , for all k < n, we get

r r r r i i i i i i kF xn − F xkk = xn − ∑(−1) CrA xn − xk + ∑(−1) CrA xk i=1 i=1

r i i i = xn − xk − ∑(−1) CrA (xn − xk) i=1 ≥ 1 − ε Fundamentals 43 because r i i i xk + ∑(−1) CrA (xn − xk) ∈ Mn−1. i=1 r This proves that the sequence (F xn)n cannot have a convergent subsequence, and contradicts the compactness of Fr. Q.E.D.

Theorem 2.2.13 Assume that the hypothesis of Lemma 2.2.4 holds. Then, A = I − F is a Fredholm operator and i(A) = 0. ♦

Let X be a Banach space. We denote by P(X) the set deﬁned by

n n k P(X) = A ∈ L (X) such that there exists a polynomial P(z) = ∑ akz k=0 o satisfying P(1) 6= 0, P(1) − a0 6= 0 and P(A) ∈ K (X) .

Lemma 2.2.5 (A. Jeribi and K. Latrach [115, Lemma 2.2]) If A ∈ P(X), then I + A ∈ Φ(X) and i(I + A) = 0. ♦

2.2.3 Basics on Unbounded Fredholm Operators

Theorem 2.2.14 (M. Schechter [137, Theorem 5.10]) Let A ∈ C (X,Y). If A ∈ Φ(X,Y) and K ∈ K (X,Y), then A + K ∈ Φ(X,Y) and i(A + K) = i(A). ♦

Theorem 2.2.15 (M. Schechter [137, Theorem 7.9, p. 161]) Let X and Y be Banach spaces. For A ∈ Φ(X,Y), there is an η > 0 such that, for every T ∈ L (X,Y) satisfying kTk < η, one has A + T ∈ Φ(X,Y),

i(A + T) = i(A) and α(A + T) ≤ α(A). ♦ 44 Linear Operators and Their Essential Pseudospectra

Let A be a closed linear from X into Y. Then, N(A) is a closed subspace of X and, hence the quotient space Xe := X/N(A) is a Banach space with respect to the norm

kxek = dist(x,N(A)) := inf{kx − yk such that y ∈ N(A)}. Since N(A) ⊂ D(A), the quotient space D(Ae) := D(A)/N(A) is contained in Xe. Defining Axe = Ax for every xe ∈ D(Ae), it follows that Ae is a well defined closed linear operator with D(Ae) ⊂ Xe and R(Ae) = R(A). Since Ae is one-to-one, the inverse Ae−1 exists on R(A). The reduced minimum modulus of A is defined by  kAxk  inf if A 6= 0, γe(A) := x∈/N(A) dist(x,N(A)) (2.13)  ∞ if A = 0. It is to remember that the reduced minimum modulus measures the closedness of the range of operators in the sense that R(A) is closed if, and −1 −1 only if, γe(A) = kAe k > 0.

Remark 2.2.1 Let A ∈ C (X). Then, γe(A) = γ1(A). ♦

Theorem 2.2.16 (T. Kato [107, Theorem 5.22, p. 236]) Let T ∈ C (X,Y) be semi-Fredholm (so that γ = γe(T) > 0). Let A be a T-bounded operator from X into Y so that we have the inequality

kAxk ≤ akxk + bkTxk x ∈ D(T) ⊂ D(A), where a < (1 − b)γ. Then, S = T + A belongs to C (X,Y), S is semi- Fredholm, α(S) ≤ α(T), β(S) ≤ β(T), and i(S) = i(T). ♦

Lemma 2.2.6 (F. Fakhfakh and M. Mnif [70, Lemma 3.1]) Let X be a

Banach space, A ∈ C (X), and K ∈ L (X). If ∆ψ (K) < Γψ (A), then A+K ∈ Φ+(X),A ∈ Φ+(X), and

i(A + K) = i(A), where ∆ψ (·) is the measure of non-strict-singularity given in (2.11) and

Γψ (·) is given in (2.10). ♦ Fundamentals 45

Lemma 2.2.7 Let A ∈ Φ+(X). Then, the following statements are equivalent (i) i(A) ≤ 0, (ii) A can be expressed in the form

A = U + K, where K ∈ K (X) and U ∈ C (X) an operator with closed range and α(U) = 0. ♦

This lemma is well known for bounded upper semi-Fredholm operators. The proof is a straightforward adaption of the proof of Theorem 3.9 in [162].

Theorem 2.2.17 (M. Schechter [137, Theorem 7.1, p. 157]) Let X and

Y be Banach spaces, and A ∈ Φ(X,Y). Then, there is an operator A0 ∈ T L (Y,X), such that N(A0) = Y0,R(A0) = X0 D(A),

\ A0A = I on X0 D(A), and

AA0 = I on R(A) i.e., there are operators F1 ∈ L (X),F2 ∈ L (Y), such that

A0A = I − F1 on D(A),

AA0 = I − F2 on Y,

R(F1) = N(A),N(F1) = X0, and R(F2) = Y0,N(F2) = R(A). ♦

Theorem 2.2.18 (T. Kato [108, Theorem 5.31, p. 241]) Let X and Y be Banach spaces. Let A ∈ C (X,Y) be semi-Fredholm and let S be an A- bounded operator from X into Y. Then, λS + A is semi-Fredholm and α(λS + A), β(λS + A) are constant for a sufﬁciently small |λ| > 0. ♦ 46 Linear Operators and Their Essential Pseudospectra

Theorem 2.2.19 (M. Schechter [141, Theorem 7.3, p. 157]) Let X and Y be Banach spaces. If A ∈ Φ(X,Y) and B ∈ Φ(Y,Z), then BA ∈ Φ(X,Z) and i(BA) = i(B) + i(A). ♦

Theorem 2.2.20 (M. Schechter [140, Theorem 3.8]) Let X, Y, Z be Banach spaces and suppose B ∈ Φb(Y,Z). Assume that A is a closed, densely deﬁned linear operator from X into Y such that BA ∈ Φ(X,Z). Then, A ∈ Φ(X,Y). ♦

Theorem 2.2.21 (M. Schechter [137, Theorem 7.12, p. 162]) Let X, Y, and Z be Banach spaces. If A ∈ Φ(X,Y) and B is a densely deﬁned closed linear operator from Y into Z such that BA ∈ Φ(X,Z), then B ∈ Φ(Y,Z). ♦

Theorem 2.2.22 (M. Schechter [137, Theorem 7.14, p. 164]) Let X, Y, and Z be Banach spaces and let A be a densely deﬁned closed linear operator from X into Y. Suppose that B ∈ L (Y,Z) with α(B) < ∞ and BA ∈ Φ(X,Z). Then, A ∈ Φ(X,Y). ♦

Proposition 2.2.1 Let A ∈ C (X,Y) and let S be a non null bounded linear operator from X into Y. Then, we have the following results

(i) ΦA,S is open,

(ii) i(λS − A) is constant on any component of ΦA,S, and

(iii) α(λS−A) and β(λS−A) are constant on any component of ΦA,S, except on a discrete set of points on which they have larger values. ♦

Proof. (i) Let λ0 ∈ ΦA,S. Then, according to Theorem 2.2.15, there exists η η > 0 such that, for all µ ∈ C with |µ| < kSk , the operator λ0S − µS −A is a Fredholm operator,

i(λ0S − µS − A) = i(λ0S − A) and

α(λ0S − µS − A) ≤ α(λ0S − A). Fundamentals 47

Consider λ ∈ C such that η |λ − λ | < . 0 kSk

Then, λS − A is a Fredholm operator,

i(λS − A) = i(λ0S − A) and

α(λS − A) ≤ α(λ0S − A).

In particular, this implies that ΦA,S is open.

(ii) Let λ1 and λ2 be any two points in ΦA,S which are connected by a smooth curve Γ whose points are all in ΦA,S. Since ΦA,S is an open set, then for each λ ∈ Γ, there exists an ε > 0 such that, for all µ ∈ C, |λ − µ| < ε, µ ∈ ΦA,S and i(µS − A) = i(λS − A).

By using the Heine-Borel theorem, there exist a ﬁnite number of such sets, which cover Γ. Since each of these sets overlaps with, at least, another set and since i(µS − A) is constant on each one, we see that

i(λ1S − A) = i(λ2S − A).

(iii) Let λ1 and λ2 be any two points in ΦA,S which are connected by a smooth curve Γ whose points are all in ΦA,S. Since ΦA,S is an open set, then for each λ ∈ Γ, there is a sufficiently small ε > 0 such that, for all µ ∈ C, |λ − µ| < ε, µ ∈ ΦA,S and by using Theorem 2.2.18, α(A + µS) and β(A+ µS) are constant for all µ ∈ C, 0 < |λ − µ| < ε. By referring to Heine-Borel’s theorem, there is a finite number of such sets which cover Γ. Since each of these sets overlaps with, at least, another set, we see that α(µS + A) and β(µS + A) are constant for all µ ∈ Γ, except for a finite number of points of Γ. Q.E.D.

The following theorem is developed by M. Gonzalez and M. O. Onieva in [77]. 48 Linear Operators and Their Essential Pseudospectra

Lemma 2.2.8 (M. Gonzalez and M. O. Onieva [77, Theorem 2.3]) Let A ∈ C (X,Y), then

(i) A ∈ Φl(X,Y) if, and only if, there exist B ∈ L (Y,X) and K ∈ K (X) such that [ R(B) R(K) ⊂ D(A) and BA = I − K on D(A),

(ii) A ∈ Φr(X,Y) if, and only if, there exist B ∈ L (Y,X) and K ∈ K (Y) such that R(B) ⊂ D(A), BA and KA are continuous, and AB = I − K. ♦

Lemma 2.2.9 (M. Gonzalez and M. O. Onieva [77, Theorem 2.5]) Let A ∈ C (Y,Z) and B ∈ C (X,Y), then

(i) if A ∈ Φl(Y,Z), B ∈ Φl(X,Y) and D(AB) = X, then AB ∈ Φl(X,Z) and i(AB) = i(A) + i(B),

(ii) if A ∈ Φr(Y,Z), B ∈ Φr(X,Y) and AB is closed, then AB ∈ Φr(X,Z) and i(AB) = i(A) + i(B). ♦

Lemma 2.2.10 (M. Gonzalez and M. O. Onieva [77, Theorem 2.7]) If A ∈

Φl(X)(resp. Φr(X)) and K ∈ K (X), then A + K ∈ Φl(X)(resp. Φr(X)) and i(A + K) = i(A). ♦

2.3 PERTURBATION 2.3.1 Fredholm and Semi-Fredholm Perturbations

Deﬁnition 2.3.1 Let X and Y be two Banach spaces, and let F ∈ L (X,Y). F is called a Fredholm perturbation, if U + F ∈ Φ(X,Y) whenever U ∈ Fundamentals 49

Φ(X,Y). F is called an upper (resp. lower) Fredholm perturbation, if U +

F ∈ Φ+(X,Y) (resp. U + F ∈ Φ−(X,Y)) whenever U ∈ Φ+(X,Y) (resp.

U ∈ Φ−(X,Y)). ♦

The sets of Fredholm, upper semi-Fredholm and lower semi-Fredholm perturbations are, respectively, denoted by F (X,Y), F+(X,Y), and F−(X,Y). In general, we have

K (X,Y) ⊆ F+(X,Y) ⊆ F (X,Y)

K (X,Y) ⊆ F−(X,Y) ⊆ F (X,Y).

If X = Y, we may write F (X), F+(X) and F−(X) instead of F (X,X), F+(X,X) and F−(X,X), respectively. In Deﬁnition 2.3.1, if b b we replace Φ(X,Y), Φ+(X,Y) and Φ−(X,Y) by Φ (X,Y), Φ+(X,Y) and b b b b Φ−(X,Y), we obtain the sets F (X,Y), F+(X,Y) and F−(X,Y), respectively. These classes of operators were introduced and investigated in [73]. In particular, it was shown that F b(X,Y) is a closed subset of L (X,Y) and F b(X) := F b(X,X) is a closed two-sided ideal of L (X). In general, we have b b K (X,Y) ⊆ F+(X,Y) ⊆ F (X,Y) b b K (X,Y) ⊆ F−(X,Y) ⊆ F (X,Y).

b b b In Ref. [72], it was shown that F (X) and F+(X) := F+(X,X) are closed two-sided ideals of L (X). It is worth noticing that, in general, the structure ideal of L (X) is extremely complicated. Most of the results on ideal structure deal with the well-known closed ideals which have arisen from applied work with operators. We can quote, for example, compact operators, weakly compact operators, strictly singular operators, strictly cosingular operators, upper semi-Fredholm perturbations, lower semi-Fredholm perturbations, and Fredholm perturbations. In general, we have 50 Linear Operators and Their Essential Pseudospectra

b b K (X) ⊂ S (X) ⊂ F+(X) ⊂ F (X) ⊂ J (X), and

b b K (X) ⊂ SC (X) ⊂ F−(X) ⊂ F (X) ⊂ J (X), b b where F−(X) := F−(X,X), and where J (X) denotes the set b J (X) = F ∈ L (X) such that I − F ∈ Φ (X) and i(I − F) = 0 .

Remark 2.3.1 J (X) is not an ideal of L (X) (since I 6∈ J (X)). ♦

If X is isomorphic to an Lp-space with 1 ≤ p ≤ ∞ or to C(Σ), where Σ is a compact Hausdorff space, then

b b b K (X) ⊂ S (X) = F+(X) = F−(X) = SC (X) = F (X).

Lemma 2.3.1 (A. Jeribi and N. Moalla [98, Lemma 2.1]) Let A ∈ C (X,Y) and let J : X −→ Y be a linear operator. Assume that J ∈ F (X,Y). Then, (i) if A ∈ Φ(X,Y), then A + J ∈ Φ(X,Y) and

i(A + J) = i(A).

Moreover,

(ii) if A ∈ Φ+(X,Y) and J ∈ F+(X,Y), then A + J ∈ Φ+(X,Y) and

i(A + J) = i(A),

(iii) if A ∈ Φ−(X,Y) and J ∈ F−(X,Y), then A + J ∈ Φ−(X,Y) and

i(A + J) = i(A),

T (iv) if A ∈ Φ±(X,Y) and J ∈ F+(X,Y) F−(X,Y), then A + J ∈ Φ±(X,Y) and i(A + J) = i(A). ♦

Lemma 2.3.2 Let X, Y and Z be three Banach spaces. Then, b b (i) if F1 ∈ F (X,Y) and A ∈ L (Y,Z), then AF1 ∈ F (X,Z), and b b (ii) if F2 ∈ F (Y,Z) and B ∈ L (X,Y), then F1B ∈ F (Y,Z). ♦ Fundamentals 51

Deﬁnition 2.3.2 An operator A ∈ L (X) is said to be polynomially Fred- holm perturbation if there exists a nonzero complex polynomial P(z) = p n ∑n=0 anz such that P(A) is a Fredholm perturbation. ♦ Lemma 2.3.3 Let X and Y be two Banach spaces. Then, F b(X,Y) = F (X,Y). ♦

An immediate consequence of Lemma 2.3.3 that F (X) is a closed two- sided ideal of L (X).

2.3.2 Semi-Fredholm Perturbations

In the beginning of this section, let us prove some results for semi- Fredholm perturbations. Proposition 2.3.1 Let X, Y and Z be three Banach spaces. (i) If the set Φb(Y,Z) is not empty, then b b b E1 ∈ F+(X,Y)and A ∈ Φ (Y,Z) imply AE1 ∈ F+(X,Z) b b b E1 ∈ F−(X,Y)and A ∈ Φ (Y,Z) imply AE1 ∈ F−(X,Z). (ii) If the set Φb(X,Y) is not empty, then b b b E2 ∈ F+(Y,Z)and B ∈ Φ (X,Y) imply E2B ∈ F+(X,Z) b b b E2 ∈ F−(Y,Z)and B ∈ Φ (X,Y) imply E2B ∈ F−(X,Z). ♦ Proof. (i) Since A ∈ Φb(Y,Z), and using Theorem 2.2.3, it follows that there exist A0 ∈ L (Z,Y) and K ∈ K (Z) such that

AA0 = I − K. b From Lemma 2.2.5, we get AA0 ∈ Φ (Z). Using Theorem 2.2.6, we have b b b b A0 ∈ Φ (Z,Y), and so A0 ∈ Φ+(Z,Y) and A0 ∈ Φ−(Z,Y). Let J ∈ Φ+(X,Z) b b (resp. Φ−(X,Z)), using Theorem 2.2.9, we deduce that A0J ∈ Φ+(X,Y) b b b (resp. Φ−(X,Y)). This implies that E1 +A0J ∈ Φ+(X,Y) (resp. Φ−(X,Y)). b b So, A(E1 + A0J) ∈ Φ+(X,Z) (resp. Φ−(X,Z)). Now, using the relation

AE1 + J − KJ = A(E1 + A0J) together with the compactness of the operator KJ, we get (AE1 + J) ∈ b b b Φ+(X,Z) (resp. Φ−(X,Z)). This implies that AE1 ∈ F+(X,Z) (resp. 52 Linear Operators and Their Essential Pseudospectra

b F−(X,Z)). (ii) The proof of (ii) is obtained in a similar way as the proof of the item (i). Q.E.D.

Theorem 2.3.1 Let X, Y and Z be three Banach spaces. Then, (i) If the set Φb(Y,Z) is not empty, then b b E1 ∈ F+(X,Y)and A ∈ L (Y,Z) imply AE1 ∈ F+(X,Z) b b E1 ∈ F−(X,Y)and A ∈ L (Y,Z) imply AE1 ∈ F−(X,Z). (ii) If the set Φb(X,Y) is not empty, then b b E2 ∈ F+(Y,Z)and B ∈ L (X,Y) imply E2B ∈ F+(X,Z) b b E2 ∈ F−(Y,Z)and B ∈ L (X,Y) imply E2B ∈ F−(X,Z). ♦

b Proof. (i) Let C ∈ Φ (Y,Z) and λ ∈ C. Let A1 = A − λC and A2 = λC. b For a sufﬁciently large λ, and using Lemma 2.2.2, we have A1 ∈ Φ (Y,Z). b From Proposition 2.3.1 (i), it follows that A1E1 ∈ F+(X,Z) (resp. b b b F−(X,Z)) and A2E1 ∈ F+(X,Z) (resp. F−(X,Z)). This implies that

A1E1 + A2E1 = AE1

b b is an element of F+(X,Z) (resp. F−(X,Z)). (ii) The proof may be achieved in a similar way as (i). It is sufﬁcient to replace Proposition 2.3.1 (i) by Proposition 2.3.1 (ii). Q.E.D.

2.3.3 Riesz Operator

Let X be a Banach space. An operator R ∈ L (X) is said to be a Riesz operators if ΦR = C\{0}. This family of Riesz operators is denoted by R(X). We know that R(X) is not an ideal of L (X) (see [49]). In general, we have K (X) ⊂ S (X) ⊂ F (X) ⊂ R(X), where n o R(X) := A ∈ L (X) such that λ − A ∈ Φb(X) for each λ 6= 0 Fundamentals 53 which is the class of all Riesz operators. The set R(X) is, not generally, a closed ideal of L (X).

Theorem 2.3.2 (P. Aiena [15, Theorem 2.3]) Let X be a Banach space. Then, (i) if A, B ∈ R(X) and AB − BA ∈ K (X), then A + B ∈ R(X), and (ii) if A ∈ R(X),B ∈ L (X) and AB−BA ∈ K (X), then AB and BA ∈ R(X). ♦

2.3.4 Some Perturbation Results

Deﬁnition 2.3.3 Let X and Y be two Banach spaces. (i) An operator A ∈ C (X) is said to have a left Fredholm inverse if there are maps Rl ∈ L (X) and F ∈ F (X) such that IX + F extends RlA.

The operator Rl is called left Fredholm inverse of A. (ii) An operator A ∈ C (X) is said to have a right Fredholm inverse if there are maps Rr ∈ L (X) such that Rr(Y) ⊂ D(A) and ARr −IY ∈ F (X). The operator Rr is called right Fredholm inverse of A. ♦

The following theorem is well known for bounded upper or lower semi-Fredholm operators. The proof is a straightforward adaption of the proof in [162].

Theorem 2.3.3 T ∈ Φ+(X,Y) (Φ−(X,Y)) either without Φ−(X,Y)

(Φ+(X,Y)) or with it with index ≥ 0 (≤ 0) if, and only if, is of the form

U1 +U2, where R(U1) is closed (U1 has range Y) and U2 is compact. ♦

Theorem 2.3.4 (S. Cˇ . Zˇ ivkovic´-Zlatanovic´, D. S. Djordjevic´, and R. E. Harte [163, Theorem 8]) Let A ∈ L (X) and E ∈ R(X), then b (i) if A ∈ Φl(X) and AE − EA ∈ F (X), then A + E ∈ Φl(X), b (ii) if A ∈ Φr(X) and AE − EA ∈ F (X), then A + E ∈ Φr(X), b (iii) if A ∈ Wl(X) and AE − EA ∈ F (X), then A + E ∈ Wl(X), and b (iv) if A ∈ Wr(X) and AE − EA ∈ F (X), then A + E ∈ Wr(X). ♦ 54 Linear Operators and Their Essential Pseudospectra

A direct consequence of Theorem 2.3.4, the ideality of F (X) and the local constancy of the index, we infer the following result.

Corollary 2.3.1 Let A ∈ L (X) and E ∈ F b(X), then

(i) if A ∈ Φl(X), then A + E ∈ Φl(X) and i(A) = i(A + E), and (ii) if A ∈ Φr(X), then A + E ∈ Φr(X) and i(A) = i(A + E). ♦

2.3.5 A-Fredholm Perturbation

Deﬁnition 2.3.4 Let X and Y be two Banach spaces, A ∈ C (X,Y), and let F be an arbitrary A-deﬁned linear operator on X. We say that F is b an A-Fredholm perturbation if Fb ∈ F (XA,Y). The operator F is called b an upper (resp. lower) A-Fredholm perturbation if Fb ∈ F+(XA,Y) (resp. b Fb ∈ F−(XA,Y)). ♦

The sets of A-Fredholm, upper A-semi-Fredholm and lower A-semi-

Fredholm perturbations are denoted by AF (X,Y), AF+(X,Y), and AF−(X,Y), respectively. If X = Y, we write AF (X), AF+(X), and AF−(X) for AF (X,X), AF+(X,X), and AF−(X,X), respectively.

Remark 2.3.2 (i) If B is bounded (resp. compact, weakly compact, strictly singular, strictly cosingular) implies that B is A-bounded (resp. A-compact, A-weakly compact, A-strictly singular, A-strictly cosingular). (ii) Notice that the concept of A-compactness and A-Fredholmness are not connected with the operator A itself, but only with its domain. (iii) Using Deﬁnition 2.3.4 and [74, page 69], we have

AK (X) ⊆ AS (X) ⊆ AF+(X) ⊆ AF (X). and

AK (X) ⊆ ASC (X) ⊆ AF−(X) ⊆ AF (X). (iv) Let B be an arbitrary A-Fredholm perturbation operator, hence we can regard A and B as operators from XA into X, they will be denoted by Fundamentals 55

Ab and B,b respectively, these operators belong to L (XA,X). Furthermore, we have the obvious relations  α(Ab) = α(A), β(Bb) = β(B), R(Ab) = R(A),  α(Ab+ Bb) = α(A + B), (2.14)   β(Ab+ Bb) = β(A + B) and R(Ab+ Bb) = R(A + B).

♦

2.3.6 A-Compact Perturbations

Theorem 2.3.5 (T. Kato [107, Theorem 5.26, p. 238]) Let X and Y be Banach spaces. Suppose that A is a semi-Fredholm operator and B is an A-compact operator from X into Y, then A + B is also semi-Fredholm with

i(A + B) = i(A). ♦

As a consequence of Theorem 2.3.5, we have the following:

Theorem 2.3.6 (M. Schechter [140, Theorem 7.26, p. 172]) ΦA+K = ΦA for all K which are A-compact, and

i(A + K − λ) = i(A − λ) for all λ ∈ ΦA. ♦

Theorem 2.3.7 (M. Schechter [140, Theorem 2.12, p. 9]) Let X and Y be two Banach spaces. If A is a closed linear operator from X into Y, and if B is A-compact, then (i) kBxk ≤ c(kAxk + kxk), x ∈ D(A), (ii) kAxk ≤ c(k(A + B)xk + kxk), x ∈ D(A), (iii) A + B is a closed operator, and (iv) B is (A + B)-compact. ♦ 56 Linear Operators and Their Essential Pseudospectra

Theorem 2.3.8 Let G be a connected open set in the complex plane, such that ρ(A)TG 6= /0. Then, each point of σ(A)TG is a pole of ﬁnite rank of (λ − A)−1 if, and only if,

α(λ − A) = β(λ − A) < ∞ at each point λ of G. ♦

2.3.7 The Convergence Compactly

Deﬁnition 2.3.5 Let (Tn)n be a sequence of bounded linear operators on c X, (Tn)n is said to be converge to zero compactly, written Tn −→ 0 if, for all x ∈ X,Tnx → 0 and (Tnxn)n is relatively compact for every bounded sequence (xn)n ⊂ X. ♦

c Clearly, kTnk → 0 implies that Tn −→ 0.

Theorem 2.3.9 (S. Goldberg [76, Theorem 4]) Let (Kn)n be a sequence of c bounded linear operators converging to zero compactly i.e., Kn −→ 0, and let T be a closed linear operator. If T is a semi-Fredholm operator, then there exists n0 ∈ N such that for all n ≥ n0, (i) T + Kn is semi-Fredholm,

(ii) α(T + Kn) ≤ α(T),

(iii) β(T + Kn) ≤ β(T), and

(iv) i(T + Kn) = i(T). ♦

We recall the following results due to M. Schechter in [139].

Theorem 2.3.10 Let (Tn)n be a sequence of bounded linear operators which converges (in norm) to a bounded operator T. Then, b b (i) if (Tn)n ∈ F (X), then T ∈ F (X), and b b (ii) if (Tn)n ∈ F−(X), then T ∈ F−(X). ♦

Deﬁnition 2.3.6 Let (Tn)n be a sequence of bounded linear operators on X and let T ∈ L (X). The sequence (Tn)n is said to be converge to T c compactly, written Tn −→ T, if (Tn − T)n converges to zero compactly. ♦ Fundamentals 57

If (Tn)n converges to T, then (Tn)n converges to T compactly. A generalization of the Theorem 2.3.10 is given by the following proposition.

Proposition 2.3.2 (A. Ammar and A. Jeribi [32]) Let (Tn)n be a sequence of bounded linear operators which converges compactly to a bounded operator T. Then, b b (i) if (Tn)n ∈ F (X), then T ∈ F (X), b b (ii) if (Tn)n ∈ F+(X), then T ∈ F+(X), and b b (iii) if (Tn)n ∈ F−(X), then T ∈ F−(X). ♦

2.4 ASCENT AND DESCENT OPERATORS 2.4.1 Bounded Operators

We will denote the set of nonnegative integers by N and if A ∈ L (X), we define the ascent and the descent of A, respectively, by n n n+ o asc(A) := min n ∈ N such that N(A ) = N(A 1) , and n n n+1 o desc(A) := min n ∈ N such that R(A ) = R(A ) . If no such integer exists, we shall say that A has infinite ascent or infinite descent. First, we will recall the following result due to A. E. Taylor.

Proposition 2.4.1 (A. E. Taylor [149, Theorem 3.6]) Let A ∈ L (X). If asc(A) and desc(A) are ﬁnite, then asc(A) = desc(A). ♦

Proposition 2.4.2 Let A be a bounded linear operator on a Banach space X. If A ∈ Φb(X) with asc(A) and desc(A) are ﬁnite. Then, i(A) = 0. ♦ 58 Linear Operators and Their Essential Pseudospectra

Proof. Since asc(A) and desc(A) are ﬁnite. Using Proposition 2.4.1, there exists an integer k such that

asc(A) = desc(A) = k.

Hence, N(Ak) = N(An+k) and R(Ak) = R(An+k) for all n ∈ N. Therefore,

i(Ak) = i(An+k).

However, since A ∈ Φb(X) and by using Theorem 2.2.5, we deduce that

i(Ak) = ki(A) = i(An+k) = (n + k)i(A), for all n ≥ 0. Hence, i(A) = 0. Q.E.D.

Theorem 2.4.1 (M. A. Kaashoek [104]) Suppose that α(A) = β(A) < ∞, and that p = asc(A) < ∞. Then, (i) desc(A) = asc(A), (ii) α(Ai) = β(Ai) < ∞ for i = 0,1,2,···, and (iii) X = R(Ap)LN(Ap). ♦

Theorem 2.4.2 Suppose λ0 is an isolated point of σ(A), and let m be a −1 positive integer. Then, λ0 is a pole of order m of (λ − A) if, and only if,

asc(λ0 − A) = desc(λ0 − A) = m

m and R[(λ0 − A) ] is closed. ♦

Lemma 2.4.1 (S. R. Caradus [50]) For any non negative k, we have (i) α(Ak) ≤ asc(A)α(A), and (ii) β(Ak) ≤ desc(A)β(A). ♦ Fundamentals 59

Lemma 2.4.2 (A. E. Taylor [149]) Suppose there exists a nonnegative integer N such that α(Ak) ≤ N when k = 0,1,2···. Then, asc(A) ≤ N. ♦

Theorem 2.4.3 (V. Rakocevi˘ c´ [132, Theorem 1]) Suppose that A, K ∈ L (X) and AK = KA. Then, b b (i) if A ∈ Φ+(X), asc(A) < ∞ and K ∈ F+(X), then asc(A + K) < ∞, and b b (ii) if A ∈ Φ−(X), desc(A) < ∞ and K ∈ F−(X), then desc(A + K) < ∞. ♦

Theorem 2.4.4 Let A ∈ L (X) and assume that there exists a nonzero complex polynomial n k P(z) = ∑ akz k=0 satisfying P(A) ∈ R(X).

b If P(λ) 6= 0 for some λ ∈ C, then λ −A ∈ Φ (X) with asc(λ −A) < ∞ and desc(λ − A) < ∞. ♦

Proof. Let λ ∈ C with P(λ) 6= 0. We have n k k P(λ) − P(A) = ∑ ak(λ − A ). k=1 Moreover, for any k ∈ {1,··· ,n}

k−1 λ k − Ak = (λ − A) ∑ λ iAk−1−i. i=0 Then, P(λ) − P(A) = (λ − A)H(A) = H(A)(λ − A), where n k−1 i k−1−i H(A) = ∑ ak ∑ λ A . k=1 i=0 60 Linear Operators and Their Essential Pseudospectra

Hence, for n ∈ N,

(P(λ) − P(A))n = (λ − A)nH(A)n = H(A)n(λ − A)n.

This implies that

N[(λ − A)n] ⊂ N[(P(λ) − P(A))n] and R[(P(λ) − P(A))n] ⊂ R[(λ − A)n] for all n ∈ N. The last inclusions gives [ [ N[(λ − A)n] ⊂ N[(P(λ) − P(A))n] (2.15) n∈N n∈N and \ \ R[(P(λ) − P(A))n] ⊂ R[(λ − A)n]. (2.16) n∈N n∈N On the other hand, since P(λ) 6= 0 and P(A) ∈ R(X), then P(λ) − P(A) ∈ Φb(X) with asc(P(λ) − P(A)) < ∞ and desc(P(λ) − P(A)) < ∞. Therefore, Proposition 2.4.1 gives

asc(P(λ) − P(A)) = desc(P(λ) − P(A)).

Let n0 this quantity. Since P(λ) commutes with P(A), Newton’s binomial formula gives

n0 (P( ) − P(A))n0 = (−1)kCk P( )n0−kP(A)k λ ∑ n0 λ k=0 = P(λ)n0 − K , where n0 = P(A) (−1)k−1Ck P( )n0−kP(A)k−1 K ∑ n0 λ k=1 is a Riesz operator on X. Then,

[ dim N[(P(λ) − P(A))n] = dimN[(P(λ) − P(A))n0 ] < ∞ n∈N Fundamentals 61 and

\ codim R[(P(λ) − P(A))n] = codimR[(P(λ) − P(A))n0 ] < ∞. n∈N It follows from Eqs. (2.15) and (2.16) that

[ dim N[(λ − A)n] < ∞ n∈N and \ codim R[(λ − A)n] < ∞. n∈N Hence, asc(λ − A) < ∞ and desc(λ − A) < ∞. This implies that α(λ − A) < ∞ and β(λ −A) < ∞ and consequently, by Lemma 2.1.3, R(λ −A) is closed. This proves that λ −A ∈ Φb(X) with asc(λ −A) < ∞ and desc(λ − A) < ∞. Q.E.D.

Corollary 2.4.1 Assume that the hypotheses of Theorem 2.4.4 hold. Then, λ − A is a Fredholm operator on X of index zero. ♦

The next corollary is a consequence of Theorem 2.4.4 and Corollary 2.4.1.

Corollary 2.4.2 Let A ∈ PR(X) i.e., there exists a nonzero complex polynomial n k P(z) = ∑ akz k=0 satisfying P(−1) 6= 0 and P(A) ∈ R(X).

Then, I + A ∈ Φb(X) and i(I + A) = 0. ♦ 62 Linear Operators and Their Essential Pseudospectra

2.4.2 Unbounded Operators

To discuss the deﬁnitions of ascent and descent for an unbounded linear operator one must consider the case in which D(A) and R(A) are in the same linear space X. Obviously,

N(An) ⊂ N(An+1) and R(An+1) ⊂ R(An) for all n ≥ 0 with the convention A0 = I (the identity operator on X). Thus, if N(Ak) = N(Ak+1) (resp. R(Ak) = R(Ak+1)), then N(An) = N(Ak) (resp. R(An) = R(Ak)) for all n ≥ k. Then, the smallest nonnegative integer n such that N(An) = N(An+1) (resp. R(An) = R(An+1)) is called the ascent (resp. the descent) of A, and denoted by asc(A) (resp. desc(A)). In case where n does not exist, we deﬁne asc(A) = ∞ (resp. desc(A) = ∞).

Proposition 2.4.3 (M. A. Kaashoek and D. C. Lay [105]) Suppose that α(A) = β(A) < ∞ and asc(A) < ∞. Then, there exists a bounded linear operator B deﬁned on X with ﬁnite-dimensional range and such that

(i) BAx = ABx for x ∈ D(A), in particular, B commutes with A, (ii) 0 ∈ ρ(A + B), i.e., A + B has a bounded inverse deﬁned on X, and (iii) if C commutes with A, then CBx = BCx for x ∈ D(A). ♦

Lemma 2.4.3 (M. A. Kaashoek and D. C. Lay [105]) If C commutes with A, then −C commutes with A +C. ♦

Theorem 2.4.5 (M. A. Kaashoek and D. C. Lay [105]) Suppose that α(A) = β(A) < ∞ and asc(A) < ∞. Let C commute with A and suppose that C satisﬁes at least one of the following conditions: (i) C is a compact linear operator on X, (ii) C is a Riesz operator, and n ∗ (iii) C is A -compact, n ∈ N . Then, A +C is closed, α(A +C) = β(A +C) < ∞, and asc(A +C) = desc(A +C)) < ∞. ♦ Fundamentals 63

2.5 SEMI-BROWDER AND BROWDER OPERATORS

For A ∈ C (X), we deﬁne the generalized kernel of A by

∞ N∞(A) = [ N(Ak) k=1 and the generalized range of A by

∞ R∞(A) = \ R(Ak). k=1

Proposition 2.5.1 (T. T. West [157]) Let A ∈ L (X). (i) If α(A) < ∞, then A has ﬁnite ascent if, and only if,

∞ ! ∞ ! [ \ \ N(An) An(X) = {0}. n=1 n=1

(ii) If β(A) < ∞, then A has ﬁnite descent if, and only if, ∞ ∞ [ N(An)M \ An(X) = X. ♦ n=1 n=1

2.5.1 Semi-Browder Operators

Two important classes of operators in Fredholm theory are given by the classes of semi-Fredholm operators which possess finite ascent or finite descent. We shall distinguish the class of all upper semi-Browder operators on a Banach space X that is defined by n o B+(X) := A ∈ Φ+(X) such that asc(A) < ∞ , and the class of all lower semi-Browder operators that is defined by n o B−(X) := A ∈ Φ−(X) such that desc(A) < ∞ . 64 Linear Operators and Their Essential Pseudospectra

The class of all Browder operators (known in the literature also as Riesz-Schauder operators) is deﬁned by

\ B(X) := B+(X) B−(X).

The class of all left Browder operators is deﬁned by

l n o B (X) = A ∈ Φl(X) such that asc(A) < ∞ and the class of all right Browder operators is deﬁned by

r n o B (X) = A ∈ Φr(X) such that desc(A) < ∞ .

Theorem 2.5.1 (S. Cˇ . Zˇ ivkovic´-Zlatanovic´, D. S. Djordjevic´, and R. E. Harte [163, Theorem 7]) Let A, E ∈ L (X). Then, (i) if A ∈ Bl(X) and E ∈ R(X) such that AE = EA, then A + E ∈ Bl(X), and (ii) if A ∈ Br(X) and E ∈ R(X) such that AE = EA, then A + E ∈ Br(X). ♦

Theorem 2.5.2 (F. Fakhfakh and M. Mnif [70, Theorem 3.1]) Let X be a Banach space, A ∈ C (X) and K ∈ L (X) such that either ρ(A) or ρ(A +

K) 6= /0. Assume that K commutes with A, and ∆ψ (K) < Γψ (A), where ψ is a perturbation function. Then,

A ∈ B+(X) if, and only if, A + K ∈ B+(X). ♦

Theorem 2.5.3 (F. Fakhfakh and M. Mnif [70, Theorem 3.2]) Let X be a Banach space, ψ be a perturbation function, A ∈ C (X),K ∈ L (X), and neither ρ(A) nor ρ(A + K) is empty. If we suppose that K commutes with

A and ∆ψ (K) < Γψ (A), then A ∈ B(X) if, and only if, A + K ∈ B(X). ♦ Fundamentals 65

Theorem 2.5.4 (F. Leon-Saavedra and V. Muller¨ [118, Theorem 10, p. 186]) An operator A ∈ L (X) is upper semi-Browder (lower semi-Browder, Browder, respectively) if, and only if, there exists a decomposition M X = X1 X2

such that dimX1 < ∞, AXi ⊂ Xi (i = 1,2), A|X1 is nilpotent and A|X2 is bounded below (onto, invertible, respectively). ♦

2.5.2 Fredholm Operator with Finite Ascent and Descent

Theorem 2.5.5 Let A ∈ PF (X) i.e., there exists a nonzero complex polynomial n i P(z) = ∑aiz i=0 b satisfying P(A) ∈ F (X). Let λ ∈ C with P(λ) 6= 0 and set B = λ − A. Then, B is a Fredholm operator on X with ﬁnite ascent and descent. ♦

We need the following lemma for the proof of the Theorem 2.5.5.

Lemma 2.5.1 Let P be a complex polynomial and λ ∈ C such that P(λ) 6= 0. Then, for all A ∈ L (X) satisfying P(A) ∈ F b(X), the operator P(λ) − P(A) is a Fredholm operator on X with ﬁnite ascent and descent. ♦

Proof. Put B = P(λ) − P(A) P(A) = P(λ) I − P(λ) = P(λ)(I − F),

P(A) b where F = P(λ) ∈ F (X). Let C = I − F, then B = P(λ)C.

It is clear that C + F ∈ B(X) and so, we can write C = C + F − F with C + F ∈ B(X) and F ∈ F b(X). On the other hand, we have

(C + F)F = F(C + F). 66 Linear Operators and Their Essential Pseudospectra

By using Theorem 2.4.3, we deduce that C ∈ B(X) and therefore, B ∈ B(X). Q.E.D.

Proof of Theorem 2.5.5 Let λ ∈ C with P(λ) 6= 0. We have n i i P(λ) − P(A) = ∑ ai(λ − A ). i=1 On the one hand, for any i ∈ {1,··· ,n}, we have

k−1 λ i − Ai = (λ − A) ∑ λ jAk−1− j. j=0 So, P(λ) − P(A) = (λ − A)Q(A) = Q(A)(λ − A), (2.17) where n k−1 j k−1− j Q(A) = ∑ ai ∑ λ A . i=1 j=0 Let p ∈ N, the Eq. (2.17) gives p P(λ) − P(A) = (λ − A)pQ(A)p = Q(A)p(λ − A)p.

Hence, p p N[(λ − A) ] ⊂ N P(λ) − P(A) , ∀p ∈ N and p p R P(λ) − P(A) ⊂ R[(λ − A) ], ∀p ∈ N. This implies that [ [ p N[(λ − A)p] ⊂ NP(λ) − P(A) , (2.18) p∈N p∈N and \ p \ RP(λ) − P(A) ⊂ R[(λ − A)p]. (2.19) p∈N p∈N On the other hand, since P(λ) 6= 0 and P(A) ∈ F b(X), then by using Lemma 2.5.1, P(λ) − P(A) is a Fredholm operator on X with ﬁnite ascent and descent. So, by Proposition 2.4.1, we have

ascP(λ) − P(A) = descP(λ) − P(A). Fundamentals 67

Let p0 this quantity, then [ p p dim NP(λ) − P(A) = dim NP(λ) − P(A) 0 < ∞, p∈N and

\ p p codim RP(λ) − P(A) = codim RP(λ) − P(A) 0 < ∞. p∈N Using Eqs. (2.18) and (2.19), we have [ dim N[(λ − A)p] < ∞, p∈N and \ codim R[(λ − A)p] < ∞. p∈N Therefore, asc(λ − A) < ∞ and desc(λ − A) < ∞. We have also, α(λ − A) < ∞ and β(λ − A) < ∞. Q.E.D.

Corollary 2.5.1 Let A ∈ PF (X), i.e., there exists a nonzero complex polynomial p r P(z) = ∑arz r=0 b satisfying P(A) ∈ F (X). Let λ ∈ C with P(λ) 6= 0. Then, the operator λ − A is a Fredholm operator on X of index zero. ♦

Proof. The proof follows immediately from both Theorem 2.5.5 and Proposition 2.4.2. Q.E.D.

2.6 MEASURE OF NONCOMPACTNESS

The notion of measure of noncompactness turned out to be a useful tool in some topological problems, in functional analysis, and in operator theory (see [1, 17, 42, 63, 121, 129]). 68 Linear Operators and Their Essential Pseudospectra

2.6.1 Measure of Noncompactness of a Bounded Subset

In order to recall the measure of noncompactness, let (X,k · k) be an inﬁnite-dimensional Banach space. We denote by MX the family of all nonempty and bounded subsets of X, while NX denotes its subfamily consisting of all relatively compact sets. Moreover, let us denote the convex hull of a set A ⊂ X by conv(A). Let us recall the following deﬁ- nition.

Deﬁnition 2.6.1 A mapping µ : MX −→ [0,+∞[ is said to be a measure of noncompactness in the space X, if it satisﬁes the following conditions:

(i) The family Ker(µ) := {D ∈ MX such that µ(D) = 0} is nonempty, and Ker(µ) ⊂ NX .

For A,B ∈ MX , we have the following: (ii) If A ⊂ B, then µ(A) ≤ µ(B). (iii) µ(A) = µ(A). (iv) µ(conv(A)) = µ(A). (v) µ(λA + (1 − λ)B) ≤ λ µ(A) + (1 − λ)µ(B), for all λ ∈ [0,1].

(vi) If (An)n is a sequence of sets from MX such that An+1 ⊂ An, An = An (n = 1,2,···) and lim µ(An) = 0, then n→+∞

∞ \ A∞ := An n=1 is nonempty and A∞ ∈ Ker(µ). ♦

The family Ker(µ), described in Deﬁnition 2.6.1 (i), is called the kernel of the measure of noncompactness µ.

Deﬁnition 2.6.2 A measure of noncompactness µ is said to be sublinear if, for all A,B ∈ MX , it satisﬁes the two following conditions: (i) µ(λA) = |λ|µ(A) for λ ∈ R (µ is said to be homogenous), and (ii) µ(A + B) ≤ µ(A) + µ(B) (µ is said to be subadditive). ♦

Deﬁnition 2.6.3 A measure of noncompactness µ is referred to as a measure with maximum property if max(µ(A), µ(B)) = µ(ASB). ♦ Fundamentals 69

Deﬁnition 2.6.4 A measure of noncompactness µ is said to be regular if

Ker(µ) = NX , sublinear and has a maximum property. ♦

For A ∈ MX , the most important examples of measures of noncompactness ( [121]) are:

• Kuratowski measure of noncompactness

γ(A) = inf{ε > 0 : A may be covered by a ﬁnite number of sets of diameter ≤ ε}.

• Hausdorff measure of noncompactness

γ(A) = inf{ε > 0 : A may be covered by a ﬁnite number of open balls of radius ≤ ε}.

Note that these measures γ(·) and γ(·) are regular. The relations between these measures are given by the following inequalities, which were obtained by J. Danes [57]

γ(A) ≤ γ(A) ≤ 2γ(A), for any A ∈ MX . The following proposition gives some frequently used properties of the Kuratowski’s measure of noncompactness.

Proposition 2.6.1 Let A and A0 be two bounded subsets of X. Then, we have the following properties: (i) γ(A) = 0 if, and only if, A is relatively compact. (ii) If A ⊆ A0, then γ(A) ≤ γ(A0). (iii) γ(A + A0) ≤ γ(A) + γ(A0). (iv) For every α ∈ C, γ(αA) = |α|γ(A). ♦

2.6.2 Measure of Noncompactness of an Operator

Deﬁnition 2.6.5 (i) Let T : D(T) ⊆ X −→ X be a continuous operator and let γ(·) be the Kuratowski measure of noncompactness in X. Let k ≥ 0. 70 Linear Operators and Their Essential Pseudospectra

T is said to be k-set-contraction if, for any bounded subset A of D(T), T(A) is a bounded subset of X and

γ(T(A)) ≤ kγ(A).

T is said to be condensing if, for any bounded subset A of D(T) such that γ(A) > 0,T(A) is a bounded subset of X and

γ(T(A)) < γ(A).

(ii) Let T : D(T) ⊆ X −→ X be a continuous operator, γ(·) being the Hausdorff measure of noncompactness in X, and k ≥ 0. T is said to be k- ball-contraction if, for any bounded subset A of D(T),T(A) is a bounded subset of X and γ(T(A)) ≤ kγ(A). ♦

Remark 2.6.1 It is well known that (i) If k < 1, then every k-set-contraction operator is condensing. (ii) Every condensing operator is 1-set-contraction. ♦

Let T ∈ L (X). We deﬁne Kuratowski measure of noncompactness, γ(T), by γ(T) := inf{k such that T is k-set-contraction}, (2.20)

and Hausdorff measure of noncompactness, γ(T), by

γ(T) := γ[T(BX )] := inf{k such that T is k-ball-contraction}, where BX denotes the closed unit ball in X, that is, the set of all x ∈ X satisfying kxk ≤ 1. In the following lemma, we give some important properties of γ(T) and γ(T).

Lemma 2.6.1 ( [38, 66]) Let X be a Banach space and T ∈ L (X). We have the following 1 (i) 2 γ(T) ≤ γ(T) ≤ 2γ(T). (ii) γ(T) = 0 if, and only if, γ(T) = 0 if, and only if, T is compact. Fundamentals 71

(iii) If T,S ∈ L (X), then γ(ST) ≤ γ(S)γ(T) and γ(ST) ≤ γ(S)γ(T). (iv) If K ∈ K (X), then γ(T + K) = γ(T) and γ(T + K) = γ(T). (v) γ(T ∗) ≤ γ(T) and γ(T) ≤ γ(T ∗), where T ∗ denotes the dual operator of T. (vi) If B is a bounded subset of X, then γ(T(B)) ≤ γ(T)γ(B). (vii) γ(T) ≤ kTk. ♦

Theorem 2.6.1 [6, Theorem 3.1] (see also [1]) Let A ∈ L (X), and P, Q be two complex polynomials satisfying Q divides P − 1. (i) If γP(A) < 1, then

b Q(A) ∈ Φ+(X). 1 (ii) If γP(A) < , then 2 Q(A) ∈ Φb(X). ♦

Proposition 2.6.2 (B. Abdelmoumen, A. Dehici, A. Jeribi, and M. Mnif [6, Corollary 3.4]) Let X be a Banach space and A ∈ L (X). If γ(An) < 1, for some n > 0, then I − A is a Fredholm operator with i(I − A) = 0. ♦

2.6.3 Measure of Non-Strict-Singularity

We recall the following deﬁnition of the measure of non-strict- singularity which is introduced by M. Schechter in [139].

Deﬁnition 2.6.6 For A ∈ L (X,Y), set

fM(A) = inf γ(AiN) (2.21) N⊂M and

f (A) = sup fM(A), (2.22) M⊂X where M, N represent inﬁnite dimensional subspaces of X and AiN denotes the restriction of A to N. ♦ 72 Linear Operators and Their Essential Pseudospectra

The semi-norm f (·) is a measure of non-strict-singularity, it was introduced by M. Schechter in [142].

Proposition 2.6.3 (V. R. Rakocevi˘ c´ [130]) Let A ∈ L (X,Y). Then, (i) A ∈ S (X,Y) if, and only if,

f (A) = 0.

(ii) A ∈ S (X,Y) if, and only if,

f (A + T) = f (T) for all T ∈ L (X,Y). (iii) If Z is a Banach space and B ∈ L (Y,Z), then f (BA) ≤ f (B) f (A). ♦

Proposition 2.6.4 (N. Moalla [122, Proposition 2.3]) Let A ∈ L (X). If f (An) < 1 for some integer n ≥ 1, then I −A ∈ Φb(X) with i(I −A) = 0. ♦

Proposition 2.6.5 (N. Moalla [122]) For A ∈ L (X,Y), we have f (A) ≤ γ(A). ♦

Lemma 2.6.2 (N. Moalla [122]) For all bounded operator ! T T T = 1 2 T3 T4 on X ×Y, we consider

n o g(T) = max f (T1) + f (T2), f (T3) + f (T4) , (2.23) where f (·) is a measure of non-strict-singularity given in (2.22). Then, g(·) deﬁnes a measure of non-strict-singularity on the space L (X ×Y). ♦ Fundamentals 73

2.6.4 γ-Relatively Bounded

Let us notice that, throughout the book, we are working on two spaces, for example X and Y with their respective measures γX (·) and γY (·).

However, and in order to simplify our reasoning, γX (·) and γY (·) will be simply called γ(·). Of course, the reader will be able to link γ(·) either to

γX (·) or to γY (·).

Deﬁnition 2.6.7 Let X and Y be Banach spaces, let γ(·) be a Kuratowski measure of noncompactness, and let S, A be two linear operators from X into Y bounded on its domains such that D(A) ⊂ D(S). The operator S is called γ-relatively bounded with respect to A (or A-γ-bounded), if there exist constants aS ≥ 0 and bS ≥ 0, such that

γ(S(D)) ≤ aSγ(D) + bSγ(A(D)), (2.24) where D is a bounded subset of D(A). The inﬁmum of the constants bS which satisfy (2.24) for some aS ≥ 0 is called the A-γ-bound of S. ♦

Theorem 2.6.2 (A. Jeribi, B. Krichen, and M. Zarai Dhahri [92, Theorem 2.1]) If S is A-γ-bounded with a bound < 1 and if S and A are closed, then A + S is closed. ♦

Lemma 2.6.3 (A. Jeribi, B. Krichen, and M. Zarai Dhahri [92, Remark 2.1]) If S is T-γ-bounded with T-γ-bound δ < 1, then S is (T + S)-γ- δ bounded with (T + S)-γ-bound ≤ 1−δ . ♦

2.6.5 Perturbation Result

We start our investigation with the following lemma which is fundamental for our purpose.

Lemma 2.6.4 Let A, K ∈ L (X) such that K commutes with A. If f (Kn) < n b b fM(A ) for some n ≥ 1, then A + K ∈ Φ+(X),A ∈ Φ+(X) and

i(A + K) = i(A), where fM(·) (resp. f (·)) is given in (2.21) (resp. (2.22)). ♦ 74 Linear Operators and Their Essential Pseudospectra

Proof. We will begin by the case n = 1. For this, we argue by contra- b diction. Suppose that A + K 6∈ Φ+(X), then by Lemma 2.2.1, there is S ∈ K (X) such that α(A + K − S) = ∞. We set M = N(A + K −S). Then, the restriction of A + K to M is compact, that is

(A + K)iM = SiM. Let N be a subspace of M such that

dimN = ∞.

So,

AiMiN = −KiMiN + SiMiN. From Proposition 2.6.3 (ii), it follows that

f (AiN) = f (KiN).

Thereby, fM(A) = inf ( f (AiN)) N⊂M ≤ f (AiN)

= f (KiN). Then,

fM(A) ≤ fM(K) ≤ f (K)

b which is absurd. Consequently, A + K ∈ Φ+(X). Let λ ∈ [0,1], then

f (λK) = λ f (K) < fM(A),

b b b and so A + λK ∈ Φ+(X). Thereby, A ∈ Φ+(X). Since A + λK ∈ Φ+(X) for all λ ∈ [0,1] and the use of Lemma 2.2.2 leads to

i(A + K) = i(A).

For n > 1, we have

An − Kn = (An−1 + An−2K + ··· + AKn−2 + Kn−1)(A − K). (2.25) Fundamentals 75

Since n n n f (−K ) = f (K ) < fM(A ), then n n b A − K ∈ Φ+(X). (2.26) In view of Theorem 2.2.10 and both (2.25) and (2.26), we get A − K ∈

Φ+(X). Let λ ∈ [−1,0]. Thus,

f (−(λK)n) = f ((λK)n) = |λ n| f (Kn) n < fM(A ) and then, by what we have just showed,

b A − λK ∈ Φ+(X).

b b Therefore, A + K ∈ Φ+(X) and A ∈ Φ+(X). It remains to show that

i(A + K) = i(A).

To do this, we reason in the same way as the case n = 1. Q.E.D.

2.7 γ-DIAGONALLY DOMINANT

We denote by L the block matrix linear operator, acting on the Banach space X ×Y, of the form ! AB L = , (2.27) CD where the operator A acts on X and has domain D(A), D is deﬁned on D(D) and acts on the Banach space Y, and the intertwining operator B (resp. C) is deﬁned on the domain D(B) (resp. D(C)) and acts on X (resp. Y). One of the problems in the study of such operators is that in general L is not closed or even closable, even if its entries are closed. 76 Linear Operators and Their Essential Pseudospectra

Deﬁnition 2.7.1 Let γ(·) be a measure of noncompactness. The block operator matrix L , given in (2.27), is called (i) γ-diagonally dominant if C is A-γ-bounded and B is D-γ-bounded. (ii) off-γ-diagonally dominant if A is C-γ-bounded and D is B-γ- bounded. ♦

Deﬁnition 2.7.2 The block operator matrix L is called (i) γ-diagonally dominant with bound δ, if C is A-γ-bounded with

A-γ-bound δC, and B is D-γ-bounded with D-γ-bound δB, and δ = max{δC,δB}. (ii) off-γ-diagonally dominant with bound δ if A is C-γ-bounded with C-γ-bound δA, and D is B-γ-bounded with B-γ-bound δD, and δ = max{δA,δD}. ♦

2.8 GAP TOPOLOGY 2.8.1 Gap Between Two Subsets

Deﬁnition 2.8.1 The gap between two subsets M and N of a normed space X is deﬁned by the following formula:

δ(M,N) = sup dist(x,N), x∈M, kxk=1 whenever M 6= {0}. Otherwise we deﬁne δ({0},N) = 0 for every subset N. We can also deﬁne n o δb(M,N) = max δ(M,N),δ(N,M) .

Sometimes the latter is called the symmetric or maximal gap between M and N in order to distinguish it from the former. The gap δ(M,N) can be characterized as the smallest number δ such that dist(x,N) ≤ δkxk, for all x ∈ M. ♦ Fundamentals 77

Remark 2.8.1 (i) The gap measures the distance between two subsets and it follows easily from the Deﬁnition 2.8.1, (a) δ(M,N) = δ(M,N) and δb(M,N) = δb(M,N), (b) δ(M,N) = 0 if, and only if, M ⊂ N, (c) δb(M,N) = 0 if, and only if, M = N, where M (resp. N) is the closure of M (resp. N). (ii) Let us notice that δb(·,·) is a metric on the set U (X) of all linear, closed subspaces of X and the convergence Mn → N in U (X) is obviously deﬁned by δb(Mn,N) → 0. Moreover, (U (X),δb) is a complete metric space. ♦

2.8.2 Gap Between Two Operators

Deﬁnition 2.8.2 Let X and Y be two normed spaces and let T, S be two closed linear operators acting from X into Y. Let us deﬁne

δ(T,S) = δG(T),G(S) and δb(T,S) = δbG(T),G(S).

δbT,S is called the gap between S and T. More explicitly, 1 2 2 2 δ G(T),G(S) = sup inf kx − ykX + kTx − SykY . y∈D(S) x ∈ D(T) 2 2 kxkX + kTxkY = 1 ♦

The function δb(·,·) deﬁnes a metric on C (X,Y) called the gap metric and the topology induced by this metric is called the gap topology. The next theorem contains some basic properties of the gap between two closed linear operator.

Theorem 2.8.1 (T. Kato [108, Chapter IV Section 2]) Let T and S be two closed densely deﬁned linear operators. Then, (i) δ(T,S) = δ(S∗,T ∗) and δb(T,S) = δb(S∗,T ∗). 78 Linear Operators and Their Essential Pseudospectra

(ii) If S and T are one-to-one, then δ(S,T) = δ(S−1,T −1) and δb(S,T) = δb(S−1,T −1). (iii) Let A ∈ L (X,Y), then δb(A + S,A + T) ≤ 2(1 + kAk2)δb(S,T). (iv) Let T be a Fredholm (resp. semi-Fredholm) operator. If δb(T,S) < −1 γ(T)(1+[γ(T)]2) 2 , then S is a Fredholm (resp. semi-Fredholm) operator, α(S) ≤ α(T) and β(S) ≤ β(T). Furthermore, there exists b > 0 such that δb(T,S) < b, which implies i(S) = i(T). 1 h i− 2 (v) Let T ∈ L (X,Y). If S ∈ C (X,Y) and δb(T,S) ≤ 1 + kTk2 , then S is a bounded operator (so that D(S) is closed). ♦

A complete discussion and properties concerning the gap may be found in T. Kato [108]. For the case of closable linear operators, the authors A. Ammar and A. Jeribi have introduced in [32] the following deﬁnition.

Deﬁnition 2.8.3 Let S and T be two closable operators. We deﬁne the gap between T and S by δ(T,S) = δ(T,S) and δb(T,S) = δb(T,S). ♦

2.8.3 Convergence in the Generalized Sense

Deﬁnition 2.8.4 Let (Tn)n be a sequence of closable linear operators from X into Y and let T be a closable linear operator from X into Y. The sequence (Tn)n is said to be converge in the generalized sense to T, written g Tn −→ T, if δb(Tn,T) converges to 0 when n → ∞. ♦

It should be remarked that the notion of generalized convergence introduced above for closed and closable operators can be thought as a generalization of convergence in norm for linear operators that may be unbounded. Moreover, an important passageway between these two notions is developed in the following theorem:

Theorem 2.8.2 (A. Ammar [32, Theorem 2.3]) Let (Tn)n be a sequence of closable linear operators from X into Y and let T be a closable linear operator from X into Y. Then, Fundamentals 79

g (i) The sequence (Tn)n converges in the generalized sense to T, (Tn −→

T), if, and only if, (Tn + S)n converges in the generalized sense to T + S, for all S ∈ L (X,Y). (ii) Let T ∈ L (X,Y). The sequence (Tn)n converges in the generalized sense to T if, and only if, Tn ∈ L (X,Y) for sufﬁciently larger n and (Tn)n converges to T. −1 (iii) If (Tn)n converges in the generalized sense to T, then T exists −1 −1 −1 and T ∈ L (Y,X), if, and only if, Tn exists and Tn ∈ L (Y,X) for −1 −1 sufﬁciently larger n and (Tn )n converges to T . ♦

2.9 QUASI-INVERSE OPERATOR

Let A be a closed operator on a Banach space X, with the property that ΦA 6= /0.If f (λ) is a complex valued analytic function of a complex variable, we denote by ∆( f ) the domain of analyticity of f .

0 Deﬁnition 2.9.1 By R∞(A) we mean the family of all analytic functions f (λ) with the following properties:

(i) C\ΦA ⊂ ∆( f ), and (ii) ∆( f ) contains a neighborhood of ∞ and f is analytic at ∞. ♦

Deﬁnition 2.9.2 A bounded operator B is called a quasi-inverse of the closed operator A if R(B) ⊂ D(A),

AB = I + K1, and

BA = I + K2, where K1,K2 ∈ K (X). ♦

If A is a closed operator such that ΦA is not empty, then by using Propo- sition 2.2.1 (i), ΦA is open. Hence, it is the union of a disjoint collection 80 Linear Operators and Their Essential Pseudospectra of connected open sets. Each of them, Φi(A), will be called a component of ΦA. In each Φi(A), a ﬁxed point λi, is chosen in a prescribed manner.

Since α(λi − A) < ∞, R(λi − A) is closed and β(λi − A) < ∞, then there exist a closed subspace Xi and a subspace Yi, such that dimYi = β(λi − A) satisfying M X = N(λi − A) Xi and M X = Yi R(λi − A).

Now, let P1i be the projection of X onto N(λi − A) along Xi and let P2i be the projection of X onto Yi along R(λi − A). P1i and P2i are bounded ﬁnite T rank operators. It is shown in [138] that (λi − A)|D(A) Xi has a bounded inverse Ai, where

\ Ai : R(λi − A) −→ D(A) Xi.

Let Ti be the bounded operator deﬁned by

Tix := Ai(I − P2i)x (2.28) satisfying

Ti(λi − A) = I − P1i on D(A) and

(λi − A)Ti = I − P2i on X.

Hence, Ti is a quasi-inverse of λi − A. Moreover, when λ ∈ Φi(A) and −1 ∈ ρ(Ti), the operator λ−λi

0 −1 Rλ (A) := Ti [(λ − λi)Ti + I]

0 is shown in [71] to be a quasi-inverse of λ − A. In fact, Rλ (A) is deﬁned 0 and analytic for all λ ∈ ΦA except for, at most, an isolated set, Φ (A), having no accumulation point in ΦA. Now, we are ready to declare this result: Fundamentals 81

Lemma 2.9.1 (F. Abdmouleh, S. Charﬁ, and A. Jeribi [12, Lemma 3.2]) 0 0 Let A ∈ C (X),B ∈ L (X), λ ∈ ΦA\Φ (A) and µ ∈ ΦB\Φ (B). If there exist a positive integer n and a Fredholm perturbation F1, such that

B : D(An) −→ D(A) and

ABx = BAx + F1x, for all x ∈ D(An). Then, there exists a Fredholm perturbation F depending analytically on λ and µ such that 0 0 0 0 Rλ (A)Rµ (B) = Rµ (B)Rλ (A) + F. ♦

Definition 2.9.3 A set D in the complex plane is called a Cauchy domain, if the following conditions are satisfied: (i) D is open, (ii) D has a finite number of components, of which the closures of any two are disjoint, and (iii) the boundary of D is composed of a finite positive number of closed rectifiable Jordan curves, of which any two are unable to intersect.

♦

Theorem 2.9.1 (A. E. Taylor [148, Theorem 3.3]) Let F and ∆ be point sets in the plane. Let F be closed, ∆ be open and F ⊂ ∆. Suppose that the boundary ∂∆ of ∆ is nonempty and bounded. Then, there exists a Cauchy domain D, such that (i) F ⊂ D, (ii) D ⊂ ∆, (iii) the curves forming ∂D are polygons, and (iv) D is unbounded if ∆ is unbounded. ♦ 82 Linear Operators and Their Essential Pseudospectra

0 Deﬁnition 2.9.4 Let f ∈ R∞(A). The class of operators F (A) will be deﬁned as follows: B ∈ F (A) if Z 1 0 B = f (∞) + f (λ)Rλ (A)dλ, 2πi +∂D where f (∞) := lim f (λ) λ→∞ and D is an unbounded Cauchy domain such that C\ΦA ⊂ D, D ⊂ ∆( f ), and the boundary of D, ∂D, does not contain any points of Φ0(A). ♦

Theorem 2.9.2 (R. M. Gethner and J. H. Shapiro [71, Theorem 7]) Let

B1,B2 ∈ F (A). Then, , B1 − B2 = K where K ∈ K (X). ♦

0 Deﬁnition 2.9.5 Let f ∈ R∞(A). By f (A) we mean an arbitrary operator in the set F (A). ♦

Theorem 2.9.3 (R. M. Gethner and J. H. Shapiro [71, Theorem 9]) Let 0 f (λ) and g(λ) be in R∞(A). Then,

f (A).g(A) = ( f .g)(A) + K, where K ∈ K (X). ♦

Deﬁnition 2.9.6 Let A ∈ L (X). By R0(A) we mean the family of all analytic functions, f (λ), such that

C\ΦA ⊂ ∆( f ). ♦

Deﬁnition 2.9.7 Let f ∈ R0(A). The class of operators F ∗(A) will be deﬁned as follows: B ∈ F ∗(A) if Z 1 0 B = f (λ)Rλ (A)dλ, 2πi +∂D where D is a bounded Cauchy domain such that C\ΦA ⊂ D, D ⊂ ∆( f ), and ∂D does not contain any points of Φ0(A). ♦ Fundamentals 83

Deﬁnition 2.9.8 Let f ∈ R0(A). By f ∗(A) we mean an arbitrary operator in the set F ∗(A). ♦

Theorem 2.9.4 (R. M. Gethner and J. H. Shapiro [71, Theorem 12]) Let ∗ B1,B2 ∈ F (A). Then,

B1 − B2 = K, K ∈ K (X). ♦

Theorem 2.9.5 (R. M. Gethner and J. H. Shapiro [71, Theorem 13]) Let A ∈ L (X), and let f (λ) = 1. Then, f ∗(A) = I + K, K ∈ K (X). ♦

Theorem 2.9.6 (R. M. Gethner and J. H. Shapiro [71, Theorem 14]) Let A ∈ L (X), and let f (λ) = λ. Then, f ∗(A) = A + K, K ∈ K (X). ♦

Lemma 2.9.2 (R. M. Gethner and J. H. Shapiro [71, Lemma 7.4]) Let µi ∈ 0 0 −1 Φ (A) (Φ (A) being the set of all λ ∈ Φi(A) such that ∈ σ(Ti) and i i λ−λi Ti is deﬁned in Eq. (2.28)). Let D be a bounded Cauchy domain with the 0 following D ⊂ Φi(A), µi ∈ D, and no other points of Φ (A) are contained in D. Then, Z 1 0 Rλ (A)dλ = K ∈ K (X). ♦ 2πi +∂D Lemma 2.9.3 (J. Shapiro and M. Snow [144, Lemma 1.1]) Let A ∈ C (X), such that ΦA is not empty, and let n be a positive integer. Then, for each 0 λ ∈ ΦA\Φ (A), there exists a subspace Vλ dense in X and depending on

λ such that, for all x ∈ Vλ , we have 0 n Rλ (A)x ∈ D(A ). ♦

2.10 LIMIT INFERIOR AND SUPERIOR

Let S be the collection of all non-empty compact subsets of C. It is well known that the convergence of a sequence in S with respect to 84 Linear Operators and Their Essential Pseudospectra the Hausdorff metric can be characterized through the concepts of limit inferior and superior. Let {En}n be a sequence of arbitrary subsets of C and, deﬁne the limits inferior and superior of {En}n, denoted respectively by liminfEn and limsupEn, as follows: n liminfEn = λ ∈ C, for every ε > 0, there exists N ∈ N such that T o B(λ,ε) En 6= /0for all n ≥ N , and n limsupEn = λ ∈ C, for every ε > 0, there exists J ⊂ N inﬁnite such T o that B(λ,ε) En 6= /0for all n ∈ J . We recall the following properties of limit inferior and superior.

Theorem 2.10.1 Let {En}n be a sequence of non-empty subsets of C. The following properties of limit inferior and superior are known: (i) liminfEn and limsupEn are closed subsets of C. (ii) λ ∈ limsupEn if, and only if, there exists an increasing sequence ∗ of natural numbers n1 < n2 < n3 < ··· and points λnk ∈ Enk , for all k ∈ N , such that

limλnk = λ.

(iii) λ ∈ liminfEn if, and only if, there exists a sequence {λn}n such that λn ∈ En for all n ∈ N, and

limλn = λ.

(iv) Suppose E, En ∈ S for all n ∈ N, and there exists K ∈ S such that En ⊂ K, for all n ∈ N. Then,

En → E in the Hausdorff metric if, and only if,

limsupEn ⊂ E and

E ⊂ liminfEn. ♦ Chapter 3

Spectra

In this chapter, we investigate the essential spectra of the closed, densely deﬁned linear operators on Banach space.

3.1 ESSENTIAL SPECTRA 3.1.1 Deﬁnitions

It is well known that if A is a self-adjoint operator on a Hilbert space, then the essential spectrum of A is the set of limit points of the spectrum of A (with eigenvalues counted according to their multiplicities), i.e., all points of the spectrum except isolated eigenvalues of finite algebraic multiplicity (see, for instance, Refs. [1, 136]). Irrespective of whether A is bounded or not on a Banach space X, there are several definitions of the essential spectrum, most of which constitute an enlargement of the continuous spectrum. Let us define the following sets: 86 Linear Operators and Their Essential Pseudospectra

\ σ j(A) := σ(A + K), K∈W ∗(X) n o σe1(A) := λ ∈ C such that λ − A 6∈ Φ+(X) := C\Φ+(X), n o σe2(A) := λ ∈ C such that λ − A 6∈ Φ−(X) := C\Φ−(X), T σe3(A) := σe1(A) σe2(A), n o σel(A) := λ ∈ C such that λ − A ∈/ Φl(X) := C\ΦlA, n o σer(A) := λ ∈ C such that λ − A ∈/ Φr(X) := C\ΦrA, n o σe4(A) := λ ∈ C such that λ − A ∈/ Φ(X) := C\ΦA, \ σe5(A) := σ(A + K), K∈K (X) n o σe6(A) := C\ λ ∈ ρ5(A) such that all scalars near λ are in ρ(A)

:= C\ρ6(A), \ σe7(A) := σap(A + K), K∈K (X) \ σewl(A) := σl(A + K), K∈K (X) \ σewr(A) := σr(A + K), K∈K (X) \ σe8(A) := σδ (A + K), K∈K (X) where W ∗(X) stands for each one of the sets W (X) or S (X),

ρ5(A) = {λ ∈ C such that λ −A is a Fredholm operator and i(λ −A) = 0}, n o σap(A) = λ ∈ C such that inf k(λ − A)xk = 0 , x∈D(A) kxk=1 n o σl(A) := λ ∈ C such that λ − A is not left invertible , n o σr(A) := λ ∈ C such that λ − A is not right invertible , and n o σδ (A) = λ ∈ C such that λ − A is not surjective .

The subset σ j(·) is the Jeribi’s essential spectrum [1]. σe1(·) and

σe2(·) are the Gustafson and Weidman’s essential spectra, respectively Spectra 87

[79]. σe3(·) is the Kato’s essential spectrum [108]. σe4(·) is the Wolf’s essential spectrum [79, 139, 160]. σe5(·) is the Schechter’s essential spectrum [79, 86–88, 139] and σe6(·) denotes the Browder’s essential spectrum [79, 96, 139]. σe7(·) was introduced by V. Rakocevi˘ c´ in [131] and designated the essential approximate point spectrum, σe8(·) is the essential defect spectrum and was introduced by V. Schmoeger [143], and σap(·) is the approximate point spectrum. Let us notice that all these sets are closed and, in general, we have

\ σe3(A) := σe1(A) σe2(A) ⊆ σe4(A) ⊆ σe5(A) ⊆ σe6(A),

[ σe5(A) = σe7(A) σe8(A),

σe1(A) ⊂ σe7(A), and

σe2(A) ⊂ σe8(A). It is proved in [1] that

σ j(A) ⊂ σe5(A). However, if X is a Hilbert space and A is self-adjoint, then all these sets coincide.

Remark 3.1.1 (i) If λ ∈ σc(A) (the continuous spectrum of A), then R(λ −A) is not closed (otherwise λ ∈ ρ(A) see [137, Lemma 5.1 p. 179]).

Therefore, λ ∈ σei(A), i = 1,··· ,6. Consequently, we have

6 \ σc(A) ⊂ σei(A). i=1 If the spectrum of A is purely continuous, then

σ(A) = σc(A) = σei(A) i = 1,··· ,6.

(ii) σe5(A + K) = σe5(A) for all K ∈ K (X). 88 Linear Operators and Their Essential Pseudospectra

(iii) Let E0 be a core of A, i.e., a linear subspace of D(A) such that the closure A|E0 of its restriction A|E0 equals A. Then, σap(A) := λ ∈ C such that inf k(λ − A)xk = 0 . kxk=1, x∈E0

(iv) Set αe(A) := inf{kAxk such that x ∈ D(A) and kxk = 1}. Whenever E0 is a core of A, then

αe(A) := inf{kAxk such that x ∈ E0 and kxk = 1} holds. Using this quantity, we obtain

σap(A) = {λ ∈ C such that αe(λ − A) = 0}. ♦

Lemma 3.1.1 Let A ∈ C (X) such that 0 ∈ ρ(A). Then, λ ∈ σei(A) if, and 1 −1 only if, λ ∈ σei(A ) i = 7, 8. ♦

3.1.2 Characterization of Essential Spectra

The following proposition gives a characterization of the Schechter essential spectrum by means of Fredholm operators.

Proposition 3.1.1 (M. Schechter [137, Theorem 5.4, p. 180]) Let X be a 0 Banach space and let A ∈ C (X). Then, λ 6∈ σe5(A) if, and only if, λ ∈ ΦA, 0 where ΦA = {λ ∈ ΦA such that i(λ − A) = 0}. ♦

∗ For n ∈ N , let

n In(X) = {K ∈ L (X) satisfying f ((KB) ) < 1 for all B ∈ L (X)}, where f (·) is a measure of non-strict-singularity given in (2.22).

Theorem 3.1.1 (N. Moalla [122]) Let A ∈ Φ(X), then for all K ∈ In(X), we have (i) K + A ∈ Φ(X) and i(K + A) = i(A),

(ii) σe5(A + K) = σe5(A). ♦ Spectra 89

Theorem 3.1.2 (N. Moalla [122]) Let A,B ∈ C (X) such that ρ(A)Tρ(B) 6= /0. If, for some λ ∈ ρ(A)Tρ(B), the operator (λ − A)−1 − (λ − B)−1 ∈

In(X), then

σe5(A) = σe5(B). ♦

We recall some properties of σe7(·) (resp. σe8(·)) due to V. Rakocevi˘ c´ (resp. C. Schmoeger), given in [143].

Proposition 3.1.2 (C. Schmoeger [143]) Let A ∈ L (X), then

(i) σe7(A) 6= /0,

(ii) σe8(A) 6= /0, and

(iii) σe7(A) and σe8(A) are compact. ♦

The following proposition gives a characterization of the essential approximate point spectrum and the essential defect spectrum by means of upper semi-Fredholm and lower semi-Fredholm operators, respectively.

Proposition 3.1.3 (A. Jeribi and N. Moalla [98, Proposition 3.1]) Let A ∈ C (X). Then,

(i) λ 6∈ σe7(A) if, and only if, λ − A ∈ Φ+(X) and i(λ − A) ≤ 0,

(ii) λ 6∈ σe8(A) if, and only if, λ − A ∈ Φ−(X) and i(λ − A) ≥ 0, and ∗ (iii) if A is a bounded linear operator, then σe8(A) = σe7(A ), where A∗ stands for the adjoint operator of A. This is equivalent that

[n o σe7(A) = σe1(A) λ ∈ C such that i(A − λ) > 0 , and [n o σe8(A) = σe4(A) λ ∈ C such that i(A − λ) < 0 .

If, in addition, ΦA is connected and ρ(A) 6= /0, then

σe1(A) = σe7(A), and

σe4(A) = σe8(A). ♦ 90 Linear Operators and Their Essential Pseudospectra

Theorem 3.1.3 (M. A. Kaashoek and D. C. Lay [105]) Let A be a closed linear operator on a Banach space X. Then, σe6(A) is the largest subset of the spectrum of A which remains invariant under perturbations of A by Riesz operators which commute with A. ♦

Theorem 3.1.4 (F. Abdmouleh and A. Jeribi [14]) Let A and B be two bounded linear operators on a Banach space X. Then, (i) If AB ∈ F b(X), then

h [ i σei(A + B)\{0} ⊂ σei(A) σei(B) \{0}, i = 4, 5.

Furthermore, if BA ∈ F b(X), then

h [ i σe4(A + B)\{0} = σe4(A) σe4(B) \{0}.

Moreover, if C\σe4(A) is connected, then h [ i σe5(A + B)\{0} = σe5(A) σe5(B) \{0}.

(ii) If the hypotheses of (i) are satisﬁed, and if C\σe5(A + B), C\σe5(A) and C\σe5(B) are connected, then S σe6(A + B)\{0} = [σe6(A) σe6(B)]\{0}. ♦

Theorem 3.1.5 Let A,B ∈ C (X) and let λ ∈ ρ(A)Tρ(B). If (λ − A)−1 − (λ − B)−1 ∈ F b(X), then

σei(A) = σei(B), i = 4,5. ♦

Theorem 3.1.6 [14,96] Let A ∈ C (X) such that ρ(A) is not empty. Then,

(i) If C\σe4(A) is connected, then

σe4(A) = σe5(A).

(ii) If C\σe5(A) is connected, then

σe5(A) = σe6(A). ♦ Spectra 91

3.2 THE LEFT AND RIGHT JERIBI ESSENTIAL SPECTRA

In this section, we will give fine description of the definition of the left and right Jeribi essential spectra of a closed densely defined linear operators. For this, let X be a Banach space and let A ∈ C (X), we denote the left Jeribi essential spectrum by

l \ σ j(A) = σl(A + K) K∈W ∗(X) and, the right Jeribi essential spectrum by

r \ σ j (A) = σr(A + K). K∈W ∗(X) For A ∈ C (X), we observe that K (X) ⊂ W ∗(X). Then,

l σ j(A) ⊆ σewl(A) and r σ j (A) ⊆ σewr(A). One of the central questions in the study of the left and right essential spectra of closed densely deﬁned linear operators on Banach space X consists of showing what are the conditions that we must impose on space X such that for A ∈ C (X),

l σewl(A) = σ j(A) and r σewr(A) = σ j (A). If X is a reﬂexive Banach space, then L (X) = W (X). So, the left or right Jeribi essential spectrum is the smallest essential spectrum in the sense of inclusion of the other essential spectra.

The set of Browder essential spectrum [1, 70] of A is characterized by n o σe6(A) := C\ λ ∈ C such that λ − A ∈ B(X) . 92 Linear Operators and Their Essential Pseudospectra

3.3 S-RESOLVENT SET, S-SPECTRA, AND S-ESSENTIAL SPECTRA 3.3.1 The S-Resolvent Set

Let X and Y be two Banach space. Let S ∈ L (X,Y), for A ∈ C (X,Y) such that A 6= S and S 6= 0, we deﬁne the S-resolvent set of A by n o ρS(A) = λ ∈ C such that λS − A has a bounded inverse .

We denote by −1 RS(λ,A) := (λS − A) and call it the S-resolvent operator of A. The name “S-resolvent” is appro- priate, since RS(λ,A) helps to solve the equation

(λS − A)x = y.

Thus, −1 x = (λS − A) y = RS(λ,A)y provided RS(λ,A) exists.

If ρS(A) is not empty, then A is closed. Indeed, let (xn)n ∈ D(A) such that xn → x and Axn → y. Since ρS(A) 6= /0, then there exist λ0 ∈ ρS(A) such −1 that (A − λ0S) ∈ L (X). Since S ∈ L (X), then

(A − λ0S)xn → y − λ0Sx.

Thus, −1 xn → (A − λ0S) (y − λ0Sx) = x, we deduce that Ax = y and x ∈ D(A).

Lemma 3.3.1 For all λ, µ ∈ ρS(A), we have

RS(λ,A) − RS(µ,A) = (µ − λ)RS(λ,A)SRS(µ,A). ♦ Spectra 93

Proof. Let λ, µ ∈ ρS(A), we have −1 −1 RS(λ,A) − RS(µ,A) = (λS − A) (µS − A) − (λS − A) (µS − A) ,

= (µ − λ)RS(λ,A)SRS(µ,A), which completes the proof of lemma. Q.E.D.

Proposition 3.3.1 Let A ∈ C (X,Y),S ∈ L (X,Y) such that S 6= 0 and S 6= A, we have

(i) the S-resolvent set ρS(A) is open, and (ii) the function

ϕ : λ −→ RS(λ,A) is holomorphic at any point of ρS(A). ♦

Proof. (i) If ρS(A) = /0,then ρS(A) is open. If ρS(A) 6= /0,then λ0 ∈ ρS(A). It sufﬁcient to ﬁnd ε > 0 such that

B(λ0,ε) ⊂ ρS(A), where B(λ0,ε) designates the open ball centered at λ0 with radius ε. We can write for any λ ∈ C

λS − A = λ0S − A + (λ − λ0)S,

= (λ0S − A)[I + (λ − λ0)RS(λ0,A)S].

If k(λ − λ0)RS(λ0,A)Sk < 1, then

I + (λ − λ0)RS(λ0,A)S is invertible. This is equivalent to say that if 1 |λ − λ0| < , kRS(λ0,A)Sk

1 then λ ∈ ρS(A). Consequently, it sufﬁces to take ε = . kRS(λ0,A)Sk 94 Linear Operators and Their Essential Pseudospectra

(ii) Let λ, µ ∈ ρS(A), we have

ϕ(µ) − ϕ(λ) R (µ,A) − R (λ,A) = S S , µ − λ µ − λ R (µ,A)(λ − µ)SR (λ,A) = S S , µ − λ = − RS(µ,A)SRS(λ,A).

Hence,

ϕ(µ) − ϕ(λ) lim = −RS(λ,A)SRS(λ,A) ∈ L (Y,X), µ−→λ µ − λ which completes the proof of proposition. Q.E.D.

Remark 3.3.1 If A ∈ L (X) and S is an invertible bounded operator, then

−1 \ −1 ρS(A) = ρ(S A) ρ(AS ).

Because λS − A = S(λ − S−1A), and λS − A = (λ − AS−1)S,

−1 T −1 it follows that λ ∈ ρS(A) if, and only if, λ ∈ ρ(S A) ρ(AS ). ♦

Theorem 3.3.1 Let A ∈ C (X,Y),S ∈ L (X,Y) and λ ∈ C. If λS − A is one-to-one and onto, then (λS − A)−1 is a bounded linear operator. ♦

− Proof. Let λ ∈ C, we shall ﬁrst prove that (λS − A) 1 is a closed linear operator. For this, we take the sequence (yn)n ⊂ Y such that ( yn → y in Y, −1 (λS − A) yn → x in X.

−1 In the following, we set xn = (λS − A) yn, then xn ∈ D(A) and

yn = (λS − A)xn ∈ Y. Spectra 95

Since (λS−A)xn → y, xn → x and λS−A is a closed linear operator which implies that x ∈ D(A) and

y = (λS − A)x.

So, y ∈ Y and x = (λS − A)−1y. This proves that (λS−A)−1 is a closed operator. Furthermore, (λS−A)−1 is a closed operator deﬁned on all Y. Using the closed graph theorem (see Theorem 2.1.1), we obtain that

(λS − A)−1 is a bounded linear operator. Q.E.D.

Proposition 3.3.2 Let A and S ∈ L (X) such that S 6= A and S 6= 0. Then,

(i) If S is an operator which commutes with A, then for any λ ∈ ρS(A), we have ARS(λ,A) = RS(λ,A)A,

RS(λ,A)S = SRS(λ,A).

(ii) If S commutes with A, then for every λ, µ ∈ ρS(A), we have

RS(λ,A)RS(µ,A) = RS(µ,A)RS(λ,A). ♦

Proof. (i) Since, by hypothesis, S commutes with A, then the last two operators commute with the operator λS−A. In addition, since λ ∈ ρS(A), we obtain that S and A commute with RS(λ,A).

(ii) Let λ, µ ∈ ρS(A), we have h i−1 RS(λ,A)RS(µ,A) = (µS − A)(λS − A) , h i−1 = λS(µS − A) − A(µS − A) , h i−1 = (λS − A)(µS − A) ,

= RS(µ,A)RS(λ,A), which completes the proof of proposition. Q.E.D. 96 Linear Operators and Their Essential Pseudospectra

Theorem 3.3.2 Let A and S ∈ L (X) such that S 6= A and S 6= 0. If

S commutes with A, then for any λ and λ0 ∈ ρS(A) with |λ − λ0| < 1 , we have kRS(λ0,A)Sk n n n+1 RS(λ,A) = ∑ (λ0 − λ) S RS(λ0,A) . ♦ n≥0

1 Proof. Let λ, λ0 ∈ ρS(A) such that |λ − λ0| < . Then, kRS(λ0,A)Sk

−1 RS(λ,A) = (λ0S − A − λ0S + λS) , h i−1 = λ0S − A I − (λ0 − λ)RS(λ0,A)S , h i−1 = I − (λ0 − λ)RS(λ0,A)S RS(λ0,A), n n = ∑ (λ0 − λ) (RS(λ0,A)S) RS(λ0,A). n≥0

According to the Proposition 3.3.2 (i),

RS(λ0,A)S = SRS(λ0,A). (3.1)

So, Eq. (3.1) implies that

n n n (RS(λ0,A)S) = S RS(λ0,A) .

Consequently,

n n n+1 RS(λ,A) = ∑ (λ0 − λ) S RS(λ0,A) , n≥0 which completes the proof of theorem. Q.E.D.

Theorem 3.3.3 Let A and S ∈ L (X) such that S 6= A and S 6= 0. If S commutes with A, then dn R (λ,A) = (−1)n n! SnR (λ,A)n+1 ∀ λ ∈ ρ (A). ♦ dλ n S S S Spectra 97

Proof. We argue by recurrence. We know that the function

ϕ : λ −→ RS(λ,A) is holomorphic at any point λ of ρS(A). Then, in view of Proposition 3.3.2 (i), we have

d RS(µ,A) − RS(λ,A) RS(λ,A) = lim , dλ µ→λ µ − λ = lim −RS(µ,A)SRS(λ,A), µ→λ 2 = −SRS(λ,A) .

For n = 1, the equality is true. We assume that is true until the order n and we prove that it remains true to the order n + 1. We will show that for any λ ∈ ρS(A),

n+1 d n+1 n+1 n+2 RS(λ,A) = (−1) (n + 1)! S RS(λ,A) . dλ n+1 In fact,

dn+1 d dn RS(λ,A) = RS(λ,A) , dλ n+1 dλ dλ n n dn d R (µ,A) − R (λ,A) dµn S n S = lim dλ , µ→λ µ − λ R (µ,A)n+1 − R (λ,A)n+1 = lim (−1)n n! Sn S S . µ→λ µ − λ

According to the Proposition 3.3.2 (ii), we have

RS(λ,A)RS(µ,A) = RS(µ,A)RS(λ,A) ∀ λ, µ ∈ ρS(A).

So, we can write

n+1 n+1 n RS(µ,A) − RS(λ,A) = RS(µ,A) − RS(λ,A) RS(µ,A) n−1 n−2 2 n +RS(µ,A) RS(λ,A) + RS(µ,A) RS(λ,A) + ··· + RS(λ,A) . This implies that 98 Linear Operators and Their Essential Pseudospectra

dn+1 RS(λ,A) dλ n+1 n n RS(µ,A) − RS(λ,A) RS(µ,A) + ··· + RS(λ,A) = lim (−1)n n! Sn , µ→λ µ − λ n n 2 n = (−1) n! S − SRS(λ,A) (n + 1) RS(λ,A) , n+1 n+1 n+2 = (−1) (n + 1)! S RS(λ,A) , which completes the proof of theorem. Q.E.D.

3.3.2 S-Spectra

Let A ∈ C (X) and S ∈ L (X) such that S 6= A and S 6= 0. We deﬁne the S-spectrum of A by σS(A) := C\ρS(A).

We see the following examples, where σS(A) can be discrete or the whole complex plane: ! ! 2 2 1 0 (i) Let A = and S = , then σS(A) = {2}. 0 3 0 0 ! ! 1 2 1 0 (ii) Let A = and S = , then σS(A) = C. 0 0 0 0 ! ! 1 2 0 0 (iii) Let A = and S = , then σS(A) = /0. 0 3 0 1 The S-left spectrum of A is deﬁned by n o σl,S(A) = λ ∈ C such that λS − A ∈/ Gl(X) and the S-right spectrum of A is deﬁned by n o σr,S(A) = λ ∈ C such that λS − A ∈/ Gr(X) .

It is clear that [ σS(A) = σl,S(A) σr,S(A). Spectra 99

Corollary 3.3.1 Let A be a linear operator and S be a non null bounded linear operator from X into Y. If A is a non closed operator, then

σS(A) = C. ♦

Proof. Let A be a linear operator which is not closed. We argue by contradiction. Suppose that ρS(A) is not empty, then there exists λ ∈ C such that −1 λ ∈ ρS(A) and consequently, (λS − A) is a bounded operator. Hence, λS − A is a closed operator. In addition, we can write

A = A − λS + λS.

We conclude that A is a closed operator, which is a contradiction. Q.E.D.

Deﬁnition 3.3.1 Let A ∈ C (X) and S ∈ L (X) such that S 6= A and S 6= 0. We deﬁne the following sets: (i) The S-point spectrum of A and it is denoted by n o σp,S(A) = λ ∈ C such that λS − A is not one-to-one .

(ii) The S-continuous spectrum of A and it is deﬁned by n σc,S(A) = λ ∈ C : λS − A is one-to-one, (λS − A)(D(A)) = X, o and (λS − A)−1 is unbounded . (iii) The S-residual spectrum of A and it is denoted by n o σR,S(A) = λ ∈ C : λS − A is one-to-one, (λS − A)(D(A)) 6= X .

(iv) The S-approximate point spectrum and it is denoted by

σap,S(A) = λ ∈ : ∃ xn ∈ D(A), kxnk = 1 and lim k(λS − A)xnk = 0 . ♦ C n→+∞

Remark 3.3.2 Let A ∈ C (X) and S ∈ L (X) such that S 6= A and S 6= 0, then the S-spectrum, σS(A), is partitioned into three disjoint sets as follows S S σS(A) = σp,S(A) σc,S(A) σR,S(A). ♦ 100 Linear Operators and Their Essential Pseudospectra

3.3.3 S-Browder’s Resolvent

Deﬁnition 3.3.2 Let A ∈ C (X) and λ0 be an isolated point of σS(A). For an admissible contour Γλ0 , I S −1 Pλ0,S = − (A − λS) dλ, 2πi Γ λ0 is called the S-Riesz integral for A, S and λ0 with range and kernel denote by Rλ,S and Kλ,S, respectively. ♦

The S-discrete spectrum of A, denoted σdS (A), is just the set of isolated points λ ∈ C of the spectrum such that the corresponding S-Riesz projec- tors Pλ,S are ﬁnite-dimensional. Another part of the spectrum, which is generally larger than σe6,S(A), is σS(A)\σdS (A). We will also use this terminology here and the notation

σe6,S(A) = σS(A)\σdS (A) and

ρ6,S(A) = C\σe6,S(A). The largest open set on which the resolvent is ﬁnitely meromorphic is precisely [ ρ6,S(A) = σdS (A) ρS(A).

For λ ∈ ρ6,S(A), let Pλ,S(A) (or Pλ,S) denotes the corresponding (ﬁnite rank) S-Riesz projector with a range and a kernel denoted by Rλ,S and

Kλ,S, respectively. Since Pλ,S is invariant, we may deﬁne the operator

Aλ,S = (A − λS)(I − Pλ,S) + Pλ,S, (3.2) with respect to the decomposition

M X = Kλ,S Rλ,S and M A = ((A − λS) ) I. λ,S |Kλ,S Spectra 101

We have just cut off the ﬁnite-dimensional part of A − λS in the S-Riesz decomposition. Since

σ(A − λS) = σ(A − λS)\{0}, |Kλ,S

Aλ,S has a bounded inverse, denoted by Rb,S(A,λ) and called the “S- Browder’s resolvent”, i.e.,

M R (A,λ) = ((A − λ) )−1 I b,S |Kλ,S with respect to M X = Kλ,S Rλ,S. Or, alternatively

R (A,λ) = (A − λS) )−1(I − P ) + P b,S |Kλ,S λ,S λ,S for λ ∈ ρ6,S(A). This clearly extends the usual resolvent RS(A,λ) from

ρS(A) to ρ6,S(A) and retains many of its important properties.

Proposition 3.3.3 Let A ∈ C (X) and S ∈ L (X) such that S 6= A and S 6= 0.

Then, for any µ, λ ∈ ρ6,S(A), we have

Rb,S(A,λ) − Rb,S(A, µ) = (λ − µ)Rb,S(A,λ)SRb,S(A, µ) + M (λ, µ), (3.3) where M (λ, µ) is a ﬁnite rank operator with the following expression M (λ, µ) = Rb,S(A,λ) (A−(λS+I))Pλ,S −(A−(µS+I))Pµ,S Rb,S(A, µ) (3.4) is a ﬁnite rank operator with rank(M (λ, µ)) = rank(Pλ,S)+rank(Pµ,S) in case λ 6= µ. ♦ 102 Linear Operators and Their Essential Pseudospectra

Proof. We have

Rb,S(A,λ) − Rb,S(A, µ) = Rb,S(A,λ)[Aµ,S − Aλ,S]Rb,S(A, µ), where

Aµ,S := (A − µS)(I − Pµ,S) + Pµ,S. So,

Aµ,S − Aλ,S = [(A − µS)(I − Pµ,S) + Pµ,S] − [(A − λS)(I − Pλ,S) + Pλ,S]

= [(A − (λS + I))Pλ,S − (A − (µS + I))Pµ,S] + (λ − µ)S.

Therefore,

Rb,S(A,λ) − Rb,S(A, µ) = (λ − µ)Rb,S(A,λ)SRb,S(A, µ) + M (λ, µ).

Q.E.D.

Proposition 3.3.4 Let X and Y be two complex Banach spaces. Let A ∈ C (X),S ∈ L (X),B ∈ L (Y,X), and C : X −→ Y be two linear operators. Then,

(i) Rb,S(A, µ)B is continuous for some µ ∈ ρ6,S(A) if, and only if, it is continuous for all µ ∈ ρ6,S(A). −1 (ii) C(A − µS) is bounded for some λ ∈ ρS(A) if, and only if,

CRb,S(A, µ) is bounded for some (hence for every) µ ∈ ρ6,S(A). (iii) If B and C satisfy the conditions (i) and (ii), respectively, and B is densely deﬁned, then CM (λ, µ), M (λ, µ)B, and CM (λ, µ)B are operators of ﬁnite rank for any µ, λ ∈ ρ6,S(A). ♦

Proof. From the resolvent identity, we have for any µ, λ ∈ ρS(A),

(A − λS)−1C = (A − µS)−1C + (λ − µ)(A − λS)−1S(A − µS)−1C, (3.5) and for any µ, λ ∈ ρ6,S(A),

Rb,S(A,λ)B = Rb,S(A, µ)B+(λ − µ)Rb,S(A,λ)SRb,S(A, µ)B+M (λ, µ)B, (3.6) Spectra 103 and

CRb,S(A,λ) = CRb,S(A, µ)+(λ −µ)CRb,S(A,λ)SRb,S(A, µ)+CM (λ, µ). (3.7) (i) Since S is bounded, then

Rb,S(A,λ)SRb,S(A, µ)B is bounded. According to Proposition 3.3.3 that the operator

(A − (λS + I))Pλ,S − (A − (µS + I))Pµ,S is bounded. Thus, M (λ, µ)B has ﬁnite dimensional range, and in view of Eq. (3.6), we have

Rb,S(A,λ)B − Rb,S(A, µ)B is bounded. Hence, Rb,S(A, µ)B is continuous for some µ ∈ ρ6,S(A) if, and only if, it is continuous for all µ ∈ ρ6,S(A).

(ii) If CRb,S(A,λ) is bounded for some λ ∈ ρ6,S(A), then clearly CRb,S(A, µ) is also bounded for any µ and, it follows from the Eq. (3.7) that

CRb,S(A, µ) is bounded for any µ. By using Eq. (3.5) we have C(A− µS)−1 is bounded for every µ ∈ ρS(A). (iii) According to Proposition 3.3.3 that the operator M (λ, µ) is a finite rank operator. So, CM (λ, µ) and M (λ, µ)B are a finite rank operator. Hence, it is clear that M (λ, µ)B is of finite rank if B is densely defined. Since CM (λ, µ)B = CRb,S(A,λ) (A − (λS + I))Pλ,S− (A − (µS + I))Pµ,S Rb,S(A, µ)B

= CRb,S(A,λ)[(A − λS)(I − Pλ,S) + Pλ,S]Rb,S(A, µ)B 104 Linear Operators and Their Essential Pseudospectra and if B and C satisfy the conditions (i) and (ii), respectively, then

CM (λ, µ)B will again be continuous and densely deﬁned with ﬁnite-dimensional range. Q.E.D.

3.3.4 S-Essential Spectra

Let A ∈ C (X) and S ∈ L (X) such that S 6= 0 and S 6= A. We are concerned with the following S-essential spectra (see [1, 10, 101]) deﬁned by the following sets

σe1,S(A) := {λ ∈ C such that λS − A ∈/ Φ+(X)} := C\Φ+A,S, σe2,S(A) := {λ ∈ C such that λS − A ∈/ Φ−(X)} := C\Φ−A,S, σe3,S(A) := {λ ∈ C such that λS − A ∈/ Φ±(X)}, σe4,S(A) := {λ ∈ C such that λS − A ∈/ Φ(X)}, \ σe5,S(A) := σS(A + K), K∈K (X) σe6,S(A) := {λ ∈ C such that λS − A ∈/ B(X)}, σel,S(A) := {λ ∈ C such that λS − A ∈/ Φl(X)}, σer,S(A) := {λ ∈ C such that λS − A ∈/ Φr(X)}, σewl,S(A) := {λ ∈ C such that λS − A ∈/ Wl(X)}, σewr,S(A) := {λ ∈ C such that λS − A ∈/ Wr(X)}, l σbl,S(A) := {λ ∈ C such that λS − A ∈/ B (X)}, r σbr,S(A) := {λ ∈ C such that λS − A ∈/ B (X)}, n σeap,S(A) := λ ∈ C such that λS − A 6∈ Φ+(X) and o i(λS − A) ≤ 0 := C\ρeap,S(A), n σeδ,S(A) := λ ∈ C such that λS − A 6∈ Φ−(X) and o i(λS − A) ≥ 0 := C\ρeδ,S(A).

They can be ordered as

\ σe3,S(A) = σe1,S(A) σe2,S(A) ⊆ σe4,S(A) ⊆ σe5,S(A) ⊆ σe6,S(A), Spectra 105

σel,S(A) ⊆ σewl,S(A) ⊆ σbl,S(A), and

σer,S(A) ⊆ σewr,S(A) ⊆ σbr,S(A). In [29], the authors proved that

[ σe4,S(A) = σel,S(A) σer,S(A),

σe1,S(A) ⊂ σel,S(A), and

σe2,S(A) ⊂ σer,S(A).

Remark 3.3.3 (i) If S = I, we recover the usual deﬁnition of the essential spectra of a closed densely deﬁned linear operator A. (ii) If A ∈ L (X) and S is invertible, then −1 S −1 σei,S(A) = σei(S A) σei(AS ), i ∈ 1,2,3,4,5,6,ap,δ . ♦

Theorem 3.3.4 (F. Abdmouleh, A. Ammar and A. Jeribi [10]) Let S and A be two bounded linear operators on a Banach space X. Then, S σe5,S(A) = σe4,S(A) λ ∈ C such that i(A − λS) 6= 0 . ♦

S Proof. Let λ 6∈ σe4,S(A) {λ ∈ C such that i(A − λS) 6= 0}. Then,

A − λS ∈ Φb(X) and i(A − λS) = 0.

By applying Theorem 2.2.11, there exists K ∈ K (X) such that A−λS+K is invertible. Hence,

λ ∈ ρS(A + K). This shows that \ λ 6∈ σS(A + K). K∈K (X) 106 Linear Operators and Their Essential Pseudospectra

Then, we have

[ σe5,S(A) ⊂ σe4,S(A) {λ ∈ C such that i(A − λS) 6= 0 . (3.8)

To prove the inverse inclusion of Eq. (3.8). Suppose λ 6∈ σe5,S(A), then there exists K ∈ K (X) such that λ ∈ ρS(A + K). Hence, A − λS + K ∈ Φb(X) and i(A − λS + K) = 0.

Now, the operator A − λS can be written in the form

A − λS = A + K − λS − K.

Since K ∈ K (X), using Lemma 2.3.1, we get A − λS ∈ Φb(X) and

i(A − λS) = i(A + K − λS) = 0.

We conclude that

[ λ 6∈ σe4,S(A) {λ ∈ C such that i(A − λS) 6= 0}.

Hence,

[ σe4,S(A) λ ∈ C such that i(A − λS) 6= 0 ⊂ σe5,S(A).

Therefore,

[ σe5,S(A) = σe4,S(A) λ ∈ C such that i(A − λS) 6= 0 .

This completes the proof. Q.E.D.

Proposition 3.3.5 Let A and S ∈ L (X) such that S 6= A. If C\σe4,S(A) is connected and ρS(A) is not empty, then

σe5,S(A) = σe4,S(A). ♦ Spectra 107

Proof. (i) By using Theorem 3.3.4, we have

[ σe5,S(A) = σe4,S(A) {λ ∈ C such that i(A − λS) 6= 0}.

Hence,

σe4,S(A) ⊂ σe5,S(A).

Conversely, let λ ∈ σe5,S(A). It sufﬁces to show that

{λ ∈ C such that i(A − λS) 6= 0} ⊂ σe4,S(A) which is equivalent to

\ [C\σe4,S(A)] {λ ∈ C such that i(A − λS) 6= 0} = /0.

Suppose that

\ [C\σe4,S(A)] {λ ∈ C such that i(A − λS) 6= 0} 6= /0 and let

\ λ0 ∈ [C\σe4,S(A)] {λ ∈ C such that i(A − λS) 6= 0}.

Since ρS(A) 6= /0,then there exists λ1 ∈ C such that λ1 ∈ ρS(A) and conse- b quently, λ1S − A ∈ Φ (X) and

i(λ1S − A) = 0.

On the other side, C\σe4,S(A) is connected, it follows from Proposition 2.2.1 (ii) that i(λS − A) is constant on any component of ΦA,S. Therefore,

i(λ1S − A) = i(λ0S − A) = 0, which is a contradiction. Then,

\ [C\σe4,S(A)] {λ ∈ C such that i(A − λS) 6= 0} = /0.

Hence,

σe5,S(A) ⊂ σe4,S(A). Q.E.D. 108 Linear Operators and Their Essential Pseudospectra

Lemma 3.3.2 (A. Jeribi, N. Moalla, and S. Yengui [101, Lemma 2.1]) Let

A ∈ C (X) and S ∈ L (X), such that ρS(A) is not empty.

(i) If C\σe4,S(A) is connected, then

σe4,S(A) = σe5,S(A).

(ii) If C\σe5,S(A) is connected, then

σe5,S(A) = σe6,S(A). ♦

3.4 INVARIANCE OF THE S-ESSENTIAL SPECTRUM

The following result gives a characterization of the S-essential spectrum by means of a Fredholm operators.

Proposition 3.4.1 Let S ∈ L (X) and A ∈ C (X). Then,

λ 6∈ σe5,S(A) if, and only if, A − λS ∈ Φ(X) and i(A − λS) = 0. ♦

Proof. Let λ 6∈ σe5,S(A). Then, there exists a compact operator K on X such that λ ∈ ρS(A + K). So, A + K − λS ∈ Φ(X) and i(A + K − λS) = 0. Now, the operator A − λS can be written in the form

A − λS = A + K − λS − K.

By Theorem 2.2.14, we have

A − λS ∈ Φ(X) and i(A − λS) = 0. Conversely, we suppose that A − λS ∈ Φ(X) and i(A − λS) = 0. Let n = α(A−λS) = β(A−λS), x1,··· ,xn being the basis for the N((A−λS)) Spectra 109

0 0 ⊥ and y1,··· ,yn being the basis for annihilator R(A − λS) . By Lemma 0 0 ∗ 2.1.1, there are functionals x1,··· ,xn in X (the adjoint space of X) and elements y1,··· ,yn such that

0 0 x j(xk) = δ jk and y j(yk) = δ jk, 1 ≤ j, k ≤ n, where δ jk = 0 if j 6= k and δ jk = 1 if j = k. Consider the operator K defined by n 0 K : X 3 x −→ Kx := ∑ xi(x)yi ∈ X. i=1 Clearly, K is a linear operator defined everywhere on X. It is bounded, since n 0 kKxk ≤ ∑ kxkkkykk kxk. k=1 Moreover, the range of K is contained in a finite dimensional subspace of X. Then, K is a finite rank operator in X (see Lemma 2.1.4). By Lemma 2.1.5, K is a compact operator in X. We may prove that \ \ N(A − λS) N(K) = {0} and R(A − λS) R(K) = {0}. (3.9)

Indeed, let x ∈ N(A − λS), then

n x = ∑ αkxk. k=1 Therefore, 0 x j(x) = α j, 1 ≤ j ≤ n. On the one hand, if x ∈ N(K), then

0 x j(x) = 0, 1 ≤ j ≤ n.

This proves the ﬁrst relation in Eq. (3.9). The second inclusion is similar. In fact, if y ∈ R(K), then n y = ∑ αkyk, k=1 and hence, 0 y j(y) = α j, 1 ≤ j ≤ n. 110 Linear Operators and Their Essential Pseudospectra

But, if y ∈ R(A − λS), then

0 y j(y) = 0, 1 ≤ j ≤ n.

This gives the second relation in Eq. (3.9). On the other hand, K is a compact operator, hence we deduce, from Theorem 2.2.14, that λS − A − K ∈ Φ(X) and i(A − λS − K) = 0.

If x ∈ N(A−λS−K), then (A−λS)x is in R(A−λS)TR(K). This implies that x ∈ N(A − λS)TN(K) and so, x = 0. Thus,

α(A + K − λS) = 0.

In the same way, one proves that

R(A + K − λS) = X.

Using Theorem 2.1.2, we get λ ∈ ρS(A + K). Also,

\ λ 6∈ σS(A + K). K∈ K (X)

So, λ 6∈ σe5,S(A). Q.E.D.

Corollary 3.4.1 Let A ∈ C (X) and S ∈ L (X). If ΦA,S is connected and ρS(A) is not empty, then σe5,S(A) = λ ∈ C such that A − λS 6∈ Φ(X) = C\ΦA,S. ♦

T Theorem 3.4.1 Let S ∈ L (X) and λ ∈ ρS(A) ρS(A + B). If

kxnk + kAxnk + kBxnk ≤ c,

−1 for all xn ∈ D(A) implies that (A − λS) Bxn has a convergent subsequence, then

σe5,S(A + B) = σe5,S(A). (3.10) ♦ Spectra 111

Proof. We employ the identities

(A+B−µS)−(A−µS)(A−λS)−1(A+B−λS) = (µ −λ)S(A−λS)−1B. (3.11)

Since, ρS(A) and ρS(A + B) are not empty, then A and A + B are closed. Hence, A + B is A-bounded. This shows that the hypotheses imply that −1 (A − λS) B is A-compact and (A + B)-compact. Let µ 6∈ σe5,S(A + B), then from Proposition 3.4.1, we get

A + B − µS ∈ Φ(X) and i(A + B − µS) = 0.

By Eq. (3.11), we have

(A − µS)(A − λS)−1(A + B − λS) ∈ Φ(X) and i (A − µS)(A − λS)−1(A + B − λS) = 0.

Since λ ∈ ρS(A + B), then

A + B − λS ∈ Φ(X) and i(A + B − λS) = 0.

Using Theorem 2.2.21, we get

(A − µS)(A − λS)−1 ∈ Φ(X) and i (A − µS)(A − λS)−1 = 0.

From this and the identity

A − µS = (A − µS)(A − λS)−1(A − λS), 112 Linear Operators and Their Essential Pseudospectra we obtain A − µS ∈ Φ(X) and i(A − µS) = 0.

Thus, in view of Proposition 3.4.1, µ 6∈ σe5,S(A) and, we conclude that

σe5,S(A) ⊂ σe5,S(A + B).

Conversely, if µ 6∈ σe5,S(A), then

A − µS ∈ Φ(X) and i(A − µS) = 0.

Since λ ∈ ρS(A + B), then by Proposition 3.4.1, we have

A + B − λS ∈ Φ(X) and i(A + B − λS) = 0.

Thus, (A − µS)(A − λS)−1(A + B − λS) ∈ Φ(X) and i (A − µS)(A − λS)−1(A + B − λS) = 0.

By Eq. (3.11), we have

A + B − µS ∈ Φ(X) and i(A + B − µS) = 0.

Then, in view of Proposition 3.4.1, µ 6∈ σe5,S(A + B), and we obtain

σe5,S(A + B) ⊂ σe5,S(A).

This proves (3.10) and completes the proof. Q.E.D. Spectra 113

Remark 3.4.1 If A and B are bounded operators. Theorem 3.4.1 remains true if we replace (A − λS)−1B by B(A − λS)−1. Indeed, it sufﬁces to replace Eq. (3.11) by (A + B − µS) − (A + B − λS)(A − λS)−1(A − µS) = (µ − λ)B(A − λS)−1S. ♦

3.5 PSEUDOSPECTRA 3.5.1 Pseudospectrum

Let us start by giving the deﬁnition of the pseudospectrum of densely closed linear operator A for every ε > 0,

[n −1 1o σε (A) := σ(A) λ ∈ C such that k(λ − A) k > , ε or by [ −1 1 Σε (A) := σ(A) λ ∈ C such that k(λ − A) k ≥ , ε with the convention k(λ −A)−1k = ∞ if, and only if, λ ∈ σ(A). The pseudospectrum, σε (A), is the open subset of the complex plane bounded by the ε−1 level curve of the norm of the resolvent. For ε > 0, it can be shown that σε (A) is a larger set and is never empty. The pseudospectra of A are a family of strictly nested closed sets, which grow to ﬁll the whole complex plane as ε → ∞ (see [80, 151, 152]). From these deﬁnitions, it follows that the pseudospectra associated with various ε are nested sets. Then, for all

0 < ε1 < ε2, we have

σ(A) ⊂ σε1 (A) ⊂ σε2 (A) and

σ(A) ⊂ Σε1 (A) ⊂ Σε2 (A), 114 Linear Operators and Their Essential Pseudospectra and that the intersections of all the pseudospectra are the spectrum

\ \ σε (A) = σ(A) = Σε (A). ε>0 ε>0

Theorem 3.5.1 Let A ∈ C (X). The following three conditions are equivalent:

(i) λ ∈ σε (A). (ii) There exists a bounded operator D such that kDk < ε and

λ ∈ σ(A + D).

(iii) Either λ ∈ σ(A) or k(λ − A)−1k < ε−1. ♦

Proof. (i) ⇒ (ii) If λ ∈ σ(A), we may put D = 0. Otherwise, let f ∈ D(A − λ), k f k = 1 and

k(A − λ) f k < ε.

Let φ ∈ X∗ satisfy kφk = 1 and φ( f ) = 1. Then, let us deﬁne the rank one operator D : X −→ X by

Dg := −φ(g)(A − λ) f .

We see immediately that kDk < ε and

(A − λ + D) f = 0.

(ii) ⇒ (iii) We derive a contradiction from the assumption that λ ∈/ σ(A) and k(A − λ)−1k ≤ ε−1.

Let B : X −→ X be the bounded operator deﬁned by the norm convergent series ∞ n B := ∑ (A − λ)−1 −D(A − λ)−1 n=0 = (A − λ)−1(I + D(A − λ)−1)−1. Spectra 115

It is immediate from these formulae that B is one-to-one with a range equal to D(A − λ). We also see that

B(I + D(A − λ)−1) f = (A − λ)−1 f , for all f ∈ X. Putting g = (A − λ)−1 f , we conclude that B(A − λ + D)g = g for all g ∈ D(A − λ). The proof that

(A − λ + D)Bh = h, for all h ∈ X is similar. Hence, A − λ + D is invertible, with an inverse B.

(iii) ⇒ (i) We assume for non-triviality that λ ∈/ σ(A). There exists g ∈ X such that k(A − λ)−1gk > ε−1kgk.

Putting f := (A − λ)−1g, we see that k(A − λ) f k < εk f k. Q.E.D.

Remark 3.5.1 From Theorem 3.5.1, it follows immediately that [ σε (A) = σ(A + D). ♦ kDk<ε

3.5.2 S-Pseudospectrum

Let A ∈ C (X) and S ∈ L (X). Consider the equation deﬁned by

(λS − A)x = b, (3.12) where λ ∈ C, b ∈ X and x ∈ D(A). For λ ∈ ρS(A), the existence of a solution of Eq. (3.12) and it is uniqueness are guaranteed. So,

x = RS(λ,A)b. (3.13) 116 Linear Operators and Their Essential Pseudospectra

Under the perturbation of b by adding r ∈ X to b such that 0 < krk < ε. We obtain the equation

(λS − A)x = b + r.

This implies that

x = RS(λ,A)(b + r). (3.14) We want the solution of the Eq. (3.12) remains stable despite the perturbation. It means that x is very close to x and so,

kx − xk < 1.

Since the two Eqs. (3.13) and (3.14) provide

kx − xk = kRS(λ,A)rk

< kRS(λ,A)kε.

For this, it sufﬁces to take 1 kR (λ,A)k ≤ . S ε In this case, we deﬁne \ 1 ρS, ε(A) = ρS(A) λ ∈ C such that kRS(λ,A)k ≤ . ε Therefore, σS, ε(A) = C\ρS, ε(A).

Deﬁnition 3.5.1 Let A ∈ C (X),S ∈ L (X) and ε > 0 such that S 6= A and S 6= 0. We deﬁne the S-pseudospectrum of A by [ 1 σS, ε(A) = σS(A) λ ∈ C such that kRS(λ,A)k > . ε

Convention: kRS(λ,A)k = +∞ if, and only if, λ ∈ σS(A). ♦

If we replace the bounded operator S by the identity operator I, we obtain the deﬁnition of pseudospectrum σε(A) (see [58, 153]). Spectra 117

Proposition 3.5.1 Let A ∈ C (X),S ∈ L (X) and ε > 0, then the S- pseudospectra veriﬁes the following properties:

(i) σS, ε(A) 6= /0, (ii) the S-pseudospectra σS, ε(A) ε>0 are increasing sets in the sense of inclusion relative to the parameter strictly positive ε, and \ (iii) σS, ε(A) = σS(A). ♦ ε>0

Proof. (i) We argue by contradiction. Suppose that σS, ε(A) = /0,then

ρS, ε(A) = C.

It means exactly that \ 1 ρS(A) λ ∈ C such that kRS(λ,A)k ≤ = C, ε and so ρS(A) = C. Hence, 1 λ ∈ C such that kRS(λ,A)k ≤ = C. ε Consider the function

ϕ : C −→ L (X) λ −→ RS(λ,A).

Since ϕ is analytic on C and for every λ ∈ C, we have 1 kϕ(λ)k ≤ , ε then ϕ is an entire bounded function. Therefore, using Liouville theorem

(see Theorem 2.1.5), we obtain that ϕ is constant. It follows that RS(λ,A) is null, which is a contradiction. 118 Linear Operators and Their Essential Pseudospectra

(ii) Let ε1, ε2 > 0 such that ε1 < ε2. If λ ∈ σS, ε1(A), then 1 1 kRS(λ,A)k > > . ε1 ε2 Therefore,

λ ∈ σS, ε2(A). (iii) We have \ \ [ 1 σS, ε(A) = σS(A) λ ∈ C such that kRS(λ,A)k > , ε ε>0 ε>0 ! S \ 1 = σS(A) λ ∈ C such that kRS(λ,A)k > . ε ε>0 It sufﬁces to prove that \ 1 λ ∈ C such that kRS(λ,A)k > = ε ε>0 {λ ∈ C such that kRS(λ,A)k = +∞}.

For the inclusion in the direct sense: let λ ∈ C such that for all ε > 0 1 kR (λ,A)k > . S ε Consequently,

lim kRS(λ,A)k = kRS(λ,A)k = +∞. ε→0+

For the other inclusion: let λ ∈ C such that 1 kR (λ,A)k = +∞ > S ε for all ε > 0. Hence,

\ [n o σS, ε(A) = σS(A) λ ∈ C such that kRS(λ,A)k = +∞ , ε>0 = σS(A), which completes the proof of proposition. Q.E.D. Spectra 119

3.5.3 Ammar-Jeribi Essential Pseudospectrum

In Ref. [29], A. Ammar and A. Jeribi introduced the deﬁnition of Ammar-Jeribi essential pseudospectrum of closed densely deﬁned linear operators by \ σe5,ε (A) = σε (A + K). K∈K (X)

Theorem 3.5.2 Let X be a Banach space, ε > 0 and A ∈ C (X). Then,

λ 6∈ σe5,ε (A) if, and only if, for all D ∈ L (X) such that kDk < ε, we have A + D − λ ∈ Φ(X) and i(A + D − λ) = 0. ♦

Proof. Let λ 6∈ σe5,ε (A). By using Theorem 3.5.1, we infer that there exists a compact operator K on X, such that

[ λ 6∈ σ(A + K + D). kDk<ε

So, for all D ∈ L (X) such that kDk < ε, we have

λ ∈ ρ(A + D + K).

Therefore, A + D + K − λ ∈ Φ(X) and

i(A + D + K − λ) = 0, for all D ∈ L (X) such that kDk < ε. From Theorem 2.2.14, we deduce that for all D ∈ L (X) such that kDk < ε, we have

A + D − λ ∈ Φ(X) and i(A + D − λ) = 0.

Conversely, we suppose that, for all D ∈ L (X) such that kDk < ε, we have

A + D − λ ∈ Φ(X) 120 Linear Operators and Their Essential Pseudospectra and i(A + D − λ) = 0.

Without loss of generality, we may assume that λ = 0. Let n = α(A+D) = 0 0 β(A + D), x1,··· ,xn being the basis for N(A + D) and y1,··· ,yn being the basis for the N((A+D)∗). By using Lemma 2.1.1, there are func- 0 0 ∗ tionals x1,··· ,xn in X (the adjoint space of X) and elements y1,··· ,yn, such that 0 x j(xk) = δ jk and

y j(yk) = δ jk,

1 ≤ j, k ≤ n, where δ jk = 0 if j 6= k and δ jk = 1 if j = k. Let K be the operator defined by n 0 Kx = ∑ xk(x)yk, k=1 x ∈ X. Clearly, K is a linear operator defined everywhere on X. It is bounded, since n 0 kKxk ≤ kxk ∑ kxkkkykk . k=1 Moreover, the range of K is contained in a finite-dimensional subspace of X. Then, K is a finite rank operator in X. So, K is a compact operator in X. Now, we may prove that

\ \ N(A + D) N(K) = {0} and R(A + D) R(K) = {0}, (3.15) for all D ∈ L (X) such that kDk < ε. Indeed, let x ∈ N(A + D), then

n x = ∑ αkxk, k=1 and therefore,

x j(x) = α j, 1 ≤ j ≤ n. Moreover, if x ∈ N(K), then

0 x j(x) = 0, Spectra 121 with 1 ≤ j ≤ n. This proves the ﬁrst relation in Eq. (3.15). The second inclusion is similar. In fact, if y ∈ R(K), then

n y = ∑ αkyk, k=1 and hence,

y j(y) = α j, with 1 ≤ j ≤ n. However, if y ∈ R(A + D), then

0 y j(y) = 0, with 1 ≤ j ≤ n. This gives the second relation in Eq. (3.15). Besides, K is a compact operator. We deduce, from Theorem 2.2.14, that 0 ∈ ΦA+K+D and i(A + D + K) = 0. If x ∈ N(A + D + K), then

(A + D)x ∈ R(A + D)\R(K).

This implies that x ∈ N(A + D)TN(K) and x = 0. Thus,

α(A + D + K) = 0.

In the same way, we can prove that

R(A + D + K) = X.

Hence, 0 ∈ ρ(A + D + K). This implies that, for all D ∈ L (X) such that kDk < ε, we have 0 6∈ σ(A + D + K). Also, \ 0 6∈ σε (A + K). K∈ K (X) So,

0 6∈ σe5,ε (A). Q.E.D. 122 Linear Operators and Their Essential Pseudospectra

Remark 3.5.2 Let X be a Banach space, ε > 0 and A ∈ C (X). (i) Using Theorem 3.5.2, we have

[ σe5,ε (A) := σe5(A + D). kDk<ε

(ii) We have \ σe5,ε (A) := σε (A + D). ♦ D∈F (X)

3.5.4 Essential Pseudospectra

Let A ∈ C (X) and ε > 0. We are concerned with the following essential pseudospectra

ε ε σe1,ε (A) := λ ∈ C such that λ − A ∈/ Φ+(X) := C\Φ+A, ε ε σe2,ε (A) := λ ∈ C such that λ − A ∈/ Φ−(X) := C\Φ−A, ε ε σe3,ε (A) := λ ∈ C such that λ − A ∈/ Φ±(X) := C\Φ±A, ε ε σe4,ε (A) := {λ ∈ C such that λ − A ∈/ Φ (X)} := C\ΦA, \ σe5,ε (A) := σε (A + K), K∈K (X) \ σeap,ε (A) := σap,ε (A + K), K∈K (X) \ Σeap,ε (A) := Σap,ε (A + K), K∈K (X) S σeδ,ε (A) := σe2,ε (A) {λ ∈ C such that i(λ − A − D) < 0, for all ||D|| < ε}, where σap,ε (A) := λ ∈ C such that inf k(λ − A)xk < ε , x∈D(A), kxk=1 and Σap,ε (A) := λ ∈ C such that inf k(λ − A)xk ≤ ε . x∈D(A), kxk=1 Note that if ε tends to 0, we recover the usual deﬁnition of the essential spectra of a closed linear operator A. Spectra 123

Remark 3.5.3 Let A ∈ C (X) and consider i ∈ {1,..,5,ap,δ}. Then,

(i) σei,ε (A) ⊂ σε (A), \ (ii) σei,ε (A) = σei(A), ε>0

(iii) if ε1 < ε2, then σei,ε1 (A) ⊂ σei,ε2 (A), and (iv) they can be ordered as T σe3,ε (A) = σe1,ε (A) σe2,ε (A) ⊆ σe4,ε (A) ⊆ σe5,ε (A) = T σeap,ε (A) σeδ,ε (A). ♦

3.5.5 Conditional Pseudospectrum

In this section we define the conditional pseudospectrum of a linear operators in infinite dimensional Banach spaces and consider some basic properties in order to put this definition in its due place. We begin with the following definition.

Definition 3.5.2 Let A ∈ L (X) and 0 < ε < 1. The condition pseudospectrum of A is denoted by Σε (A) and is defined as, ε [ −1 1 Σ (A) := σ(A) λ ∈ C : kλ − Akk(λ − A) k > , ε with the convention that kλ − Akk(λ − A)−1k = ∞, if λ − A is not invertible. The condition pseudoresolvent of A is denoted by ρε (A) and is defined as, ε T −1 1 ρ (A) := ρ(A) λ ∈ C : kλ − Akk(λ − A) k ≤ . ♦ ε For 0 < ε < 1, it can be shown that ρε (A) is a larger set and is never empty. Here also ρε (A) ⊆ ρ(A) for 0 < ε < 1. Recall that the usual condition pseudospectral radius rε (A) of A ∈ L (X) by

n ε o rε (A) := sup |λ| : λ ∈ Σ (A) , and the spectral radius of A ∈ L (X) by n o rσ (A) = sup |λ| : λ ∈ σ(A) . 124 Linear Operators and Their Essential Pseudospectra

3.6 STRUCTURED PSEUDOSPECTRA 3.6.1 Structured Pseudospectrum

We refer to E. B. Davies [58] which deﬁned the structured pseudospectrum, or spectral value sets of a closed densely deﬁned linear operator A on X by [ σ(A,B,C,ε) = σ(A +CDB), kDk<ε where B ∈ L (X,Y) and C ∈ L (Z,X). It is clear that

σ(A,I,I,ε) = σε(A).

We ﬁrst prove the following proposition which is basic for our purpose.

Proposition 3.6.1 Let A ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X),G(s) := B(s−A)−1C and ε > 0. Then, σ(A,B,C,ε)\σ(A) is a bounded open subset of C such that n 1o σ(A,B,C,ε) = σ(A)S s ∈ ρ(A) such that kG(s)k > . ♦ ε Proof. We will discuss these two cases: 1st case : If B or C = 0, then σ(A,B,C,ε) = σ(A,ε). Hence,

σ(A,B,C,ε)\σ(A) = /0 is bounded. 2nd case : If B and C 6= 0. We consider the following function ( ϕ : ρ(A) −→ R s −→kG(s)k.

ϕ is continuous. It is clear that

1 σ(A,B,C,ε)\σ(A) = ϕ−1 ,+∞ . ε Spectra 125

So, we can deduce that σ(A,B,C,ε)\σ(A) is open. Furthermore, we argue by contradiction so we suppose that there exists s ∈ σ(A,B,C,ε)\σ(A) such that |s| → +∞. Since

lim kG(s)k = 0, |s|→+∞

1 then 0 ≥ ε which is impossible. As a consequence we obtain that

σ(A,B,C,ε)\σ(A) is a bounded set. Q.E.D.

Remark 3.6.1 As a consequence of Proposition 3.6.1, that the structured pseudospectra, σ(A,B,C,ε), of a bounded operator A is a bounded subset of C. ♦

The following identity was established in [58, Theorem 9.2.18].

Theorem 3.6.1 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, Sn −1 1o σ(A,B,C,ε) = σ(A) s ∈ C such that kB(s − A) Ck > ε . ♦

3.6.2 The Structured Essential Pseudospectra

In this section, we are interested in studying the following structured essential pseudospectra of a closed, densely deﬁned linear operator A introduced in [67] with respect to the perturbation structure (B,C), where B ∈ L (X,Y), C ∈ L (Z,X) and uncertainty level ε > 0. We note that the structured essential pseudospectrum consists of large sets of all complex numbers to which at least one essential spectrum can be shifted by disturbance operator D of norm kDk < ε.

Deﬁnition 3.6.1 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X), and ε > 0. 126 Linear Operators and Their Essential Pseudospectra

(i) The structured Jeribi essential pseudospectrum is deﬁned by

\ σ j(A,B,C,ε) := σ(A + K,B,C,ε). K∈W ∗(X) (ii) The structured Wolf essential pseudospectrum is deﬁned by

[ σe4(A,B,C,ε) := σe4(A +CDB). kDk<ε

(iii) The structured Ammar-Jeribi essential pseudospectrum is deﬁned by [ σe5(A,B,C,ε) := σe5(A +CDB). kDk<ε (iv) The structured Browder essential pseudospectrum is deﬁned by [ σe6(A,B,C,ε) := σe6(A +CDB). ♦ kDk<ε

Remark 3.6.2 All these sets satisfy the following inclusions:

σe4(A,B,C,ε) ⊆ σe5(A,B,C,ε) ⊆ σe6(A,B,C,ε). ♦

By Theorem 3.1.6, we have the following.

Lemma 3.6.1 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Suppose that for every kDk < ε, we have ρ(A +CDB) 6= /0. Then,

(i) If for all kDk < ε, we have C\σe4(A +CDB) is connected, then

σe4(A,B,C,ε) = σe5(A,B,C,ε).

(ii) If C\σe5(A +CDB) is connected for every kDk < ε, then

σe5(A,B,C,ε) = σe6(A,B,C,ε). ♦

3.6.3 The Structured S-Pseudospectra

Deﬁnition 3.6.2 Let A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. We deﬁne the structured S-pseudospectrum by [ σS(A,B,C,ε) := σS(A +CDB). ♦ kDk<ε Spectra 127

Remark 3.6.3 If S is an invertible operator, then −1 −1 σS(A,B,C,ε) = σ(S A,B,S C,ε). ♦

Definition 3.6.3 Let A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. (i) The structured S-point pseudospectrum is defined by [ σp,S(A,B,C,ε) := σp,S(A +CDB). kDk<ε (ii) The structured S-continuous pseudospectrum is defined by [ σc,S(A,B,C,ε) := σc,S(A +CDB). kDk<ε (iii) The structured S-residual pseudospectrum is defined by [ σR,S(A,B,C,ε) := σR,S(A +CDB). kDk<ε (iv) The structured S-approximate point pseudospectrum is defined by [ σap,S(A,B,C,ε) := σap,S(A +CDB). ♦ kDk<ε Remark 3.6.4 Let A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0, then the structured S-pseudospectra is partitioned into three disjoint sets as follows: S S σS(A,B,C,ε) = σp,S(A,B,C,ε) σc,S(A,B,C,ε) σR,S(A,B,C,ε). ♦

Lemma 3.6.2 Let A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0, then

σp,S(A,B,C,ε) ⊂ σap,S(A,B,C,ε). ♦

Proof. Let λ ∈ σp,S(A,B,C,ε). Then, there exists D ∈ L (Y,Z) such that kDk < ε and

λ ∈ σp,S(A +CDB). Therefore, there exists x ∈ D(A)\ 0 with (λS − A −CDB)x = 0. In this case, it is sufﬁcient to take x = x . Consequently, n kxk 128 Linear Operators and Their Essential Pseudospectra

λ ∈ σap,S(A,B,C,ε). Q.E.D.

3.6.4 The Structured S-Essential Pseudospectra

Deﬁnition 3.6.4 Let A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0, we deﬁne the structured S-essential pseudospectra in the following way

[ σe1,S(A,B,C,ε) := σe1,S(A +CDB), kDk<ε [ σe2,S(A,B,C,ε) := σe2,S(A +CDB), kDk<ε [ σe3,S(A,B,C,ε) := σe3,S(A +CDB), kDk<ε [ σe4,S(A,B,C,ε) := σe4,S(A +CDB), kDk<ε

[ σe5,S(A,B,C,ε) := σe5,S(A +CDB). ♦ kDk<ε

Remark 3.6.5 Let A, S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X), λ ∈ C and ε > 0. If SA = AS, then we can write

(λS − A)(λS −CDB) = ACDB + λS(λS − A −CDB) (3.16) and

(λS −CDB)(λS − A) = CDBA + λ(λS − A −CDB)S. (3.17)

♦ Chapter 4

Perturbation of Unbounded Linear Operators by γ-Relative Boundedness

In Ref. [61], T. Diagana studied some sufﬁcient conditions such that if S, T and K are three unbounded linear operators with S being a closed operator, then their algebraic sum S + T + K is also a closed operator. The main focus of this chapter is to extend these results to the closable operator by adding a new concept of the gap and the γ-relative boundedness inspired by the work of A. Jeribi, B. Krichen and M. Zarai Dhahri in Ref. [92]. After that, we apply the obtained results to study the speciﬁc properties of some block operator matrices. All results of this chapter are given in Ref. [22].

4.1 SUM OF CLOSABLE OPERATORS 4.1.1 Norm Operators

The goal of this part consists in establishing some properties of closable operators in Banach spaces. 130 Linear Operators and Their Essential Pseudospectra

Theorem 4.1.1 Let X and Y be two Banach spaces and let S, T and K be three operators from X into Y such that D(T) ⊂ D(S) ⊂ D(K). If (i) there exist two constants a, b > 0 such that

kSxk ≤ akxk + bkTxk, x ∈ D(T),

(ii) there exist two constants e, d > 0 such that µ = b(1 + d) < 1 and

kKxk ≤ ekxk + dkSxk, x ∈ D(S).

Then, S + T + K is closable if, and only if, T is also closable.

In this case, D(S + T + K) = D(T) and

−1 2 2 1 δb(S + T + K,T) ≤ (1 − µ) (ν + µ ) 2 , (4.1) where ν = (a(1 + d) + e)(ν > 0) and δb(·,·) is the gap between two operators. ♦

Proof. Let x ∈ D(S + T + K) = D(T). Then,

k(S + T + K)xk ≤ (1 + b)kTxk + (a + e)kxk + dakxk + bkTxk, and hence,

k(S + T + K)xk ≤ (a + e + da)kxk + (1 + b + bd)kTxk. (4.2)

Similarly, we also have, for all x ∈ D(T),

k(S + K)xk ≤ bkTxk + (a + e)kxk + d(akxk + bkTxk) and hence,

k(S + K)xk ≤ (a + e + da)kxk + b(1 + d)kTxk. (4.3)

Combining (4.2) and (4.3), it follows that for all x ∈ D(T), we have

k(S + T + K)xk ≥ −(a + e + da)kxk + [1 − b(1 + d)]kTxk, Perturbation of Unbounded Linear Operators 131 which yields

1 − b(1 + d)kTxk ≤ k(S + T + K)xk + (a + e + da)kxk.

Setting µ = b(1 + d)(0 < µ < 1) and ν = a + e + da, we deduce that kTxk ≤ (1 − µ)−1 k(S + T + K)xk + νkxk . (4.4)

Let (xn)n be a sequence in D(S+T +K) = D(T) such that xn → 0 and

(S + T + K)xn → χ.

By using (4.4), it follows that (Txn)n is a Cauchy sequence in the Banach space Y and therefore, there exists ψ1 ∈ Y such that

Txn → ψ1.

Since T is closable, then

ψ1 = 0. It follows from (4.2) that

(S + T + K)xn → 0.

Then, χ = 0 and S+T +K is closable. Similarly, T is closable if S+T +K is also closable. Now, we may prove that

D(S + T + K) = D(T).

Let x ∈ D(S + T + K), then there exists a sequence (xn)n such that (xn)n is (S + T + K)-convergent to x. By using (4.4), we deduce that (xn)n is T-convergent to x, then x ∈ D(T) so that

D(S + T + K) ⊂ D(T).

The opposite inclusion is proved similarly. In order to prove (4.1). Let ϕ = (u,(S + T + K)u) ∈ G(S + T + K) with kϕk = 1 so that, we have

kuk2 + k(S + T + K)uk2 = kϕk2 = 1. (4.5) 132 Linear Operators and Their Essential Pseudospectra

Since G(S + T + K) = G(S + T + K), then there exists a sequence (ϕn)n in G(S + T + K), such that

ϕn → ϕ. Hence,

ϕn = (un,(S + T + K)un), where un ∈ D(S + T + K), un → u and

(S + T + K)un → (S + T + K)u.

Now, we take ψn = (un,Tun) ∈ G(T). By using (4.4), we have (un)n is T-convergent to u. Then,

ψn → ψ0 ∈ G(T).

Combining both (4.3) and (4.4), we have

kϕn − ψnk = k(S + K)unk −1 ≤ (1 − µ) νkunk + µk(S + T + K)unk .

If n → ∞, then we obtain

−1 kϕ − ψ0k ≤ (1 − µ) νkuk + µk(S + T + K)uk .

From Schwarz inequality and Eq. (4.5), it follows that

−1 2 2 1 kϕ − ψ0k ≤ (1 − µ) (ν + µ ) 2 .

Hence, −1 2 2 1 dist(ϕ,G(T)) ≤ (1 − µ) (ν + µ ) 2 .

Now, since ϕ is an arbitrary element of the unit sphere of G(S + T + K), then −1 2 2 1 δ(S + T + K,T) ≤ (1 − µ) (ν + µ ) 2 .

Similarly, we can estimate the quantity δ(T,S + T + K). As a result, we obtain (4.1). Q.E.D. Perturbation of Unbounded Linear Operators 133

Remark 4.1.1 If the hypotheses of Theorem 4.1.1 are satisﬁed, then (i) S + T + K is closed if, and only if, T is also closed. (ii) If S + K ∈ L (X,Y), then δb(S + T + K,T) ≤ kS + Kk. ♦

Theorem 4.1.2 Let S, T and K be three operators from X into Y such that D(T) ⊂ D(S) ⊂ D(K) and (a) there exist two constants a, b > 0

kSxk ≤ akxk + bkTxk, x ∈ D(T),

(b) there exist two constants e, d > 0 such that

kKxk ≤ ekxk + dkSxk, x ∈ D(S), and γe(T)(1 − ν) > µ, (4.6) where µ = b(1 + d) < 1, ν = a(1 + d) + e (ν > 0) and γe(T) is given in (2.13). If T is a semi-Fredholm operator, then S +T +K is semi-Fredholm and it satisﬁes the following properties: (i) α(S + T + K) ≤ α(T), (ii) β(S + T + K) ≤ β(T), and (iii) i(S + T + K) = i(T). ♦

Proof. Referring to (4.6), we have ν < 1. Then, by virtue of Remark 4.1.1 (i), we have S + T + K is closed. Let us introduce the new norm in D(T) by [k xk] = (ν + ε)kxk + (µ + ε)kTxk, x ∈ D(T), where ε is ﬁxed and γ(T)(1 − ν) − µ 0 < ε < e . (4.7) 1 + γe(T)

It is clear that [k · k] and k · kT (see Eq. (2.5)) are equivalent and, if we denote by Xb = (D(T), [k · k]), then Xb is a Banach space and we have the following inequalities −1 kTbk ≤ (µ + ν) 134 Linear Operators and Their Essential Pseudospectra and kSb+ Kbk ≤ 1. Since ( α(Tb) = α(T), β(Tb) = β(T), R(Tb) = R(T), β(S +\T + K) = β(S + T + K) and R(S +\T + K) = R(S + T + K), then we deduce that Tb ∈ Φ±(X,Y) and, it is sufﬁcient to show that S +\T + K ∈ Φ±(X,Y). It is clear that

γe(T) γe(Tb) = . (4.8) ν + ε + (µ + ε)γe(T)

By using (4.7) and Eq.(4.8), we have γ(Tb) > 1. Thus,

kSb+ Kbk < γe(Tb).

Accordingly, we have

S +\T + K ∈ Φ±(X,Y),

α(S +\T + K) ≤ α(Tb), and β(S +\T + K) ≤ β(Tb), which completes the proof of theorem. Q.E.D.

Corollary 4.1.1 Let ε > 0 such that

γ(T) kS + Kk ≤ e , p 2 1 + (εγe(T)) where γe(T) is given in (2.13). If T is a semi-Fredholm operator, then S + T + K is semi-Fredholm and it satisﬁes the following properties: (i) α(S + T + K) ≤ α(T), (ii) β(S + T + K) ≤ β(T), and (iii) i(S + T + K) = i(T). ♦ Perturbation of Unbounded Linear Operators 135

4.1.2 Kuratowski Measure of Noncompactness

In this part, we provide some sufﬁcient conditions for three closed operators S, T and K to have their algebaric sum also closed.

Theorem 4.1.3 Let γ(·) be the Kuratowski measure of noncompactness and let S, T and K be three closed operators such that D(S) ⊂ D(T) ⊂ D(K). If (i) there exist two constants as,bs > 0 such that

γ(S(D)) ≤ asγ(D) + bsγ(T(D)), D ⊂ D(T),

(ii) there exist two constants ak, bk > 0 such that bk < 1, bs(1+bk) < 1 and

γ(K(D)) ≤ akγ(D) + bkγ(S(D)), D ⊂ D(T). Then, the operator S + T + K is closed. ♦

Proof. Let D ⊂ D(T). The fact that

γ(T(D)) = γ ((T + S + K − S − K)(D)) (4.9) allows us to

γ((T + S + K − S − K)(D)) ≤ γ ((T + S + K)(D)) + γ((S + K)(D)). (4.10) By using Eq. (4.9) and the relation (4.10), we have

γ((T + S + K)(D)) ≥ γ(T(D)) − γ((S + K)(D)). (4.11)

Moreover,

γ((S + K)(D)) ≤ γ(S(D)) + γ(K(D))

≤ asγ(D) + bsγ(T(D)) + akγ(D) + bk(asγ(D) + bsγ(T(D)))

≤ asγ(D) + bsγ(T(D)) + akγ(D) + bkasγ(D) + bkbsγ(T(D))

≤ (as + ak + bkas)γ(D) + bs(1 + bk)γ(T(D)), we infer, from the last equation and Eq. (4.11), that 136 Linear Operators and Their Essential Pseudospectra

γ((T + S + K)(D)) ≥ γ(T(D)) − γ((S + K)(D))

≥ γ(T(D)) − (as + ak + bkas)γ(D) − bs(1 + bk)γ(T(D))

≥ −(as + ak + bkas)γ(D) + (1 − bs(1 + bk))γ(T(D)). Then,

γ((T + S + K)(D)) ≥ −(as + ak + bkas)γ(D) + (1 − bs(1 + bk))γ(T(D)) (4.12) provided that 0 < 1−bs(1+bk) < 1 and as +ak +bkas > 0. Let (xn)n be a sequence in D(T + S + K) = D(T) such that xn → x and

(T + S + K)xn → y.

This implies that (xn)n and ((T + S + K)xn)n are relatively compact. Thus,

γ((xn)n) = γ(((T + S + K)xn)n) = 0.

By using (4.12), we deduce

γ(Txn) = 0,

and there exist a subsequence xnk → x ∈ D(T) such that

Txnk → α.

Since T is closed, then Tx = α.

In view of Theorem 2.6.2, we have S + K is closed. Since

(S + K)xnk → y − α, then x ∈ D(T) and (T + S + K)x = y.

We conclude that the algebaric sum T +S +K is a closed operator. Q.E.D. Perturbation of Unbounded Linear Operators 137

4.2 BLOCK OPERATOR MATRICES 4.2.1 2 × 2 Block Operator Matrices

We denote by L the linear block operator matrices, acting on the Banach space X ×Y, of the form ! AB L = , CD where the operator A acts on X and has domain D(A), D is deﬁned on D(D) and acts on the Banach space Y, and the intertwining operator B (resp. C) is deﬁned on the domain D(B) (resp. D(C)) and acts on X (resp. Y). One of the problems in the study of such operators is that in general L is not closed or even closable, even if its entries are closed.

Proposition 4.2.1 Consider the block operator matrices acting on the Banach space X ×Y by ! A 0 T = , 0 D ! 0 B S = , 0 0 and ! 0 0 K = . C 0 If the operator T + S + K is γ-diagonally dominant with bound δ, then (i) S is T -γ-bounded with T -γ-bound δ, and (ii) K is S -γ-bounded with S -γ-bound δ. ♦ 138 Linear Operators and Their Essential Pseudospectra

Proof. Let πi(·), i = 1,2 be denote the natural projections on X and Y, respectively. (i) For D ⊂ D(K ), we get " ! !# 0 B π1(D) γ = γ(B(π2 (D))). 0 0 π2(D)

According to these assumptions, there exist two constants aB, bB ≥ 0 such that

γ(B(π2(D))) ≤ aBγ(π2(D)) + bBγ(D(π2(D))). Since

γ(π2(D)) ≤ γ(π2)γ(D), then

γ(B(π2(D))) ≤ aBγ(π2)γ(D) + bBγ(D(π2(D))). Hence, " ! !# 0 B π (D) γ 1 0 0 π2(D)

≤ aBγ(π2)γ(D) + bBγ(D(π2(D))) " ! !# A 0 π1(D) ≤ aBγ(π2)γ(D) + bB maxγ . 0 D π2(D) (ii) Also, " ! !# 0 0 π1(D) γ = γ(C(π1 (D))). C 0 π2(D)

According to these assumptions, there exist two constants aC,bC ≥ 0 such that

γ(C(π1(D))) ≤ aCγ(π1(D)) + bCγ(A(π1(D))). Since

γ(π1(D)) ≤ γ(π1)γ(D), then

γ(C(π1(D))) ≤ aCγ(π1)γ(D) + bCγ(A(π1(D))). Hence, we get Perturbation of Unbounded Linear Operators 139

" ! !# 0 0 π (D) γ 1 C 0 π2(D)

≤ aCγ(π1)γ(D) + bCγ(A(π1(D))) " ! !# A 0 π1(D) ≤ aCγ(π1)γ(D) + bC maxγ . 0 D π2(D) This completes the proof. Q.E.D.

4.2.2 3 × 3 Block Operator Matrices

In what follows, we are concerned with the block operator matrices

 ABC    M =  DEF , GHK which acts on the product of Banach spaces X ×Y ×Z, where the entire of the matrix are linear operators. The operator A, having the domain D(A), acts on X, the operator E, having the domain D(E), acts on Y, and the operator K, having the domain D(K), acts on Z. Similarly, the operator B, H, C, F, D, and G are deﬁned, respectively, by the domains D(B) ⊂ Y into X, D(H) ⊂ Y into Z, D(C) ⊂ Z into X, D(F) ⊂ Z into Y, D(D) ⊂ X into Y, and D(G) ⊂ X to Z. Let M be the operator

         u1 v1    G(M ) =   u , v   ∈ (X ×Y × Z)2 such that    2   2      u3 v3    v = Au + Bu +Cu  1 1 2 3  v2 = Du1 + Eu2 + Fu3    v3 = Gu1 + Hu2 + Ku3     T T T T  D(M ) = (D(A) D(D) D(G)) × (D(B) D(E) D(H))×   (D(C)TD(F)TD(K)). 140 Linear Operators and Their Essential Pseudospectra

We denote by  A 0 0    T =  0 E 0 , 0 0 K

 0 B 0    S =  0 0 F , G 0 0 and  0 0 C    K =  D 0 0 . 0 H 0 Then, it is clear that M = T + S + K .

Remark 4.2.1 It is clear that T is closed if, and only if, A, E and K are closed. ♦

Theorem 4.2.1 (a) Let us consider the following conditions

(i) B is E-γ-bounded with E-γ-bound δ1,

(ii) F is K-γ-bounded with K-γ-bound δ2, and

(iii) G is A-γ-bounded with A-γ-bound δ3. Then, S is T -γ-bounded with T -γ-bound

δ = max{δ1,δ2,δ3}.

(b) Let us consider the following conditions

(i) C is F-γ-bounded with F-γ-bound δ1,

(ii) D is G-γ-bounded with G-γ-bound δ2, and

(iii) H is B-γ-bounded with B-γ-bound δ3.

Then, K is S -γ-bounded with S -γ-bound δ = max{δ1,δ2,δ3}. ♦ Perturbation of Unbounded Linear Operators 141

Proof. (a) Let πi(·), i = 1,2,3 be the natural projections on X, Y and Z, respectively, and let ε > 0. Consider the constants, aB, aG, aF , bB, bG, and bF such that

δ1 ≤ b1 ≤ δ1 + ε,

δ2 ≤ b2 ≤ δ2 + ε,

δ3 ≤ b3 ≤ δ3 + ε,

a1 = {aGγ(π1),aBγ(π2),aF γ(π3)}, and

a2 = {bG,bB,bF }.

For D ⊂ D(S ), we get    0 B 0 π1(D)    γ  0 0 F  π2(D)  = G 0 0 π3(D)

max{γ(G(π1 (D))),γ(B(π2 (D))),γ(F(π3 (D)))}. Hence,  γ(G(π (D))) ≤ a γ(π (D)) + b γ(A(π (D))),  1 G 1 G 1 γ(B(π2(D))) ≤ aBγ(π2(D)) + bBγ(E(π2(D))),   γ(F(π3(D))) ≤ aF γ(π3(D)) + bF γ(K(π3(D))).

Since,

γ(πi(D)) ≤ γ(πi)γ(D), i = 1,2,3, then  γ(G(π (D))) ≤ a γ(π )γ(D) + b γ(A(π (D))),  1 G 1 G 1 γ(B(π2(D))) ≤ aBγ(π2)γ(D) + bBγ(E(π2(D))),   γ(F(π3(D))) ≤ aF γ(π3)γ(D) + bF γ(K(π3(D))). Hence, we get    0 B 0 π1(D)    γ  0 0 F  π2(D)  G 0 0 π3(D) 142 Linear Operators and Their Essential Pseudospectra

≤ maxa2 max{γ(A(π1(D))),γ(E(π2(D))),γ(K(π3(D)))}+

γ(D)maxa1    A 0 0 π1(D)    ≤ γ(D)maxa1 + γ  0 E 0  π2(D) maxa2. 0 0 K π3(D) Since

max{bG,bB,bF } = {δ1 + ε,δ2 + ε,δ3 + ε} = δ + ε, then S is T -γ-bounded with T -γ-bound < δ. The proof of (b) may be checked in the same way as in the proof of the item (a). Q.E.D. Chapter 5

Essential Spectra

In this chapter, we characterize the essential spectra of the closed, densely deﬁned linear operators on Banach space.

5.1 CHARACTERIZATION OF ESSENTIAL SPECTRA

The goal of this section consists in establishing some preliminary results which will be needed in the sequel.

5.1.1 Characterization of Left and Right Weyl Essential Spectra

Theorem 5.1.1 Let A ∈ C (X). Then,

(i) λ ∈/ σewl(A) if, and only if, λ − A ∈ Φl(X) and i(λ − A) ≤ 0. (ii) λ ∈/ σewr(A) if, and only if, λ − A ∈ Φr(X) and i(λ − A) ≥ 0. S (iii) σe5(A) = σewl(A) σewr(A). ♦

Proof. (i) If λ − A ∈ Φl(X) and i(λ − A) ≤ 0. Then, by Lemma 2.2.7, λ − A can be expressed in the form

λ − A = U + K, 144 Linear Operators and Their Essential Pseudospectra where K ∈ K (X) and U ∈ C (X) such that R(U) is closed and U is injective. Furthermore, by Lemma 2.2.10, R(λ − A − K) is complemented. So,

λ ∈/ σl(A + K) and hence,

λ ∈/ σewl(A).

Conversely, if λ ∈/ σewl(A), then there exists K ∈ K (X) such that λ −A− K is left invertible. Hence, λ − A − K is left Fredholm and

α(λ − A − K) = 0.

Using Lemma 2.2.10, we get λ − A ∈ Φl(X) and

i(λ − A) ≤ 0.

(ii) Let λ ∈ ΦrA such that

i(λ − A) ≥ 0.

Using Theorem 2.3.3, there exist K ∈ K (X) and V ∈ C (X) such that

λ − A = V + K, V(X) = X.

Thus, λ − A − K is surjective. Furthermore, by Lemma 2.2.10, N(λ − A −

K) is complemented. So, λ ∈/ σr(A + K). This gives

λ ∈/ σewr(A).

(iii) From Proposition 3.1.1, λ ∈/ σe5(A) if, and only if, λ − A ∈ Φ(X) and i(λ − A) = 0. Then, the result follows from (i) and (ii). Q.E.D.

Remark 5.1.1 Let A ∈ C (X). Then, \ σc(A) ⊂ σewl(A) σewr(A).

In fact, let λ ∈ σc(A), then R(λ − A) is not closed (otherwise λ ∈ ρ(A) see [137, Lemma 5.1 p. 179]). Therefore, T λ ∈ σewl(A) σewr(A). ♦ Essential Spectra 145

Lemma 5.1.1 Let A ∈ C (X) such that 0 ∈ ρ(A). Then, for λ 6= 0, we have 1 −1 λ ∈ σei(A) if, and only if, ∈ σei(A ), i = l, r, wl, wr. ♦ λ

1 Proof. For λ 6= 0, assume that λ ∈ ΦA−1 , then

λ −1 − A−1 ∈ Φb(X).

The operator λ − A can be written in the form

λ − A = −λ(λ −1 − A−1)A. (5.1)

Since A is one-to-one and onto, then by Eq. (5.1), we have N(λ − A) and N(λ −1 − A−1) are isomorphic and, R(λ − A) and R(λ −1 − A−1) are isomorphic. This shows that λ ∈ Φ+A resp. Φ−A and R(λ − A) resp. −1 N(λ − A) is complemented if, and only if, λ ∈ Φ+A−1 resp. Φ−A−1 and R(λ −1 − A−1) resp. N(λ −1 − A−1) is complemented. Therefore,

1 λ ∈ Φl resp. Φr if, and only if, ∈ Φl resp. Φr . A A λ A−1 A−1 Since 0 ∈ ρ(A), then i(A) = 0.

Using both Theorem 2.2.5 and Eq. (5.1), we conclude that

i(λ − A) = i(A) + i(λ −1 − A−1) = i(λ −1 − A−1).

l r So, λ ∈ ΦA resp. ΦA and i(λ − A) ≤ 0 resp. i(λ − A) ≥ 0 if, and only 1 if, ∈ Φl resp. Φr and i(λ −1 −A−1) ≤ 0 resp. i(λ −1 −A−1) ≥ 0. λ A−1 A−1 Hence,

1 −1 λ ∈ σei(A), if, and only if, ∈ σei(A ), i = l, r, wl, wr, λ which completes the proof. Q.E.D. 146 Linear Operators and Their Essential Pseudospectra

Now, we discuss the left and right Weyl essential spectra by means of the polynomially Riesz operators. Let A ∈ C (X), with a non-empty resolvent set, we will give a reﬁnement of the deﬁnition of the left Weyl essential spectrum and the right Weyl essential spectrum of A, respectively, by

l \ σe(A) = σl(A + R) R∈G(X) and r \ σe (A) = σr(A + R), R∈G(X) where

G(X) = {R ∈ L (X) such that (λ − A − R)−1R ∈ PR(X) for all λ ∈ ρ(A + R)}.

For A ∈ L (X), we observe that

K (X) ⊂ G(X).

Indeed, let K ∈ K (X). If we take P(z) = z, then for all λ ∈ ρ(A + K),

P((λ − A − K)−1K) ∈ K (X), so P((λ − A − K)−1K) ∈ R(X). Evidently, the following inclusions hold

l σe(A) ⊆ σewl(A) and r σe (A) ⊆ σewr(A). Note that if R and R0 are two operators in G(X), we have not neces- 0 sarily R + R ∈ G(X). So, we can note deduce the stability of σewl(A) and σewr(A) by perturbations of operators in the class G(X). This bring us to introduce the following subset of G(X). Let Ie(X) = R ∈ L (X) satisfying SR ∈ PR(X) for all S ∈ L (X) .

By [165, Theorem 2.18], Ie(X) is equivalent to R ∈ L (X) satisfying RS ∈ PR(X) for all S ∈ L (X) . Essential Spectra 147

Proposition 5.1.1 Let A ∈ C (X). Then,

(i) If A ∈ Φl(X) and R ∈ Ie(X), then A + R ∈ Φl(X) and

i(A + R) = i(A).

(ii) If A ∈ Φr(X) and R ∈ Ie(X), then A + R ∈ Φr(X) and i(A + R) = i(A). ♦

Proof. Let A ∈ Φl(X), then by Lemma 2.2.8 (i), there exist Al ∈ L (X) and K ∈ K (X) such that

Al(A + R) = I − K + AlR on D(A).

Since R ∈ Ie(X), then AlR ∈ PR(X). By applying Corollary 2.4.2, we b get I + AlR ∈ Φ (X) and

i(I + AlR) = 0.

Since K is compact, then

Al(A + R) ∈ Φ(X) and [ R(Al) R(AlR − K) ⊂ D(A). Using Lemma 2.2.8 (i), we have

A + R ∈ Φl(X).

Since A+R ∈ C (X), we can make D(A+R) = D(A) into a Banach space by introducing the norm

kxkA = kxk + kAxk.

Let XA = (D(A),k · kA) be the Banach space for the graph norm k · kA. We can regard A as operator from XA into X. This will be denoted by Ab. Clearly 148 Linear Operators and Their Essential Pseudospectra that Ab+ Rb and Rb are bounded operators from XA into X, Kb ∈ L (XA) and Abl ∈ L (X,XA). Clearly,

AblAb = IXA − Kb is a Fredholm operator satisfying

i(AblAb) = 0 and

i(Abl(Ab+ Rb)) = i(Al(A + R)).

It is clear Abl is Fredholm if, and only if, Ab is Fredholm if, and only if, Ab+Rb is Fredholm, and therefore, by (2.6)

i(A + R) = i(Ab+ Rb) = i(Ab) = i(A).

(ii) A same reasoning allows us to reach the result of (ii). Q.E.D.

Theorem 5.1.2 Let A ∈ C (X). If R ∈ Ie(X), then

σewl(A) = σewl(A + R) (5.2) and

σewr(A) = σewr(A + R). (5.3) ♦

Proof. Let λ ∈ ΦlA such that i(λ − A) ≤ 0. By Proposition 5.1.1 (i), we have

λ − A − R ∈ Φl(X) and i(λ − A − R) = i(λ − A) ≤ 0.

The opposite inclusion follows by symmetry and, we obtain (5.2). The proof of (5.3) may be checked in a similar way to that (5.2). Q.E.D. Essential Spectra 149

Theorem 5.1.3 Let A and B ∈ C (X) such that ρ(A)Tρ(B) 6= /0. If, for some λ ∈ ρ(A)Tρ(B), the operator (λ −A)−1 −(λ −B)−1 ∈ Ie(X), then

σewl(A) = σewl(B) and

σewr(A) = σewr(B). ♦

Proof. Without loss of generality, we may assume that λ = 0. Hence, 0 ∈ ρ(A)Tρ(B) and therefore, we can write for µ 6= 0

µ − A = −µ(µ−1 − A−1)A.

Since A is one-to-one and onto, then

α(µ − A) = α(µ−1 − A−1) and R(µ − A) = R(µ−1 − A−1).

−1 This shows that µ ∈ ΦlA (resp. ΦrA) if, and only if, µ ∈ ΦlA−1 (resp.

ΦrA−1 ), in this case we have

i(µ − A) = i(µ−1 − A−1).

Therefore, it follows from Proposition 5.1.1 that ΦlA = ΦlB (resp. ΦrA = ΦrB) and i(µ − A) = i(µ − B)

−1 −1 for each µ ∈ ΦlA (resp. ΦrA), since A − B ∈ Ie(X). Hence, to use both Lemma 5.1.1 and Theorem 5.1.1 makes us conclude that σewl(A) =

σewl(B)(resp. σewr(A) = σewr(B)), which completes the proof. Q.E.D.

5.1.2 Characterization of Left and Right Jeribi Essential Spectra

We begin with the following theorem which gives a reﬁnement of the deﬁnition of the left Jeribi essential spectrum on L1-spaces. 150 Linear Operators and Their Essential Pseudospectra

Theorem 5.1.4 Let (Ω,Σ, µ) be an arbitrary positive measure space. Let

A be a closed, densely deﬁned and linear operator on L1(Ω, µ). Then, l σ j(A) = σewl(A). ♦ Proof. We ﬁrst claim that

l σewl(A) ⊆ σ j(A).

l ∗ Indeed, if λ ∈/ σ j(A), then there exists F ∈ W (L1(Ω, µ)) such that λ ∈/ σl(A + F). Then, λ − A − F ∈ Φl(X) and

i(λ − A − F) ≤ 0. (5.4)

∗ Since F ∈ W (L1(Ω, µ)), we have

−1 ∗ (λ − A − F) F ∈ W (L1(Ω, µ)).

Hence, by applying both Eqs. (2.8) and (2.9), we get

−1 2 [(λ − A − F) F] ∈ K (L1(Ω, µ)).

−1 So, I + (λ − A − F) F ∈ Φl(X) and

i(I + (λ − A − F)−1F) = 0. (5.5)

By using the equality

λ − A = (λ − A − F)(I + (λ − A − F)−1F), together with Lemma 2.2.9 and both Eqs. (5.4) and (5.5), we have λ −A ∈

Φl(X) and i(λ − A) = i(λ − A − F) ≤ 0.

Thus, λ ∈/ σewl(A). This proves the claim. Moreover, since

∗ K (L1(Ω, µ)) ⊂ W (L1(Ω, µ)), we infer that l σ j(A) ⊂ σewl(A), which completes the proof of theorem. Q.E.D. Essential Spectra 151

The following theorem will allow us to show that the equalities between the right Weyl essential spectrum and the right Jeribi essential spectrum.

Theorem 5.1.5 Let (Ω,Σ, µ) be an arbitrary positive measure space. Let

A be a closed, densely deﬁned and linear operator on L1(Ω, µ). Then, we have r σ j (A) = σewr(A). ♦

∗ Proof. Since K (L1(Ω, µ)) ⊂ W (L1(Ω, µ)), then

r σ j (A) ⊂ σewr(A).

It remains to show that r σewr(A) ⊆ σ j (A). r ∗ For this, we consider λ ∈/ σ j (A), then there exists F ∈ W (L1(Ω, µ)) such that

λ ∈/ σl(A + F).

∗ Since F ∈ W (L1(Ω, µ)), we have

−1 ∗ (λ − A − F) F ∈ W (L1(Ω, µ)).

Hence, by applying Eqs. (2.8) and (2.9), we get

−1 2 [(λ − A − F) F] ∈ K (L1(Ω, µ)).

Using Lemma 2.2.5, we get

I + (λ − A − F)−1F ∈ Φ(X) and i(I + (λ − A − F)−1F) = 0.

Writing λ − A = (λ − A − F)(I + (λ − A − F)−1F). 152 Linear Operators and Their Essential Pseudospectra

Since λ ∈/ σr(A + F), then λ − A − F is right Fredholm and

i(λ − A) ≥ 0.

Therefore, by Lemma 2.2.9 (ii),

−1 (λ − A − F)(I + (λ − A − F) F) ∈ Φr(X) and i(λ − A) = i(λ − A − F) ≥ 0, which completes the proof. Q.E.D.

Remark 5.1.2 (i) Theorems 5.1.4 and 5.1.5 provide a uniﬁed deﬁnition of the left and right Weyl spectrum on L1-spaces. l r (ii) We have seen in the previous that σ j(A) and σewl(A) (resp. σ j (A) and σewr(A)) are not equal. As we observed in Theorem 5.1.4 that

l σ j(A) = σewl(A) and in Theorem 5.1.5 that

r σ j (A) = σewr(A) on L1-spaces. ♦

As a consequence of Theorems 5.1.4 and 5.1.5, we obtain the equalities l r between σewl(A) (resp. σewr(A)) and σ j(A) (resp. σ j (A)) for A ∈ C (X) with X satisﬁes the Dunford-Pettis property.

Corollary 5.1.1 If X satisﬁes the Dunford-Pettis property and A is a closed densely deﬁned and linear operator on X, then

l σ j(A) = σewl(A), and r σ j (A) = σewr(A). ♦ Essential Spectra 153

Proof. The proof is obtained in the same way of the proof of Theorems 5.1.4 and 5.1.5. Q.E.D.

Remark 5.1.3 If X satisﬁes the Dunford-Pettis property and A is a closed densely deﬁned and linear operator on X, then

l l σ j(A) = σ j(A + K) and r r σ j (A) = σ j (A + K) for all K ∈ W (X). ♦

Recall that the relationship between the left-right Jeribi essential spectra and the left-right Weyl essential spectra is the inclusion strict on a Banach space. This suggests the following theorem.

Theorem 5.1.6 If A is a closed densely deﬁned and linear operator on ∗ Lp(Ω,dµ) with p ∈ [1,∞). If W (Lp(Ω, µ)) = S (Lp(Ω, µ)), then

l σ j(A) = σewl(A), and r σ j (A) = σewr(A). ♦

Proof. The fact that K (Lp(Ω, µ)) ⊂ S (Lp(Ω, µ)), then

l σ j(A) ⊂ σewl(A).

l Conversely, let λ ∈/ σ j(A), then there exists F ∈ S (Lp(Ω, µ)) such that

λ ∈/ σl(A + F).

This implies that λ ∈ ΦlA+F and

i(λ − A − F) ≤ 0. 154 Linear Operators and Their Essential Pseudospectra

Since F ∈ S (Lp(Ω, µ)) and S (Lp(Ω, µ)) is a two-sided ideal of Lp(Ω, µ), we have

−1 (λ − A − F) F ∈ S (Lp(Ω, µ)).

Hence, by applying Lemma 2.1.6, we get

−1 2 [(λ − A − F) F] ∈ K (Lp(Ω, µ)).

Using Lemma 2.2.5, we get

I + (λ − A − F)−1F ∈ Φb(X) and i(I + (λ − A − F)−1F) = 0. Using the equality λ − A as

λ − A = (λ − A − F)(I + (λ − A − F)−1F), together with Lemma 2.2.9 (i), we have

−1 (λ − A − F)(I + (λ − A − F) F) ∈ Φl(X) and i(λ − A) = i(λ − A − F) ≤ 0. Finally, the use of Theorem 5.1.1 shows

λ ∈/ σewl(A), which completes the proof of theorem. Q.E.D.

Corollary 5.1.2 Let A ∈ C (Lp(Ω,dµ)) with p ∈ [1,∞). Then,

(i) If A ∈ Φl(Lp(Ω,dµ)) and R ∈ S (Lp(Ω,dµ)), then A + R ∈

Φl(Lp(Ω,dµ)) and i(A + R) = i(A).

(ii) If A ∈ Φr(Lp(Ω,dµ)) and R ∈ S (Lp(Ω,dµ)), then A + R ∈ Φr(Lp(Ω,dµ)) and i(A + R) = i(A). ♦ Essential Spectra 155

T Theorem 5.1.7 Let A, B ∈ C (Lp(Ω,dµ)) such that ρ(A) ρ(B) 6= /0 (p ∈ [1,∞)). If for some λ ∈ ρ(A)Tρ(B), the operator (λ −A)−1 −(λ −B)−1 ∈

S (Lp(Ω, µ)), then

l l σ j(A) = σ j(B), and r r σ j (A) = σ j (B). ♦

Proof. Without loss of generality, we suppose that λ = 0. Hence, 0 ∈ ρ(A)Tρ(B). Therefore, we can write for µ 6= 0

µ − A = −µ(µ−1 − A−1)A.

Since, A is one-to-one and onto, then

α(µ − A) = α(µ−1 − A−1) and R(µ − A) = R(µ−1 − A−1).

−1 This shows that µ ∈ ΦlA (resp. ΦrA) if, and only if, µ ∈ ΦlA−1 (resp.

ΦrA−1 ), in this case, we have

i(µ − A) = i(µ−1 − A−1).

Therefore, it follows, from Corollary 5.1.2, that ΦlA = ΦlB (resp. ΦrA = ΦrB) and i(µ − A) = i(µ − B)

−1 −1 for each µ ∈ ΦlA (resp. ΦrA), since A − B ∈ S (Lp(Ω, µ)). Hence, l to use from Theorems 5.1.1 and 5.1.6 makes us conclude that σ j(A) = l r r σ j(B)(resp. σ j (A) = σ j (B)). Q.E.D. 156 Linear Operators and Their Essential Pseudospectra

5.2 STABILITY OF ESSENTIAL APPROXIMATE POINT SPECTRUM AND ESSENTIAL DEFECT SPECTRUM OF LINEAR OPERATOR 5.2.1 Stability of Essential Spectra

The reader interested in the results of this section may also refer to [28], which constitutes the real basis of our work. The purpose of this section, is to present the following useful stability of essential spectra.

Theorem 5.2.1 Let X be a Banach space, A and B be two operators in L (X).

(i) Assume that for each λ ∈ Φ+A, there exists a left Fredholm inverse

Aλl of λ − A such that kBAλlk < 1. Then,

σe7(A + B) = σe7(A).

(ii) Assume that for each λ ∈ Φ−A, there exists a right Fredholm inverse Aλr of λ − A such that kAλrBk < 1. Then,

σe8(A + B) = σe8(A). ♦

Proof. Let

n Pγ (X) = {A ∈ L (X) such that γ(A ) < 1, for some n > 0}, where γ(·) is the Kuratowski measure of noncompactness. Since kBAλlk <

1 (resp. kAλrBk < 1), then γ(BAλl) < 1 (resp. γ(AλrB) < 1). So, BAλl ∈

Pγ (X) (resp. AλrB ∈ Pγ (X)). Applying Proposition 2.6.2, we have

b I − BAλl ∈ Φ (X) and i(I − BAλl) = 0, (5.6) and b I − AλrB ∈ Φ (X) and i(I − AλrB) = 0. (5.7)

(i) Let λ 6∈ σe7(A), then by Proposition 3.1.3 (i), we get

b λ − A ∈ Φ+(X) Essential Spectra 157 and i(λ − A) ≤ 0.

b As, Aλl is a left Fredholm inverse of λ − A, then there exists F ∈ F (X) such that

Aλl(λ − A) = I − F on X. (5.8) By Eq. (5.8), the operator λ − A − B can be written in the form

λ −A−B = λ −A−B(Aλl(λ −A)+F) = (I −BAλl)(λ −A)−BF. (5.9)

b According to the Eq. (5.6), we have I − BAλl ∈ Φ+(X), and by using b Theorem 2.2.5, we have (I − BAλl)(λ − A) ∈ Φ+(X) and

i[(I − BAλl)(λ − A)] = i(I − BAλl) + i(λ − A) = i(λ − A) ≤ 0. By using both Eq. (5.9) and Lemma 2.3.1 (ii), we get

λ − A − B ∈ Φ+(X) and i(λ − A − B) = i(λ − A) ≤ 0.

Hence, λ 6∈ σe7(A + B). Conversely, let λ 6∈ σe7(A + B), then by Proposi- tion 3.1.3 (i), we have

b λ − A − B ∈ Φ+(X) and i(λ − A − B) ≤ 0.

Since kBAλlk < 1, and by Eq. (5.9), the operator λ − A can be written in the form −1 λ − A = (I − BAλl) (λ − A − B + BF). (5.10) Using Lemma 2.3.1 (ii), we get

b λ − A − B + BF ∈ Φ+(X) 158 Linear Operators and Their Essential Pseudospectra and i(λ − A − B + BF) = i(λ − A − B) ≤ 0.

Since I − BAλl is boundedly invertible, then by using Eq. (5.10), we have

b λ − A ∈ Φ+(X) and i(λ − A) ≤ 0.

This proves that λ 6∈ σe7(A) and, we conclude that

σe7(A + B) = σe7(A).

(ii) Let λ 6∈ σe8(A), then according to Proposition 3.1.3 (ii), we get

b λ − A ∈ Φ−(X) and i(λ − A) ≥ 0.

Since Aλr is a right Fredholm inverse of λ − A, then there exists F ∈ F b(X) such that

(λ − A)Aλr = I − F on X. (5.11) By using Eq. (5.11), the operator λ − A − B can be written in the form

λ − A − B = λ − A − ((λ − A)Aλr + F)B = (λ − A)(I − AλrB) − FB. (5.12) b A similar proof as (i), it sufﬁces to replace Φ+(·), σe7(·), Eqs. (5.6), (5.9) b and Lemma 2.3.1 (ii) by Φ−(·), σe8(·), Eqs. (5.7), (5.12) and Lemma 2.3.1 (iii), respectively. Hence, we show that

σe8(A + B) ⊂ σe8(A).

Conversely, let λ 6∈ σe8(A + B), then by Proposition 3.1.3 (ii), we have

λ − A − B ∈ Φ−(X) Essential Spectra 159 and i(λ − A − B) ≥ 0.

Since kAλrBk < 1, and in view of Eq. (5.12), the operator λ − A can be written in the form

−1 λ − A = (λ − A − B + FB)(I − AλrB) . (5.13) According to Lemma 2.3.1 (iii), we get

b λ − A − B + FB ∈ Φ−(X) and i(λ − A − B + FB) = i(λ − A − B) ≥ 0.

So, I − AλrB is boundedly invertible. Hence, by Eq. (5.13), we have λ − b A ∈ Φ−(X) and i(λ − A) ≥ 0. This proves that λ 6∈ σe8(A). Thus,

σe8(A + B) = σe8(A), which completes the proof. Q.E.D.

5.2.2 Invariance of Essential Spectra

The purpose of this section is to show the following useful stability result for the essential approximate point spectrum and the essential defect spectrum of a closed, densely deﬁned linear operator on a Banach space X. We begin with the following useful result.

Theorem 5.2.2 Let X be a Banach space and let A, B ∈ Φ(X). Assume that there are A0, B0 ∈ L (X) and F1,F2 ∈ PF (X) such that

AA0 = I − F1, (5.14) and

BB0 = I − F2. (5.15) T b (i) If 0 ∈ ΦA ΦB,A0 − B0 ∈ F+(X) and i(A) = i(B), then

σe7(A) = σe7(B). T b (ii) If 0 ∈ ΦA ΦB,A0 − B0 ∈ F−(X) and i(A) = i(B), then 160 Linear Operators and Their Essential Pseudospectra

σe7(A) = σe7(B). ♦

Proof. Let λ ∈ C, Eqs. (5.14) and (5.15) imply

(λ − A)A0 − (λ − B)B0 = F1 − F2 + λ(A0 − B0). (5.16)

(i) Let λ 6∈ σe7(B), then by Proposition 3.1.3 (i), we have

λ − B ∈ Φ+(X) and i(λ − B) ≤ 0.

It is clearly that B ∈ L (XB,X), where XB = (D(B),k · kB) is a Banach space for the graph norm k · kB. We can regard B as operator from XB into X. This will be denoted by Bb. Then,

b λ − Bb ∈ Φ+(XB,X) and i(λ − Bb) ≤ 0.

Moreover, as F2 ∈ PF (X), Eq. (5.15), and both Theorems 2.5.5 and b 2.2.22 imply that B0 ∈ Φ (X,XB), and consequently,

b (λ − Bb)B0 ∈ Φ+(XB,X).

b Using both Eq. (5.16) and Lemma 2.3.1, the operator A0 − B0 ∈ F+(X) imply that

b (λ − Ab)A0 ∈ Φ+(X) (5.17) and

i((λ − Ab)A0) = i((λ − Bb)B0). (5.18) A similar reasoning as before by combining Eq. (5.14), Theorem 2.5.5 and b Theorem 2.2.22, we show that A0 ∈ Φ (X,XA), where XA = (D(A),k·kA). Now, according to Theorem 2.2.3, we can write

A0T = I − F on XA, (5.19) Essential Spectra 161 where

T ∈ L (XA,X) and F ∈ F (XA). By Eq. (5.19), we have

(λ − Ab)A0T = λ − Ab− (λ − Ab)F. (5.20) b By using Eq. (5.19) and Theorem 2.2.22, we have T ∈ Φ (XA,X). According to both Eq. (5.17) and Theorem 2.2.5, we have

b (λ − Ab)A0T ∈ Φ+(XA,X). Using both Eq. (5.20) and Lemma 2.3.1 (ii), we get

b λ − Ab ∈ Φ+(XA,X). Hence, Eq. (2.6) gives

λ − A ∈ Φ+(X). (5.21)

As, F1, F2 ∈ PF (X), Eqs. (5.14), (5.15) and both Theorems 2.5.5 and 2.2.5 give

i(A) + i(A0) = i(I − F1) = 0 and

i(B) + i(B0) = i(I − F1) = 0.

Since i(A) = i(B), then i(A0) = i(B0). Using Eq. (5.18), we can write

i(λ − A) + i(A0) = i(λ − B) + i(B0). Therefore, i(λ − A) ≤ 0. (5.22)

Using both Eqs. (5.21) and (5.22), we get λ 6∈ σe7(A). Hence,

σe7(A) ⊂ σe7(B). The opposite inclusion follows by symmetry and, we obtain

σe7(A) = σe7(B).

(ii) The proof of (ii) may be checked in a similar way to that in (i). It sufﬁces to replace σe7(·), Φ+(·), and i(·) ≤ 0 by σe8(·), Φ−(·), and i(·) ≥ 0, respectively. Q.E.D. 162 Linear Operators and Their Essential Pseudospectra

5.3 CONVERGENCE

In this section we gather some notation and results of each of convergence compactly and the convergence in generalized sense, that we need to prove our results later.

5.3.1 Convergence Compactly

In this section we investigate the essential spectra, σei(·), i = 1,··· ,5, of the sequence of linear operators in a Banach space X.

Theorem 5.3.1 Let (Tn)n be a bounded linear operators mapping on X, and let T and B be two operators in L (X), λ0 ∈ C, and U ⊆ C is open.

(i) If ((λ0 −Tn −B)−(λ0 −T −B))n converges to zero compactly, and 0 ∈ U , then there exists n0 ∈ N such that, for all n ≥ n0,

σei(λ0 − Tn − B) ⊆ σei(λ0 − T − B) + U , i = 1,··· ,5 and,

δ σei(λ0 − Tn − B),σei(λ0 − T − B) = 0, i = 1,··· ,5, where δ(·) is the gap between two sets.

(ii) If (λ0 − Tn − B)n converges to zero compactly, then there exists n0 ∈ N such that for all n ≥ n0

σei((λ0 − T − B) + (λ0 − Tn − B)) ⊆ σei(λ0 − T − B), i = 1,··· ,5 and,

δ σei((λ0 − T − B) + (λ0 − Tn − B)),σei(λ0 − T − B) = 0, i = 1,··· ,5.

♦ Essential Spectra 163

Proof. (i) For i = 1. Assume that the assertion fails. Then, by passing to a subsequence, it may be deduced that, for each n, there exists λn ∈

σe1(λ0 − Tn − B) such that

λn 6∈ σe1(λ0 − T − B) + U .

It is clear (if necessary pass to a subsequence) that

lim λn = λ n→+∞ since (λn)n is bounded. This implies that

λ 6∈ σe1(λ0 − T − B) + U .

By using the fact that 0 ∈ U , we infer that λ 6∈ σe1(λ0 − T − B). Therefore,

b λ − (λ0 − T − B) ∈ Φ+(X).

Let An = λn − λ + (λ0 − T − B) − (λ0 − Tn − B). Since An converges to zero compactly, writing

λn − (λ0 − Tn − B) = λ − (λ0 − T − B) + An and according to Theorem 2.3.9, we infer that, there exists n0 ∈ N such b that for all n ≥ n0, we have λn − (λ0 − Tn − B) ∈ Φ+(X) and

i(λn − (λ0 − Tn − B)) = i(λ − (λ0 − T − B) + An)

= i(λ − (λ0 − T − B)).

So, λn 6∈ σe1(λ0 − Tn − B), which is a contradiction. Then, for all n ≥ n0, we have

σe1(λ0 − Tn − B) ⊆ σe1(λ0 − T − B) + U . Since 0 ∈ U , we obtain

σe1(λ0 − Tn − B) ⊆ σe1(λ0 − T − B).

Hence, in view of Remark 2.8.1 (i)(b), we get δ σe1(λ0 − Tn − B),σe1(λ0 − T − B) = 0, for all n ≥ n0. 164 Linear Operators and Their Essential Pseudospectra

For i = 2, 3, 4, by using a similar reasoning as the proof of the case i = 1, b b by replacing σe1(·), and Φ+(X) by σe2(·), or σe3(·), or σe4(·), and Φ−(X), b S b b or Φ−(X) Φ+(X), or Φ (X), respectively, we get if

((λ0 − Tn − B) − (λ0 − T − B))n converges to zero compactly, and 0 ∈ U , then there exists n0 ∈ N such that, for all n ≥ n0.

σei(λ0 − Tn − B) ⊆ σei(λ0 − T − B) + U , i = 2,··· ,4.

Furthermore,

δ σei(λ0 − Tn − B),σei(λ0 − T − B) = 0, i = 2,··· ,4.

For i = 5. Assume that the assertion fails. Then, by passing to a subsequence, it may be deduced that, for each n, there exists λn ∈

σe5(λ0 − Tn − B) such that

λn 6∈ σe5(λ0 − T − B) + U .

It is clear (if necessary pass to a subsequence) that

lim λn = λ n→+∞ since (λn)n is bounded, this implies that

λ 6∈ σe5(λ0 − T − B) + U .

Using the fact that 0 ∈ U , we have λ 6∈ σe5(λ0 − T − B) and therefore, b λ − (λ0 − T − B) ∈ Φ (X) and

i(λ − (λ0 − T − B)) = 0.

Let An = λn − λ + (λ0 − T − B) − (λ0 − Tn − B). Since (An)n converges to zero compactly, writing

λn − (λ0 − Tn − B) = λ − (λ0 − T − B) + An Essential Spectra 165 and according to Theorem 2.3.9, we infer that, there exists n0 ∈ N such b that for all n ≥ n0, we have λn − (λ0 − Tn − B) ∈ Φ (X) and

i(λn − (λ0 − Tn − B)) = i(λ − (λ0 − T − B) + An)

= i(λ − (λ0 − T − B)) = 0.

So, λn 6∈ σe5(λ0 − Tn − B), which is a contradiction. Then, for all n ≥ n0, we have

σe5(λ0 − Tn − B) ⊆ σe5(λ0 − T − B) + U . Since 0 ∈ U , we have

σe5(λ0 − Tn − B) ⊆ σe5(λ0 − T − B).

Hence, by Remark 2.8.1 (i)(b), we have for all n ≥ n0, δ σe5(λ0 − Tn − B),σe5(λ0 − T − B) = 0.

(ii) For i = 1. Let λ 6∈ σe1(λ0 − T − B). Then, λ − (λ0 − T − B) ∈ b Φ+(X). Since (λ0 − Tn − B)n converges to zero compactly and applying Theorem 2.3.9 to the operators λ0 − T − B and λ0 − Tn − B, we prove that, there exists n0 ∈ N such that

b λ − (λ0 − T − B) + (λ0 − Tn − B) ∈ Φ+(X) for all n ≥ n0. Hence,

λ 6∈ σe1((λ0 − T − B) + (λ0 − Tn − B)).

We conclude that

σe1((λ0 − T − B) + (λ0 − Tn − B)) ⊆ σe1(λ0 − T − B).

Now applying Remark 2.8.1 (i)(b), we obtain for all n ≥ n0, δ σe1((λ0 − T − B) + (λ0 − Tn − B)),σe1(λ0 − T − B) = 0.

For i = 2, 3, 4, by using a similar reasoning as the proof of the case i = 1, b b by replacing σe1(·), and Φ+(X) by σe2(·), or σe3(·), or σe4(·), and Φ−(X), 166 Linear Operators and Their Essential Pseudospectra

b S b b or Φ−(X) Φ+(X), or Φ (X), respectively, we get if (λ0 − Tn − B)n converges to zero compactly, then there exists n0 ∈ N such that for all n ≥ n0,

σei((λ0 − T + B) + (λ0 − Tn − B)) ⊆ σei(λ0 − T − B).

Furthermore, for all n ≥ n0, we have δ σei((λ0 − T − B) + (λ0 − Tn − B)),σei(λ0 − T − B) = 0.

b For i = 5. Let λ 6∈ σe5(λ0 − T − B). Then, λ − (λ0 − T − B) ∈ Φ (X) and

i(λ − (λ0 − T − B)) = 0.

Since (λ0 − Tn − B)n converges to zero compactly and by applying the

Theorem 2.3.9 to the both operators λ0 − T − B and λ0 − Tn − B, we prove that, there exists n0 ∈ N such that for all n ≥ n0, we have

b λ − (λ0 − T − B) + (λ0 − Tn − B) ∈ Φ (X).

Hence,

λ 6∈ σe5((λ0 − T − B) + (λ0 − Tn − B)). We conclude that

σe5((λ0 − T − B) + (λ0 − Tn − B)) ⊆ σe5(λ0 − T − B).

Now applying Remark 2.8.1 (i)(b), we have for all n ≥ n0 δ σe5((λ0 − T − B) + (λ0 − Tn − B)),σe5(λ0 − T − B) = 0. Q.E.D. The following results may be found in [28].

Theorem 5.3.2 Let (Tn)n be a sequence in L (X) and let T be a bounded linear operator on X. Essential Spectra 167

(i) If (Tn)n converges to T compactly, U ⊆ C is open and 0 ∈ U , then there exists n0 ∈ N such that, for every n ≥ n0

σe7(Tn) ⊆ σe7(T) + U , (5.23) and

σe8(Tn) ⊆ σe8(T) + U . (5.24)

(ii) If (Tn)n converges to zero compactly, then there exists n0 ∈ N such that, for every n ≥ n0

σe7(T + Tn) ⊆ σe7(T), (5.25) and

σe8(T + Tn) ⊆ σe8(T). (5.26) ♦

Proof. (i) The proof by contradiction, assume that the inclusion is fails. Then, by passing to a subsequence (if necessary) it may be assumed that, for each n, there exists λn ∈ σe7(Tn) such that

λn ∈/ σe7(T) + U , since (λn)n is bounded, we suppose (if necessary pass to a subsequence) that

lim λn = λ, n→+∞ which implies that λ ∈/ σe7(T) + U . Using the fact that 0 ∈ U , we have b λ ∈/ σe7(T). Therefore, λ − T ∈ Φ+(X) and i(λ − T) ≤ 0.

c In other hand (λn −Tn)−(λ −T) −→ 0, which implies, by Theorem 2.3.9 b (i) and (iv), that λn − Tn ∈ Φ+(X), and

i(λn − Tn) = i(λ − T) ≤ 0.

Hence, λn ∈/ σe7(Tn), which is a contradiction. So, the inclusion (5.23) holds. The statement (5.24) for the essential defect spectrum can be proved similarly. 168 Linear Operators and Their Essential Pseudospectra

b (ii) Let λ ∈/ σe7(T). Then, λ − T ∈ Φ+(X), and

i(λ − T) ≤ 0.

Since (Tn)n converge to 0 compactly, then if we apply Theorem 2.3.9 (i) and (iv), we obtain that there exists n0 ∈ N such that for all n > n0,

(λ − T) − Tn = λ − (T + Tn) ∈ Φ+(X), and i(λ − T) = i λ − (T + Tn) ≤ 0, which implies that λ ∈/ σe7(T + Tn). Hence, the inclusion (5.25) is valid. For the inclusion (5.26), the proof is similarly. Q.E.D.

From the above result, we have the following.

Corollary 5.3.1 Let T be a closed linear operator and let (Tn)n be a T sequence of closed linear operators on X such that ρ(Tn) ρ(T) 6= /0, T −1 −1 and let η ∈ ρ(Tn) ρ(T). If (Tn − η) − (T − η) converges to zero compactly, then there exists n0 ∈ N such that, for all n ≥ n0

σe7(Tn − η) ⊆ σe7(T − η), and

σe8(Tn − η) ⊆ σe8(T − η). ♦

−1 −1 Proof. If we put Kn = (Tn − η) − (T − η) , then

−1 −1 (Tn − η) = (T − η) + Kn, and Kn converges to zero compactly. From the inclusions (5.25) and (5.26), respectively, we have

−1 −1 −1 σe7 (Tn − η) = σe7 (T − η) + Kn ⊆ σe7 (T − η) , and

−1 −1 −1 σe8 (Tn − η) = σe8 (T − η) + Kn ⊆ σe8 (T − η) . Essential Spectra 169

Then, by using Lemma 3.1.1, we have

σe7(Tn − η) ⊆ σe7(T − η), and

σe8(Tn − η) ⊆ σe8(T − η), which completes the proof. Q.E.D.

5.3.2 Convergence in the Generalized Sense

The ﬁrst main result is embodied in the following theorem.

Theorem 5.3.3 Let (Tn)n be a sequence of closed linear operators mapping on Banach spaces X and let T ∈ C (X), and let B and L be two operators in L (X), λ0 ∈ C such that (Tn)n converges in the generalized sense to T, λ0 ∈ ρ(T + B), and U ⊆ C is open. (i) If 0 ∈ U , then there exists n0 ∈ N such that, for every n ≥ n0, we have

σei(λ0 − Tn − B) ⊆ σei(λ0 − T − B) + U , i = 1,··· ,5. (5.27)

Furthermore, δ σei(λ0 − Tn − B),σei(λ0 − T − B) = 0, i = 1,··· ,5.

(ii) Let i = 1,··· ,5. Then, there exist ε > 0 and n ∈ N such that, for all kLk < ε, we have

σei(λ0 − Tn − B + L) ⊆ σei(λ0 − T − B) + U , for all n ≥ n0. (5.28)

Furthermore, for i = 1,··· ,5, we have δ σei(λ0 − Tn − B + L),σei(λ0 − T − B) = δ σei(λ0 − T − B + L),σei(λ0 − T − B) = 0. ♦ 170 Linear Operators and Their Essential Pseudospectra

Proof. (i) For i = 1. In view of both B − λ0 is a bounded operator and

λ0 ∈ ρ(T + B) and according to Theorem 2.8.2 (i) and (iii), the sequence

(λ0 − Tn − B)n converges in the generalized sense to λ0 − T − B, and −1 λ0 ∈ ρ(Tn + B) for a sufficiently large n and (λ0 − Tn − B) converges to −1 (λ0 − T − B) . Now, to prove Eq. (5.27), it suffices to prove the existence of n0 ∈ N, such that for all n ≥ n0, we have −1 −1 σe1 (λ0 − Tn − B) ⊆ σe1 (λ0 − T − B) + U . (5.29) In first step, by an indirect proof, we suppose that the (5.29) does not hold, −1 and for each n ∈ N there exists λn ∈ σe1 (λ0 − Tn − B) such that −1 λn 6∈ σe1 (λ0 − T − B) + U . It is clear (if necessary pass to a subsequence) that

lim λn = λ n→+∞ since (λn)n is bounded. This implies that −1 λ 6∈ σe1 (λ0 − T − B) + U .

−1 In view of 0 ∈ U , we have λ 6∈ σe1 (λ0 − T − B) . Therefore,

−1 b λ − (λ0 − T − B) ∈ Φ+(X). Applying Theorem 2.8.2 (ii), we conclude that

−1 −1 δb(λn − (λ0 − Tn − B) ,λ − (λ0 − T − B) ) → 0, as n → ∞.

−1 Let δ = γe(λ − (λ0 − T − B) ) > 0. Then, there exists N ∈ N such that, for all n ≥ N, we have

−1 −1 δ δb(λn − (λ0 − Tn − B) ,λ − (λ0 − T − B) ) ≤ √ . 1 + δ 2 According to Theorem 2.8.1 (iv), we infer that

−1 b λn − (λ0 − Tn − B) ∈ Φ+(X). Then, we obtain −1 λn ∈/ σe1((λ0 − Tn − B) ), which contradicts our assumption. Hence, (5.29) holds. Essential Spectra 171

Now, let λ ∈ σe1(λ0 − Tn − B), then

1 −1 ∈ σ ((λ − Tn − B) ). λ e1 0 In view of (5.29), we conclude that

1 −1 ∈ σe ((λ − T − B) ) + U . (5.30) λ 1 0 Since 0 ∈ U , then (5.30) implies that 1 ∈ σ ((λ − T − B)−1). λ e1 0 We have to prove

λ ∈ σe1(λ0 − T − B) + U . (5.31) We will proceed by contradiction, we suppose that

λ 6∈ σe1(λ0 − T − B) + U .

The fact that 0 ∈ U implies that λ 6∈ σe1(λ0 − T − B) and so, 1 6∈ σ ((λ − T − B)−1) λ e1 0 which contradicts our assumption. This proves Eq. (5.31). Therefore, (5.27) holds. Since U is an arbitrary neighborhood of 0 and by using the relation (5.27), we have

σe1(λ0 − Tn − B) ⊆ σe1(T + B − λ 0), for all n ≥ n0. Hence, by Remark 2.8.1 (i)(b), we have for all n ≥ n0, δ σe1(λ0 − Tn − B),σe1(λ0 − T − B) = δ σe1(λ0 − Tn − B),σe1(λ0 − T + B) = 0. This ends the proof for i = 1. 172 Linear Operators and Their Essential Pseudospectra

For i = 2, 3, 4, by using a similar reasoning as the proof of ((i) for i = 1), by replacing σe1(·), and Φ+(X) by σe2(·), or σe3(·), or σe4(·), and S Φ−(X), or Φ−(X) Φ+(X), or Φ(X), respectively, we get

σei(λ0 − Tn − B) ⊆ σei(λ0 − T − B) + U .

Furthermore, for all n ≥ n0, we have δ σei(λ0 − Tn − B),σei(λ0 − T − B) = 0.

For i = 5. In view of both λ0 −B is a bounded operator and λ0 ∈ ρ(T +B), and according to Theorem 2.8.2 (i) and (iii), the sequence

(λ0 − Tn − B)n converges in the generalized sense to λ0 − T − B, and λ0 ∈ ρ(Tn + B) for −1 −1 a sufﬁciently large n and (λ0 − Tn − B) converges to (λ0 − T − B) . Now, to prove Eq. (5.27), it sufﬁces to prove there exist n0 ∈ N, such that for all n ≥ n0, we have

−1 −1 σe5 (λ0 − Tn − B) ⊆ σe5 (λ0 − T − B) + U . (5.32)

In ﬁrst step, by an indirect proof, we suppose that the inclusion (5.32) does −1 not hold, and for each n ∈ N, there exists λn ∈ σe5 (λ0 − Tn − B) such that −1 λn 6∈ σe5 (λ0 − T − B) + U . It is clear (if necessary pass to a subsequence) that

lim λn = λ n→+∞ since (λn)n is bounded. This implies that

−1 λ 6∈ σe5 (λ0 − T − B) + U .

In view of 0 ∈ U , we have

−1 λ 6∈ σe5 (λ0 − T − B) . Essential Spectra 173

−1 b Therefore, λ − (λ0 − T − B) ∈ Φ (X) and

−1 i(λ − (λ0 − T − B) ) = 0.

Applying Theorem 2.8.2 (ii), we conclude that

−1 −1 δb(λn − (λ0 − Tn − B) ,λ − (λ0 − T − B) ) → 0 as n → ∞.

−1 Let δ = γe(λ − (λ0 − T − B) ) > 0. Then, there exists N ∈ N such that, for all n ≥ N, we have

−1 −1 δ δb(λn − (λ0 − Tn − B) ,λ − (λ0 − T − B) ) ≤ √ . 1 + δ 2 According to Theorem 2.8.1 (iv), we infer

−1 b λn − (λ0 − Tn − B) ∈ Φ (X) and −1 −1 i(λn − (λ0 − Tn − B) ) = i(λ − (λ0 − T − B) ) = 0.

−1 Then, we obtain λn ∈/ σe5((λ0 −Tn −B) ), which is a contradicts. Hence,

(5.32) holds. Now, if λ ∈ σe5(λ0 − Tn − B), then

1 −1 ∈ σ ((λ − Tn − B) ). λ e5 0 In view of Eq. (5.32), we conclude that 1 ∈ σ ((λ − T − B)−1) + U . (5.33) λ e5 0 Since 0 ∈ U , then (5.33) implies that 1 ∈ σ (λ − T − B)−1. λ e5 0 We have to prove

λ ∈ σe5(λ0 − T − B) + U . (5.34) We will proceed by contradiction, we suppose that

λ 6∈ σe5(λ0 − T − B) + U . 174 Linear Operators and Their Essential Pseudospectra

The fact that 0 ∈ U implies that

λ 6∈ σe5(λ0 − T − B)

1 −1 and so, λ 6∈ σe5((λ0 − T − B) ) which contradicts our assumption. This proves Eq. (5.34). Therefore (5.27) holds. Since U is an arbitrary neighborhood of 0 and by using (5.27), we have for all n ≥ n0

σe5(λ0 − Tn − B) ⊆ σe5(λ0 − T − B).

Hence, by Remark 2.8.1 (i)(b), we have for all n ≥ n0, δ σe5(λ0 − Tn − B),σe5(λ0 − T − B) = δ σe5(λ0 − Tn − B),σe5(λ0 − T − B) = 0. This ends the proof of (i). −1 (ii) For i = 1. Since λ0 ∈ ρ(T + B), then (T + B − λ0) exists and 1 bounded. Let ε1 = −1 and let L ∈ L (X) such that kLk < ε1. k(λ0−T−B) k Then, −1 kL(λ0 − T − B) k < 1.

According to Theorem 2.8.2 (i), the sequence (λ0 −Tn −B+L)n converges in the generalized sense to λ0 − T − B + L, and the Neumann series ∞ −1 k ∑(−L(λ0 − T − B) ) k=0

−1 −1 converges to (I + L(λ0 − T − B) ) and

−1 −1 1 k(I + L(λ0 − T − B) ) k < −1 . 1 − kLkk(λ0 − T − B) k Since

−1 λ0 − T − B + L = (I + L(λ0 − T − B) )(λ0 − T − B), then λ0 ∈ ρ(T + B + L). Now, applying (i) for i = 1, we deduce that there exists n0 ∈ N such that for all n ≥ n0, we have

σe1(λ0 − Tn − B + L) ⊆ σe1(λ0 − T − B + L) + U . Essential Spectra 175

Let λ 6∈ σe1(λ0 − T − B). Then,

λ − (λ0 − T − B) ∈ Φ+(X).

By applying Theorem 2.2.15, there exists ε2 > 0 such that for kLk < ε2, one has

λ − (λ0 − T − B) − L ∈ Φ+(X) and, this implies that λ 6∈ σe1(λ0 − T − B + L). From above and if we take ε = min(ε1,ε2), then for all kLk < ε, there exists n0 ∈ N such that for all n ≥ n0, we have

σe1(λ0 − Tn − B + L) ⊆ σe1(λ0 − T − B) + U .

Therefore, Eq. (5.28) holds for i = 1. Since 0 ∈ U , then we have δ σe1(λ0 − Tn − B + L),σe1(λ0 − T − B) = 0 and δ σe1(λ0 − T − B + L),σe1(λ0 − T − B) = 0. For i = 2, 3, 4. By using a similar reasoning as the proof of (ii) for i = 1, by replacing σe1(·), and Φ+(X) by σe2(·), or σe3(·), or σe4(·), and S Φ−(X), or Φ−(X) Φ+(X), or Φ(X), respectively, we get there exist ε > 0 and n ∈ N such that, for all kLk < ε, we have for all n ≥ n0,

σei(λ0 − Tn − B + L) ⊆ σei(λ0 − T − B) + U .

Furthermore, δ σei(λ0 − Tn − B + L),σei(λ0 − T − B) = δ σei(λ0 − T − B + L),σei(λ0 − T − B) = 0. 176 Linear Operators and Their Essential Pseudospectra

−1 For i = 5. Since λ0 ∈ ρ(T + B), then (λ0 − T − B) exists and 1 bounded. We put ε1 = −1 . Let L ∈ L (X) such that kLk < ε1. k(λ0−T−B) k This implies −1 kL(λ0 − T − B) k < 1.

By according to Theorem 2.8.2 (i), we have (λ0 − Tn − B + L)n converges in the generalized sense to λ0 − T − B + L, and the Neumann series ∞ −1 k ∑(−L(λ0 − T − B) ) k=0

−1 −1 converges to (I + L(λ0 − T − B) ) and

−1 −1 1 k(I + L(λ0 − T − B) ) k < −1 . 1 − kLkk(λ0 − T − B) k Since

−1 λ0 − T − B + L = (I + L(λ0 − T − B) )(λ0 − T − B), then λ0 ∈ ρ(T + B + L). Now, applying (i) for i = 5, we deduce that there exists n0 ∈ N such that for all n ≥ n0, we have

σe5(λ0 − Tn − B + L) ⊆ σe5(λ0 − T − B + L) + U .

Let λ 6∈ σe5(λ0 − T − B). Then, λ − (λ0 − T − B) ∈ Φ(X) and

i(λ − (λ0 − T − B)) = 0.

By applying Theorem 2.2.15, there exists ε2 > 0 such that for kLk < ε2, one has

λ − (λ0 − T − B) − L ∈ Φ(X) and

i(λ − (λ0 − T − B − L)) = i(λ − (λ0 − T − B) = 0.

This implies that λ 6∈ σe5(λ0 − T − B + L). From above and if we take ε = min(ε1,ε2), then for all kLk < ε, there exists n0 ∈ N such that

σe5(λ0 − Tn − B + L) ⊆ σe5(λ0 − T − B) + U , for all n ≥ n0. Since 0 ∈ U , then we have Essential Spectra 177

δ σe5(λ0 − Tn − B + L),σe5(λ0 − T − B) = δ σe5(λ0 − T − B + L),σe5(λ0 − T − B) = 0. Therefore, (ii) holds for i = 5. Q.E.D. The next main results in the rest of this section is embodied in the following theorem, when we discuss and study the essential approximate point spectrum, and the essential defect spectrum of a sequence of closed linear operators perturbed by a bounded operator, and converges in the generalized sense to a closed linear operator in a Banach space.

Theorem 5.3.4 Let X be Banach space, let (Tn)n be a sequence of closed linear operators which converges in the generalized sense, in C (X), to a closed linear operator T, and let B be a bounded linear operator on X such that ρ(T + B) 6= /0. If λ0 ∈ ρ(T + B), then

(i) there exists n0 ∈ N such that, for every n ≥ n0, we have

σe7(Tn + B − λ 0) ⊆ σe7(T + B − λ 0) + U (5.35) and

σe8(Tn + B − λ 0) ⊆ σe8(T + B − λ 0) + U , (5.36) where U ⊂ C is an open containing 0. In particular, for all n ≥ n0 δ σe7(Tn + B − λ0),σe7(T + B − λ0) = δ σe8(Tn + B − λ0),σe8(T + B − λ0) = 0,

(ii) there exist ε > 0, and n0 ∈ N, such that, for all S ∈ L (X), and kSk < ε, we have for all n ≥ n0,

σe7(Tn + B + S − λ 0) ⊆ σe7(T + B − λ 0) + U , and

σe8(Tn + B + S − λ 0) ⊆ σe8(T + B − λ 0) + U , where U ⊂ C is an open containing 0. In particular, for all n ≥ n0 δ σe7(Tn + B + S − λ0),σe7(T + B − λ0) = δ σe8(Tn + B + S − λ0),σe8(T + B − λ0) = 0. ♦ 178 Linear Operators and Their Essential Pseudospectra

Proof. For (i), before proof, we make some preliminary observations. g Since Tn −→ T, then by Theorem 2.8.2 (i),

g Tn + B − λ0 −→ T + B − λ0.

−1 Furthermore, we have (T + B − λ0) ∈ L (X), which implies, according to Theorem 2.8.2 (iii), that λ0 ∈ ρ(Tn + B) for a sufﬁciently large n and −1 −1 (Tn + B − λ0) converges to (T + B − λ0) . We recall that the essential approximate point spectrum of a bounded operator is compact. But this property is not valid for the case of unbounded operators, for this reason, −1 −1 using the compactness of σe7((T + B − λ0) ) because (T + B − λ0) is bounded, as a ﬁrst step, we will prove the existence of n0 ∈ N, such that for all n ≥ n0, we have

−1 −1 σe7((Tn + B − λ0) ) ⊆ σe7((T + B − λ0) ) + U . (5.37)

The proof by contradiction. Suppose that (5.37) does not hold. Then, by studying a subsequence (if necessary), we may assume that, for each n, there exists −1 λn ∈ σe7((Tn + B − λ0) ) such that −1 λn ∈/ σe7((T + B − λ0) ) + U .

Since (λn)n is bounded, we may assume (if necessary pass to a subsequence) that

lim λn = λ, n→+∞ which implies that

−1 λ 6∈ σe7(T + B − λ0) + U .

Since 0 ∈ U , then −1 λ 6∈ σe7((T + B − λ0) ). Therefore, −1 b λ − (T + B − λ0) ∈ Φ+(X) Essential Spectra 179 and −1 i λ − (T + B − λ0) ≤ 0.

−1 −1 In other hand, λn − (Tn + B − λ0) converges to λ − (T + B − λ0) , we deduce that

−1 −1 δb λn − (Tn + B − λ0) ,λ − (T + B − λ0) → 0

−1 as n → ∞. Let δ = γe λ − (T + B − λ0) > 0. Then, there exists n0 ∈ N such that, for all n ≥ n0, we have

−1 −1 δ δb λn − (Tn + B − λ0) ,λ − (T + B − λ0) ) ≤ √ . 1 + δ 2 By using Theorem 2.8.1 (iv), we infer that

−1 b λn − (Tn + B − λ0) ∈ Φ+(X) and −1 i λn − (Tn + B − λ0) ≤ 0. Furthermore, there exists b > 0 such that

−1 −1 δb λn − (Tn + B − λ0) ,λ − (T + B − λ0) < b, which implies

−1 −1 i λn − (Tn + B − λ0) = i λ − (T + B − λ0) ≤ 0.

Then, we obtain −1 λn ∈/ σe7 (Tn + B − λ0) , which is a contradiction. Hence, (5.37) holds. Now, we assume that λ ∈

σe7(Tn + B − λ0), then

1 −1 ∈ σ (Tn + B − λ ) . λ e7 0 By using the inclusion (5.37), we have

1 −1 ∈ σe (T + B − λ ) + U , λ 7 0 180 Linear Operators and Their Essential Pseudospectra which implies that 1 ∈ σ (T + B − λ )−1 λ e7 0 because 0 ∈ U , Now, we claim that λ ∈ σe7(T + B − λ0) + U . In fact, let us assume that λ ∈/ σe7(T +B−λ0)+U . The fact that 0 ∈ U implies that

λ 6∈ σe7(T + B − λ0) and so,

1 −1 6∈ σ (Tn + B − λ ) λ e7 0 which is a contradiction. This proves the claim and so,

λ ∈ σe7(T + B − λ0) + U .

This implies that (5.35) holds. Since U is an arbitrary neighborhood of 0 and by using (5.35), we get

σe7(Tn + B − λ 0) ⊆ σe7(T + B − λ 0), for all n ≥ n0. Hence, by Remark 2.8.1 (ii), δ σe7(Tn + B − λ 0),σe7(T + B − λ 0) = δ σe7(Tn + B − λ 0),σe7(T + B − λ 0) = 0,

for all n ≥ n0. With the same procession as we do for (5.35), and using Proposition 3.1.3 (ii), the inclusion (5.36) yields. Therefore, (i) holds. (ii) Since S is bounded, it is clear by using Theorem 2.8.2 (i), that the sequence

An = Tn + B + S − λ 0 converges in the generalized sense to the operator A = T + B + S − λ 0, then we need, for applying (i), to prove that

ρ(T + B) ⊂ ρ(T + B + S).

Let λ0 ∈ ρ(T + B), for S ∈ L (X) such that 1 kSk < −1 = ε1, k(T + B − λ 0) k Essential Spectra 181 we have −1 kS(T + B − λ 0) k < 1,

−1−1 which gives that I + S(T + B − λ 0) exists and bounded, when the existence is given by the convergence of the Neumann serie

∞ −1k ∑ − S(T + B − λ 0) , k=0 and the boundedness is immediately from the inequality

−1−1 1 k I + S(T + B − λ 0) k < −1 , 1 − kSkk(T + B − λ 0) k which implies that the operator

−1 −1 −1−1 (T + B − λ 0) + S = (T + B − λ 0) I + S(T + B − λ0) exists and bounded, then 0 ∈ ρ(T + B + S − λ0). Now, applying (i) to An and A, we deduce that there exists n0 ∈ N, such that

σe7(Tn + B + S − λ0) ⊆ σe7(T + B + S − λ 0) + U , for all n ≥ n0, where U ⊂ C is an open set containing 0. Now, we will prove, by contradiction, the following

σe7(T + B + S − λ 0) ⊆ σe7(T + B − λ 0).

Let λ 6∈ σe7(T + B − λ 0), then λ − (T + B − λ 0) ∈ Φ+(X) and i λ − (T + B − λ 0) ≤ 0.

From Theorem 2.2.16, we deduce that there exists ε2 > 0 such that for kSk < ε2, one has λ − (T + B + S − λ0) ∈ Φ+(X) and i λ − (T + B + S − λ0) = i λ − (T + B − λ0) ≤ 0.

This implies that λ 6∈ σe7(T + B + S − λ0). Then, by transitivity

σe7(Tn + B + S − λ 0) ⊆ σe7(T + B − λ 0). 182 Linear Operators and Their Essential Pseudospectra

From what has been mentioned and if we take ε = min(ε1,ε2), then for all kSk < ε, there exists n0 ∈ N such that

σe7(Tn + B + S − λ 0) ⊆ σe7(T + B − λ 0) + U , for all n ≥ n0. With the same procedure of the previous prove, the inclusion concerned in σe8(·) yields. Since U is an arbitrary neighborhood of the origin, then we have δ σe7(Tn + S + B − λ0),σe7(T + B + S − λ0) = δ σe7(T + S + B − λ0),σe7(T + B − λ0) = 0. Therefore, (ii) holds. Q.E.D.

A particular case is obtained from Theorem 5.3.4, if we replace B by 0, and λ0 by 0, which requires that 0 ∈ ρ(T), then we have the following corollary.

Corollary 5.3.2 Let (Tn)n be a sequence of closed linear operators and T g be a closed operator such that (Tn)n −→ T, we suppose that 0 ∈ ρ(T). If U ⊂ C is open and 0 ∈ U , then there exists n0 ∈ N such that, for every n ≥ n0, we have

σe7(Tn) ⊆ σe7(T) + U , and

σe8(Tn) ⊆ σe8(T) + U .

In particular, for all n ≥ n0 δ σe7(Tn),σe7(T) = δ σe8(Tn),σeδ (T) = 0. ♦ Chapter 6

S-Essential Spectra of Closed Linear Operator on a Banach Space

In this chapter, we give a characterization of S-essential spectra of the closed, densely deﬁne linear operator A on a Banach space X.

6.1 S-ESSENTIAL SPECTRA 6.1.1 Characterization of S-Essential Spectra

In the next, we will suppose that S is not invertible.

Lemma 6.1.1 Let A ∈ C (X) and S ∈ L (X). Then, \ \ (i) σe5,S(A) = σS(A + K) = σS(A + K), K∈F0(X) K∈F (X) \ \ (ii) σeap,S(A) = σap,S(A + K) = σap,S(A + K), and K∈F0(X) K∈F+(X) \ \ (iii) σeδ,S(A) = σδ,S(A + K) = σδ,S(A + K), K∈F0(X) K∈F−(X) where F0(X) stands for the ideal of ﬁnite rank operators and σδ,S(A) := λ ∈ C such that λS − A is not surjective . ♦ 184 Linear Operators and Their Essential Pseudospectra

\ Proof. (i) Let λ 6∈ O = σS(A+K). Then, there exists K ∈ F0(X) K∈F0(X) such that λ ∈ ρS(A + K). Hence,

A + K − λS ∈ Φ(X) and i(A + K − λS) = 0.

Now, the operator A − λS can be written in the form

A − λS = A + K − λS − K.

By Theorem 2.2.14, we have A − λS ∈ Φ(X) and

i(A − λS) = 0.

Then,

λ 6∈ σe5,S(A).

Conversely, we suppose that λ 6∈ σe5,S(A), then

A − λS ∈ Φ(X) and i(A − λS) = 0. Let n = α(A − λS) = β(A − λS), x1,··· ,xn being the basis for N((A − ∗ 0 0 ⊥ λS) ) and y1,··· ,yn being the basis for annihilator R(A − λS) . By 0 0 ∗ Lemma 2.1.1, there are functionals x1,··· ,xn in X (the adjoint space of X) and elements y1,··· ,yn such that

0 0 x j(xk) = δ jk and y j(yk) = δ jk, 1 ≤ j, k ≤ n, where δ jk = 0 if j 6= k and δ jk = 1 if j = k. The operator K is deﬁned by

n 0 Kx = ∑ xk(x)yk, x ∈ X. k=1 S-Essential Spectra of Closed Linear Operator 185

Clearly, K is a linear operator defined everywhere on X. It is bounded, since n 0 kKxk ≤ kxk ∑ kxkkkykk . k=1 Moreover, the range of K is contained in a finite dimensional subspace of X. Then, K is a finite rank operator in X. In the next, we will prove that

\ \ N(A − λS) N(K) = {0} and R(A − λS) R(K) = {0}. (6.1)

Let x ∈ N(A − λS), then

n x = ∑ αkxk, k=1 therefore 0 x j(x) = α j, 1 ≤ j ≤ n. On the other hand, if x ∈ N(K), then

0 x j(x) = 0, 1 ≤ j ≤ n.

This proves the ﬁrst relation in Eq. (6.1). The second inclusion is similar. In fact, if y ∈ R(K), then n y = ∑ αkyk, k=1 and hence,

y j(y) = α j, 1 ≤ j ≤ n.

But, if y ∈ R(A − λS), then

0 y j(y) = 0, 1 ≤ j ≤ n.

This gives the second relation in Eq. (6.1). Using the fact that K is a compact operator, we deduce from Theorem 2.2.14 that λ ∈ ΦA+K,S. If 186 Linear Operators and Their Essential Pseudospectra x ∈ N(A − λS + K), then (A − λS)x is in R(A − λS)TR(K) which implies that x ∈ N(A − λS)TN(K) and hence, x = 0. Thus,

α(A − λS + K) = 0.

In the same way, one proves that

R(A − λS + K) = X.

We conclude that λ 6∈ O. Also, \ σe5,S(A) = σS(A + K). K∈F0(X) \ Now, let O1 := σS(A + F). Since F0(X) ⊂ F (X), we infer that F∈F (X)

O ⊂ σe5,S(A).

Conversely, let λ 6∈ O1, then there exist F ∈ F (X) such that

λ 6∈ σS(A + F).

Then, λ ∈ ρS(A + F). So,

A + F − λS ∈ Φ(X) and i(A + F − λS) = 0. The use of Lemma 2.3.1 makes us conclude that

A − λS ∈ Φ(X) and i(A − λS) = 0. Then,

λ 6∈ σe5,S(A). So, \ \ σe5,S(A) = σS(A + K) = σS(A + K). K∈F0(X) K∈F (X)

Using the following relations F0(X) ⊂ K (X) ⊂ F (X), we have S-Essential Spectra of Closed Linear Operator 187

\ \ σe5,S(A) = σS(A + K) ⊂ σS(A + K) ⊂ K∈F (X) K∈K (X) \ σS(A + K) = σe5,S(A). K∈F0(X) Statements (ii) and (iii) can be checked in a similar way as the assertion (i). Q.E.D.

Lemma 6.1.2 Let A ∈ C (X) and S ∈ L (X). If ΦA,S is connected and ρS(A) 6= /0, then

(i) σe1,S(A) = σeap,S(A).

(ii) σe2,S(A) = σeδ,S(A). ♦

Proof. (i) It is easy to check that

σe1,S(A) ⊂ σeap,S(A).

For the second inclusion we take λ ∈ C\σe1,S(A), then [ λ ∈ ΦA,S (Φ+A,S\ΦA,S).

Hence, we will discuss the following two cases:

st 1 case : If λ ∈ ΦA,S, then

i(A − λS) = 0.

Indeed, let λ0 ∈ ρS(A), then λ0 ∈ ΦA,S and

i(A − λ0S) = 0.

It follows from Proposition 2.2.1 that i(A−λS) is constant on any component of ΦA,S. Therefore,

ρS(A) ⊆ ΦA,S. Hence, i(A − λS) = 0 for all λ ∈ ΦA,S. This shows that λ ∈ ρeap,S(A). 188 Linear Operators and Their Essential Pseudospectra

nd 2 case : If µ ∈ Φ+A,S\ΦA,S, then

α(A − λS) < ∞ and β(A − µS) = +∞. So, i(A − λS) = −∞ < 0. Thus, we obtain from the above

σeap,S(A) ⊂ σe1,S(A).

Statement (ii) can be checked similarly from the assertion (ii). Q.E.D.

6.1.2 Stability of S-Essential Spectra of Closed Linear Operator

Theorem 6.1.1 Let A ∈ C (X),S ∈ L (X), λ ∈ C, and let B be a bounded operator on X. Then, (i) If A − λS ∈ Φ(X) and B ∈ AF (X), then A + B − λS ∈ Φ(X) and

i(A + B − λS) = i(A − λS).

(ii) If A − λS ∈ Φ+(X) and B ∈ AF+(X), then A + B − λS ∈ Φ+(X). (iii) If A − λS ∈ Φ−(X) and B ∈ AF−(X), then A + B − λS ∈ Φ−(X). T (iv) A − λS ∈ Φ±(X) and B ∈ AF+(X) AF−(X), then

A + B − λS ∈ Φ±(X). ♦

Proof. Assume that A − λS ∈ Φ(X). Then, using (2.14), we infer that

b Ab− λSb∈ Φ (XA,X).

Hence, it follows from Theorem 2.2.3 that there exist A0 ∈ L (X,XA), K1 ∈

K (X) and K2 ∈ K (XA) such that

(Ab− λSb)A0 = I − K1, on X, (6.2) S-Essential Spectra of Closed Linear Operator 189

A0(Ab− λSb) = I − K2, on XA. (6.3) Thus,

A0(Ab+ Bb− λSb) = I − K2 + A0Bb, on X. (6.4) Next, using both Eq. (6.2) and Theorem 2.2.13, we get

b (Ab− λSb)A0 ∈ Φ (X) and i (Ab− λSb)A0 = 0. So, by using Eqs. (6.2), (6.3), Lemma 2.2.3 and both Theorems 2.2.13 and b 2.2.5, we deduce that A0 ∈ Φ (XA,X) and

i(Ab− λSb) = −i(A0).

Since B ∈ AF (X) and A0 ∈ L (X), and by applying Lemma 2.3.2, we have b A0Bb ∈ F (X). So, b K2 − A0Bb ∈ F (X). Using Eq. (6.4), we get

b A0(Ab+ Bb− λSb) ∈ Φ (X) and

i(A0(Ab+ Bb− λSb)) = 0. b As, A0 ∈ Φ (X,XA), and according to both Theorems 2.2.7 and 2.2.5, we have b Ab+ Bb− λSb∈ Φ (XA+B,X) and

i(Ab+ Bb− λSb) = −i(A0). (6.5) Now, by using Eqs. (2.14), (6.2) and (6.5), we afﬁrm that

i(A + B − λS) = i(A − λS), which completes the proof of (i). 190 Linear Operators and Their Essential Pseudospectra

The assertion (ii), the ﬁrst part of (iii) and (iv) are immediate. To prove the second part of (iii) we proceed as follows. Let A − λS ∈ Φ−(X). By Theorem 2.2.2, we infer that

∗ ∗ ∗ ∗ (A − λS) = A − λS ∈ Φ+(X ),

∗ ∗ the fact that B ∈ AF+(X ) implies that

∗ ∗ ∗ ∗ ∗ (A + B − λS) = A + B − λS ∈ Φ+(X ).

According to Theorem 2.2.2, we get A + B − λS ∈ Φ−(X). Q.E.D.

Corollary 6.1.1 Let A ∈ C (X), S ∈ L (X), λ ∈ C, and let B be a bounded operator on X. Then,

(i) If A − λS ∈ Φ+(X) and B ∈ AS (X), then A + B − λS ∈ Φ+(X). (ii) If A−λS ∈ Φ−(X) and B ∈ ASC (X), then A+B−λS ∈ Φ−(X). ♦

Theorem 6.1.2 Let A ∈ C (X) and let B be an operator on X. The following statements are satisﬁed.

(i) If B ∈ AF+(X), then

σe1,S(A + B) = σe1,S(A).

If, in addition, we suppose that the sets ΦA,S and ΦA+B,S are connected and the sets ρS(A) and ρS(A + B) are not empty, then

σeap,S(A + B) = σeap,S(A).

(ii) If B ∈ AF−(X), then

σe2,S(A + B) = σe2,S(A).

If, in addition, we suppose that the sets ΦA,S and ΦA+B,S are connected and both the sets ρS(A) and ρS(A + B) are not empty, then

σeδ,S(A + B) = σeδ,S(A). S-Essential Spectra of Closed Linear Operator 191

T (iii) If B ∈ AF+(X) AF−(X), then

σe3,S(A + B) = σe3,S(A).

(iv) If B ∈ AF (X), then

σei,S(A + B) = σei,S(A), i = 4, 5.

Moreover, if C\σe5,S(A) is connected. If neither ρS(A) nor ρS(A + B) is empty, then

σe6,S(A + B) = σe6,S(A). ♦

Proof. (i) Let λ 6∈ σe1,S(A), then

λ ∈ Φ+A,S.

Since B ∈ AF+(X), applying Theorem 6.1.1 (ii), we infer that

λS − A − B ∈ Φ+(X).

Thus,

λ 6∈ σe1,S(A + B).

Conversely, let λ 6∈ σe1,S(A + B), then

λS − A − B ∈ Φ+(X).

Using both Theorem 6.1.1 (ii) and the fact that −B ∈ AF+(X), we get

λ ∈ Φ+A,S. So,

λ 6∈ σe1,S(A). We deduce that

σe1,S(A + B) = σe1,S(A).

Now, if ΦA,S and ΦA+B,S are connected and the sets ρS(A) and ρS(A + B) are not empty, then by Lemma 6.1.2, we have

σeap,S(A) = σe1,S(A) 192 Linear Operators and Their Essential Pseudospectra and

σeap,S(A + B) = σe1,S(A + B). Hence,

σeap,S(A + B) = σeap,S(A). The proof of (ii) is analogues to the previous one. (iii) This statement is an immediate consequence of the item (i).

(iv) For i = 5. Let λ 6∈ σe5,S(A), then λ ∈ ΦA,S and

i(λS − A) = 0.

Since B ∈ AF (X), applying Theorem 6.1.1 (i), we infer that λ ∈ ΦA+B,S and i(λS − A − B) = 0, and therefore

λ 6∈ σe5,S(A + B). Thus,

σe5,S(A + B) ⊆ σe5,S(A).

Similarly, If λ 6∈ σe5,S(A + B), then using Theorem 6.1.1 (i) and arguing as above we derive the opposite inclusion

σe5,S(A) ⊆ σe5,S(A + B).

Now, we get

C\σe5,S(A + B) = C\σe5,S(A), which is connected by hypothesis. Thus, by Lemma 3.3.2, we have

σe5,S(A) = σe6,S(A) and σe5,S(A + B) = σe6,S(A + B).

We deduce that

σe6,S(A + B) = σe6,S(A). Q.E.D. S-Essential Spectra of Closed Linear Operator 193

Theorem 6.1.3 Let A ∈ C (X) and let Ii(X), i ∈ {1, 2, 3} be any subset of operators satisfying

(i) K (X) ⊆ I1(X) ⊆ AF (X). Then, \ σe5,S(A) = σS(A + B). B∈I1(X)

(ii) K (X) ⊆ I2(X) ⊆ AF+(X). Then, \ σeap,S(A) = σap,S(A + B). B∈I2(X)

(iii) K (X) ⊆ I3(X) ⊆ AF−(X). Then, \ σeδ,S(A) = σδ,S(A + B). ♦ B∈I3(X) \ Proof. (i) Let O = σS(A + B). According to Remark 2.3.2, we B∈I1(X) have K (X) ⊆ AK (X) ⊆ AF (X). Hence,

O ⊆ σe5,S(A). So, we have only to prove that

σe5,S(A) ⊆ O.

Let λ0 6∈ O, then there exists B ∈ I1(X) such that

λ0 ∈ ρS(A + B).

Let x ∈ X and put −1 y = (λ0S − A − B) x. It follows from the estimate

kykA+B = kyk + k(Ab+ Bb)yk = kyk + kx − λ0Syb k −1 −1 = k(λ0Sb− Ab− Bb) xk + kx − λ0Sb(λ0Sb− Ab− Bb) xk −1 ≤ 1 + (1 + |λ0|kSbk)k(λ0S − Ab− Bb) k kxk. 194 Linear Operators and Their Essential Pseudospectra

Thus, −1 (λ0Sb− Ab− Bb) ∈ L (X,XA+B).

Since B ∈ I1(X) ⊆ AF (X), and the use of Lemma 2.3.2 allows us to conclude that −1 b (λ0Sb− Ab− Bb) Bb ∈ F (XA,XA+B).

Let I denote the imbedding operator which maps every x ∈ XA onto the same element x ∈ XA+B. Clearly, we have N(I) = 0 and R(I) = XA+B. So,

kI(x)k = kxkA+B ≤ kxk + kAxk + kBxk ≤ 1 + kBkL (X,XA+B) kxkXA , ∀x ∈ XA. b −1 Thus, I ∈ Φ (XA,XA+B) and i(I) = 0. Next, since (λ0Sb− Ab − Bb) Bb ∈ b F (XA,XA+B), we get by Theorem 6.1.1 (i),

−1 b −1 I + (λ0Sb− Ab− Bb) Bb ∈ Φ (XA,XA+B) and i(I + (λ0Sb− Ab− Bb) Bb) = 0. (6.6)

On the other hand, since λ0 ∈ ρS(A + B), it follows from Eq. (2.14) that

b λ0Sb− Ab− Bb ∈ Φ (XA,XA+B) and i(λ0Sb− Ab− Bb) = 0. (6.7)

Writing λ0Sb− Ab in the from −1 λ0Sb− Ab = (λ0Sb− Ab− Bb)(I + (λ0Sb− Ab− Bb) Bb). (6.8) Using Eqs. (6.6), (6.7), and (6.8) together with Theorem 2.2.5, we get

b λ0Sb− Ab ∈ Φ (XA,X) and i(λ0Sb− Ab) = 0. Now, using (2.14) we infer that

λ0S − A ∈ Φ(X) and i(λ0S − A) = 0. Hence,

σe5,S(A) ⊆ O.

The proof of (ii) is analogues to the previous one. (iii) This statement is an immediate consequence of the item (i). Q.E.D. S-Essential Spectra of Closed Linear Operator 195

6.2 S-LEFT AND S-RIGHT ESSENTIAL SPECTRA 6.2.1 Stability of S-Left and S-Right Fredholm Spectra

This section concerns the stability of S-left (resp. S-right) Fredholm spectrum of a bounded linear operator on a Banach space X.

Lemma 6.2.1 Let A ∈ L (X) and S be an invertible operator on X such that 0 ∈ ρ(A). Then, for λ 6= 0, we have 1 −1 λ ∈ σ (A) if, and only if, ∈ σ −1 (A ), i = l, r, wl, wr. ♦ ei,S λ ei,S 1 Proof. For λ 6= 0, assume that λ ∈ ΦA−1,S−1 , then λ −1S−1 − A−1 ∈ Φb(X).

The operator λS − A can be written in the form

λS − A = −λS(λ −1S−1 − A−1)A. (6.9)

Since A and S are one-to-one and onto, then by Eq. (6.9), we have N(λS − A) and N(λ −1S−1 − A−1) are isomorphic and, R(λS − A) and −1 −1 −1 R(λ S − A ) are isomorphic. This shows that λ ∈ Φ+A,S resp. Φ−A,S and R(λS − A) resp. N(λS − A) is complemented if, and −1 −1 −1 −1 only if, λ ∈ Φ+A−1,S−1 resp. Φ−A−1,S−1 and R(λ S − A ) resp. N(λ −1S−1 − A−1) is complemented. Therefore, 1 λ ∈ Φl resp. Φr if, and only if, ∈ Φl resp. Φr . A,S A,S λ A−1,S−1 A−1,S−1 Since 0 ∈ ρ(A) and S is invertible, then

i(A) = i(S) = 0.

Using both Theorem 2.2.5 and Eq. (6.9), we conclude that i(λS − A) = i(A) + i(S) + i(λ −1S−1 − A−1) = i(λ −1S−1 − A−1).

l r So, λ ∈ ΦA,S resp. ΦA,S and i(λS − A) ≤ 0 resp. i(λS − A) ≥ 0 if, −1 l r i −1S−1 A−1 and only if, λ ∈ ΦA−1,S−1 resp. ΦA−1,S−1 and (λ − ) ≤ 0 196 Linear Operators and Their Essential Pseudospectra

−1 −1 −1 1 resp. i(λ S − A ) ≥ 0 . Hence, λ ∈ σei,S(A), if, and only if, λ ∈ −1 σei,S−1 (A ), i = l, r, wl, wr, which completes the proof. Q.E.D.

Theorem 6.2.1 Let A, B ∈ L (X) and S be an invertible operator on X. T If, for some, λ ∈ ρS(A) ρS(B),

(λS − A)−1 − (λS − B)−1 ∈ F b(X), then

σei,S(A) = σei,S(B), i = l, r, wl, wr. ♦

Proof. Without loss of generality, we suppose that λ = 0. Since

A−1 − B−1 ∈ F b(X), then it follows from Corollary 2.3.1 (i) that

l l ΦA−1,S−1 = ΦB−1,S−1

l l resp. WA−1,S−1 = WB−1,S−1 . Thus, the use of Lemma 6.2.1 makes us conclude that

l l ΦA,S = ΦB,S

l l resp. WA,S = WB,S . Hence,

σei,S(A) = σei,S(B), i = l, wl. Similarly, we can show that

σei,S(A) = σei,S(B), i = r, wr, which completes the proof. Q.E.D.

The Fredholm perturbation operators in the previous theorem can be replaced by ﬁnite rank operators or compact operators but, in general, it is not true for Riesz operators. S-Essential Spectra of Closed Linear Operator 197

Theorem 6.2.2 Let A, B ∈ L (X) and S be an invertible operator on X T such that S 6= A and S 6= B. Suppose λ0 ∈ ρS(A) ρS(B) and

−1 −1 (λ0S − A) − (λ0S − B) ∈ R(X).

−1 −1 −1 −1 b (λ0S − A) (λ0S − B) − (λ0S − B) (λ0S − A) ∈ F (X), then

σei,S(A) = σei,S(B), i = l, r, wl, wr. ♦

Proof. Without loss of generality, we suppose that λ0 = 0. Then, 0 ∈ T ρS(A) ρS(B) such that

R = A−1 − B−1 ∈ R(X), therefore RB−1 − B−1R = A−1B−1 − B−1A−1.

Since A−1B−1 − B−1A−1 ∈ F b(X), we deduce that RB−1 − B−1R ∈ F b(X). From Theorem 2.3.4 (i), we infer that

l l ΦA−1,S−1 = ΦB−1,S−1 and, applying Lemma 6.2.1, we conclude that

σel,S(A) = σel,S(B).

Similarly, we can show that

σei,S(A) = σei,S(B), i = r, wl, wr, which completes the proof. Q.E.D. 198 Linear Operators and Their Essential Pseudospectra

Theorem 6.2.3 Let A, S ∈ L (X) such that S 6= A and E ∈ R(X) such that AE − EA ∈ F b(X) and SE − ES ∈ F b(X). Then,

(i) σel,S(A) = σel,S(A + E), (ii) σer,S(A) = σer,S(A + E),

(iii) σewl,S(A) = σewl,S(A + E), and (iv) σewr,S(A) = σewr,S(A + E). ♦

Proof. (i) Let λ ∈/ σel,S(A), then

λS − A ∈ Φl(X).

The operator (λS − A)E − E(λS − A) can be written in the form

(λS − A)E − E(λS − A) = EA + λSE − λES − AE = EA − AE + λ(SE − ES). (6.10)

Since AE −EA ∈ F b(X) and SE −ES ∈ F b(X), then from Theorem 2.3.4 (i), it follows that

λS − (E + A) ∈ Φl(X). Hence,

λ ∈/ σel,S(A + E). Therefore,

σel,S(A + E) ⊆ σel,S(A). (6.11) To prove the reverse inclusion of Eq. (6.11), it sufﬁces to replace A and E by A + E and −E, respectively. The proof of (ii) is analogues to the previous one.

(iii) Assume that λ ∈/ σewl,S(A), then

λS − A ∈ Wl(X).

Now, we can written the operator

(λS − A)E − E(λS − A) S-Essential Spectra of Closed Linear Operator 199 in the form (6.10). Since

AE − EA ∈ F b(X) and SE − ES ∈ F b(X), then from Theorem 2.3.4 (iii), it follows that

λS − (A + E) ∈ Wl(X) and hence

λ ∈/ σewl,S(A + E). Therefore,

σewl,S(A + E) ⊆ σewl,S(A). (6.12) The opposite inclusion of Eq. (6.12) follows by symmetry, it sufﬁces to replace A and E by A + E and −E, respectively. The proof of (iv) may be checked in a similar way to that in (iii). Q.E.D.

Corollary 6.2.1 Let A, S ∈ L (X) such that S 6= A and E ∈ F b(X). Then,

(i) σel,S(A) = σel,S(A + E), (ii) σer,S(A) = σer,S(A + E),

(iii) σewl,S(A) = σewl,S(A + E), and (iv) σewr,S(A) = σewr,S(A + E). ♦

Remark 6.2.1 The perturbation assumption in the Corollary 6.2.1 which cannot be relaxed, even nilpotent operators. For example, let A , T , and

K in L (l2 × l2) are deﬁned by ! ! ! V 0 U 0 0 I A = , T = , and K = , 0 U 0 V 0 0 where U and V are the forward and the backward unilateral shifts. U and

V are linear bounded operators on l2 deﬁned by

U(x1,x2,x3,···) = (x2,x3,···) 200 Linear Operators and Their Essential Pseudospectra and

V(x1,x2,x3,···) = (0,x1,x2,···).

Then, A and T are both Weyl, while T +K is not right Weyl and A −K is not left Weyl. Indeed, ! ! I 0 0 0 (A − K )(T + K ) = − 0 I 0 I −UV and ! ! I 0 I −UV 0 (T + K )(A − K ) = − . 0 I 0 0 Since (A − K )(T + K ) and (T + K )(A − K ) are Fredholm operators of index zero. Hence, the products are Weyl. We can check that T +K is one-to-one and A − K is onto, with i(A − K ) = 1 = −i(T + K ). ♦

The concept of the following theorem gives a reﬁnement of the deﬁni- tion of some S-essential spectra on a Banach space.

Theorem 6.2.4 Let A ∈ L (X) and S ∈ L (X) such that S 6= 0. Then, (i) \ σewl,S(A) = σl,S(A + E), (6.13)

E∈ReA,S(X) \ (ii) σewr,S(A) = σR,S(A + E), and

E∈ReA,S(X) \ (iii) σe5,S(A) = σS(A + E),

E∈ReA,S(X) b where ReA,S(X) = E ∈ R(X) such that AE −EA ∈ F (X) and SE −ES ∈ b F (X) . ♦ S-Essential Spectra of Closed Linear Operator 201

Proof. (i) To prove the inclusion ” ⊆ ” in Eq. (6.13). Suppose that

\ λ ∈/ σl,S(A + E),

E∈ReA,S(X) then there exists E ∈ ReA,S(X) such that

λ ∈/ σl,S(A + E), so

λS − (E + A) ∈ Gl(X) and hence

λS − (E + A) ∈ Wl(X). It follows from Theorem 6.2.3 (iii) that

λS − A ∈ Wl(X), i.e.,

λ ∈/ σewl,S(A).

To prove the inverse inclusion in Eq. (6.13), assume that λ ∈/ σewl,S(A), then λS − A is left Weyl. Using Lemma 2.2.7, there exists a compact operator E such that λS − (E + A) is injective, and by Theorem 2.2.12, it follows that λS − (E + A) is left invertible and hence,

λ 6∈ σl,S(A + E).

The proof of (ii) is analogues to the previous one. (iii) This statement is an immediate consequence of both the items (i) and (ii). Q.E.D.

6.2.2 Stability of S-Left and S-Right Browder Spectra

The purpose of this section is to discuss the S-left and S-right Browder spectra of a bounded linear operator on a Banach space X. We ﬁrst prove the following theorem. 202 Linear Operators and Their Essential Pseudospectra

Theorem 6.2.5 Let A, S ∈ L (X) such that S 6= A and E be a Riesz operator commuting with A and S, then

(i) σbl,S(A) = σbl,S(A + E), and

(ii) σbr,S(A) = σbr,S(A + E). ♦

Proof. (i) Let λ ∈/ σbl,S(A), then

λS − A ∈ Bl(X).

Since AE = EA and SE = ES, then

E(λS − A) = (λS − A)E.

To use of Theorem 2.5.1 (i) leads to

λS − (E + A) ∈ Bl(X).

Thus,

λ ∈/ σbl,S(A + E). Hence,

σbl,S(A + E) ⊆ σbl,S(A). The opposite inclusion follows by symmetry, it sufﬁces to replace A and E by A + E and −E, respectively, to ﬁnd

σbl,S(A) ⊆ σbl,S(A + E).

(ii) Using the same reasoning, we can show that

σbr,S(A) = σbr,S(A + E). Q.E.D. S-Essential Spectra of Closed Linear Operator 203

Remark 6.2.2 The commutativity assumption in Theorem 6.2.5 cannot generally be relaxed even bounded operators. For example, let 1 A = I − U, 2 E = I +U, and S = U, where U is the forward unilateral shift on l2. Then,

AE = EA,

SE = ES and E is not Riesz operator. On the other hand, since 1 S − (A + E) = −2I, 2 then 1 ∈/ σ (A + E). 2 bl,S So, 1 ∈ σ (A), 2 bl,S since 1 S − A = U − I ∈/ Bl(X), 2 which implies that

σbl,S(A) 6= σbl,S(A + E). ♦

In the following theorem, we extend to S-left and S-right Browder spectra.

Theorem 6.2.6 Let A, S ∈ L (X). Then, \ (i) σbl,S(A) ⊆ σl,S(A+E) such that E ∈ R(X), AE = EA and SE = ES , 204 Linear Operators and Their Essential Pseudospectra

\ (ii) σbr,S(A) ⊆ σR,S(A+E) such that E ∈ R(X), AE = EA and SE = ES , and \ (iii) σe6,S(A) ⊆ σS(A+E) such that E ∈ R(X), AE = EA and SE = ES . ♦

Proof. (i) Assume that λ ∈ σbl,S(A), so by Theorem 6.2.5 (i), λ ∈

σbl,S(A + E) for all E ∈ R(X) such that AE = EA and SE = ES. On the other hand, we have

σbl,S(A + E) ⊂ σl,S(A + E), then

λ ∈ σl,S(A + E) for all E ∈ R(X) such that AE = EA and SE = ES. Therefore, \ σbl,S(A) ⊆ σl,S(A+E) such that E ∈ R(X), AE = EA and SE = ES .

(ii) Can be checked in the same way as (i). The assertion (iii) is an immediate consequence of (i) and (ii). Q.E.D. Chapter 7

S-Essential Spectrum and Measure of Non-Strict-Singularity

One of the central questions in the study of the S-essential spectra of closed densely deﬁned linear operators consists in showing when different notions of essential spectrum coincide with the studying of the invariance of a certain class of perturbations. The purpose of this chapter is to characterize the S-essential spectrum by mean of measure of non-strict-singularity and give some application to matrix operator.

7.1 A CHARACTERIZATION OF THE S-ESSENTIAL SPECTRUM

In this section we will give a ﬁne description of the S-essential spectrum of a closed densely deﬁned linear operator by mean of the measure of non-strict-singularity.

Let A ∈ C (X) and S, K ∈ L (X) and let f (·) be a measure of non- strict-singularity, given in (2.22). Consider the following sets 206 Linear Operators and Their Essential Pseudospectra

1 n −1 n SA,S(X) = K ∈ L (X) such that f [(λS − A − K) K] < 1, o n ∈ N, ∀λ ∈ ρS(A + K) and

2 n −1 n SA,S(X) = K ∈ L (X) such that f [K(λS − A − K) ] < 1, o n ∈ N, ∀λ ∈ ρS(A + K) . (i) If (λS − A)−1K ∈ S (X) (resp. K(λS − A)−1 ∈ S (X)) for some −1 −1 λ ∈ ρS(A), then (λS − A) K ∈ S (X) (resp. K(λS − A) ∈ S (X)) for all λ ∈ ρS(A). Indeed, for all λ, µ ∈ ρS(A), we have

(λS − A)−1K − (µS − A)−1K = (µ − λ)(µS − A)−1S(λS − A)−1K, resp. K(λS−A)−1 −K(µS−A)−1 = (µ −λ)K(µS−A)−1S(λS−A)−1 .

(ii) Now, if we consider the following sets

n −1 HA,S(X) = K ∈ L (X) such that (λS − A) K ∈ S (X) o for some (hence for all) λ ∈ ρS(A) and

n −1 FA,S(X) = K ∈ L (X) such that K(λS − A) ∈ S (X) o for some (hence for all) λ ∈ ρS(A) . Then, 1 HA,S(X) ⊂ SA,S(X) and 2 FA,S(X) ⊂ SA,S(X). −1 Indeed, let K ∈ HA,S(X), then there exists λ ∈ ρS(A) such that (λS−A) K is strictly singular. For µ ∈ ρS(A + K), we have h ih i (µS−A−K)−1K = I +(µS−A−K)−1((λ − µ)S+K) (λS−A)−1K .

By the ideal’s propriety of S (X), we deduce that (µS − A − K)−1K is strictly singular. Then, by using Proposition 2.6.3, we have

f ((µS − A − K)−1K) = 0, S-Essential Spectrum and Measure of Non-Strict-Singularity 207 where f (·) is a measure of non-strict-singularity, given in (2.22). There- fore, 1 K ∈ SA,S(X). So, 1 HA,S(X) ⊂ SA,S(X).

2 A similar reasoning allows us to deduce that FA,S(X) ⊂ SA,S(X). We begin with the following theorem which gives a reﬁned deﬁnition of S-Schechter essential spectrum.

Theorem 7.1.1 Let A ∈ C (X), and S ∈ L (X). Then, \ σe5,S(A) = σS(A + K). ♦ 1 K∈SA,S(X) Proof. We ﬁrst claim that

\ σe5,S(A) ⊂ σS(A + K). 1 K∈SA,S(X)

Indeed, if \ λ 6∈ σS(A + K), 1 K∈SA,S(X) 1 then there exists K ∈ SA,S(X) such that

λ 6∈ σS(A + K).

So, f ([(λS − A − K)−1K]n) < 1 for some n ∈ N, where f (·) is a measure of non-strict-singularity, given in (2.22). Hence, by Proposition 2.6.4, we get

I + (λS − A − K)−1K ∈ Φb(X) and i I + (λS − A − K)−1K = 0. 208 Linear Operators and Their Essential Pseudospectra

Writing h i λS − A = λS − A − K I + (λS − A − K)−1K , we can deduce that λS − A ∈ Φ(X) and i(λS − A) = 0. This shows that

λ 6∈ σe5,S(A). 1 Conversely, since K (X) ⊂ SA,S(X), then \ σS(A + K) ⊂ σe5,S(A). 1 K∈SA,S(X) Hence, \ σe5,S(A) = σS(A + K), 1 K∈SA,S(X) which completes the proof of theorem. Q.E.D.

Theorem 7.1.2 Let A ∈ C (X) and S ∈ L (X). Then, \ σe5,S(A) = σS(A + K). ♦ 2 K∈SA,S(X)

2 Proof. Since K (X) ⊂ SA,S(X), then \ σS(A + K) ⊂ σe5,S(A). 2 K∈SA,S(X) Now, we may prove that \ σe5,S(A) ⊂ σS(A + K). 2 K∈SA,S(X) Indeed, if \ λ 6∈ σS(A + K), 2 K∈SA,S(X) S-Essential Spectrum and Measure of Non-Strict-Singularity 209

2 then there exists K ∈ SA,S(X) such that

λ 6∈ σS(A + K).

So, f ([K(λS − A − K)−1]n) < 1 for some n ∈ N, where f (·) is a measure of non-strict-singularity, given in (2.22). Hence, by applying Proposition 2.6.4, we get

I + K(λS − A − K)−1 ∈ Φb(X) and i(I + K(λS − A − K)−1) = 0. Writing λS − A = [I + K(λS − A − K)−1](λS − A − K), we can deduce that λS − A ∈ Φ(X) and i(λS − A) = 0.

This shows that λ 6∈ σe5,M(A). Then, \ σe5,S(A) ⊂ σS(A + K). 2 K∈SA,S(X) Hence, \ σe5,S(A) = σS(A + K), 2 K∈SA,S(X) which completes the proof of theorem. Q.E.D.

Corollary 7.1.1 Let A ∈ C (X),S ∈ L (X), and M (X) be any subset of L (X) satisfying 1 K (X) ⊂ M (X) ⊂ SA,S(X) or 2 K (X) ⊂ M (X) ⊂ SA,S(X). Then, 210 Linear Operators and Their Essential Pseudospectra

\ σe5,S(A) = σS(A + K). ♦ K∈M (X)

7.2 THE S-ESSENTIAL SPECTRA OF 2 × 2 BLOCK OPERATOR MATRICES

During the last years, the following papers [39, 111, 120, 146] were devoted to the study of the I -essential spectra of operators deﬁned by a 2 × 2 block operator matrices that ! AB L0 = (7.1) CD acts on the product X ×Y of Banach spaces, where I is the identity operator deﬁned on the product space X ×Y by ! I 0 I = . 0 I

Let S be a bounded operator formally defined on the product space X ×Y by ! M M S = 1 2 , M3 M4 where operator M1 acts on X and everywhere defined and the intertwining operator M2 (resp. M3) acts on the Banach space Y (resp. on X) everywhere defined and are strictly singular. The operator M4 acts on Y and everywhere defined and L0 is given by Eq. (7.1), where the operator A acts on X and has domain D(A), D is defined on D(D) and acts on the Banach space Y and the intertwining operator B (resp. C) is defined on the domain D(B) (resp. D(C)) and acts on Y into X (resp. Y into Y). The purpose of this section is to discuss the S-essential spectrum of the 2 × 2 matrix operator

L0. In what follows, we will assume that the following conditions, introduced by M. Faierman, R. Mennicken and M. Muller in [69], are hold: S-Essential Spectrum and Measure of Non-Strict-Singularity 211

(H1) A is closed, densely deﬁned linear operator on X with non empty

M1-resolvent set ρM1 (A).

(H2) The operator B is densely deﬁned linear operator on X and for −1 some (hence for all) µ ∈ ρM1 (A), the operator (A − µM1) B is closable.

(H3) The operator C satisﬁes D(A) ⊂ D(C), and for some (hence for −1 all) µ ∈ ρM1 (A), the operator C(A − µM1) is bounded (in particular, if −1 C is closable, then C(A − µM1) is bounded).

(H4) The lineal D(B)TD(D) is dense in Y, and for some (hence −1 for all) µ ∈ ρM1 (A), the operator D − C(A − µM1) B is closable, we will denote by S(µ) the closure of the operator D − (C − µM3)(A − −1 µM1) (B − µM2).

Remark 7.2.1 (i) It follows, from the closed graph theorem that the operator −1 G(µ) := (A − µM1) (B − µM2) is bounded on Y.

(ii) We emphasize that neither the domain of S(µ) nor the property of being closable depend on µ. Indeed, consider λ, µ ∈ ρM1 (A), then we have h i S(λ) − S(µ) = (µ − λ) M3G(µ) + F(λ)M2 + F(λ)M1G(µ) , where −1 F(λ) = (C − λM3)(A − λM1) .

Since the operators F(λ) and G(µ) are bounded (see the condition (H3) and (i), respectively), then the difference S(λ) − S(µ) is bounded. There- fore, neither the domain of S(µ) nor the property of being closable depend on µ. ♦

We recall the following result which describes the closure of the operator L0. 212 Linear Operators and Their Essential Pseudospectra

Theorem 7.2.1 (M. Faierman, R. Mennicken, and M. Muller [69]) Let conditions (H1)-(H3) be satisﬁed and the lineal D(B)TD(D) be dense in Y. Then, the operator L0 is closable if, and only if, the operator −1 D −C(A − µM1) B, is closable in Y, for some µ ∈ ρM1 (A). Moreover, the closure L of L0 is given by ! ! ! I 0 A − µM1 0 IG(µ) L = µS + . F(µ) I 0 S(µ) − µM4 0 I (7.2) ♦

For n ∈ N, let

n n o In(X) = K ∈ L (X) satisfying f (KB) < 1 for all B ∈ L (X) , where f (·) is a measure of non-strict-singularity, given in (2.22). We have the following inclusion

S (X) ⊂ In(X).

Theorem 7.2.2 (N. Moalla [122, Theorem 3.2 (i)]) Let A ∈ Φ(X), then for all K ∈ In(X), we have A + K ∈ Φ(X) and i(A + K) = i(A). ♦

Remark 7.2.2 (i) If K ∈ In(X) and A ∈ L (X), then KA ∈ In(X). (ii) If K ∈ In(X) and S ∈ S (X), then

K + S ∈ In(X). ♦

Let f (·) be a measure of non-strict-singularity, given in (2.22). In all that follows we will make the following assumption  f M G(µ)HM G(µ)K < 1 , f F(µ)M HF(µ)M K < 1  1 1 4 1 1 4    (H5) : 1 1 f M1G(µ)HF(µ)M1K < 4 , f F(µ)M1HM1G(µ)K < 4     for some µ ∈ ρM1 (A) and for all bounded operators H and K. S-Essential Spectrum and Measure of Non-Strict-Singularity 213

Remark 7.2.3 (i) Note that if G(µ) and F(µ) are strictly singular operators, then hypothesis (H5) is satisﬁed. (ii) If the hypothesis

1 f F(µ)M HM G(µ)K < (7.3) 1 1 4 for all bounded operators H and K, then F(µ)M1G(µ) is strictly singular. Indeed, since Eq. (7.3) is valid for all bounded operators K and H, we can consider 2 K = n IX and

HM1 = IY

∗ (where n ∈ N ,IX and IY denote the identity operator on X and Y, respectively). We obtain 1 f F(µ)M G(µ) < . 1 4n2 So, f F(µ)M1G(µ) = 0 and this implies that F(µ)M1G(µ) is strictly singular. ♦

Theorem 7.2.3 Let the matrix operator L0 satisfy conditions (H1)-(H4) and assume that hypothesis (H5) is satisﬁed, then

[ σe5,S(L ) ⊆ σe5,M1 (A) σe5,M4 (S(µ)).

Moreover, if ΦA,M1 is connected, then S σe5,S(L ) = σe5,M1 (A) σe5,M4 (S(µ)). ♦ 214 Linear Operators and Their Essential Pseudospectra

Proof. Let µ ∈ ρM1 (A) be such that hypothesis (H5) is satisﬁed and set λ be a complex number. It follows from Eq. (7.2) that ! 0 M1G(µ) − M2 λS − L = UV(λ)W − (λ − µ) , F(µ)M1 − M3 F(µ)M1G(µ) (7.4) where ! I 0 U = , F(µ) I ! IG(µ) W = , 0 I and ! λM − A 0 V(λ) = 1 . 0 λM4 − S(µ) Let ! K1 K2 K = K3 K4 be a bounded operator, formally deﬁned on the product space X ×Y. Then,

" ! #2 ! 0 M1G(µ) J1 J2 K = F(µ)M1 0 J3 J4 where 2 J1 = (M1G(µ)K3) + M1G(µ)K4F(µ)M1K1,

J2 = M1G(µ)K3M1G(µ)K4 + M1G(µ)K4F(µ)M1K2,

J3 = F(µ)M1K1M1G(µ)K3 + F(µ)M1K2F(µ)M1K1, and 2 J4 = F(µ)M1K1M1G(µ)K4 + (F(µ)M1K2) . It follows from both hypothesis (H5) and Lemma 2.6.2 that

" ! #2 0 M1G(µ) g K  < 1, F(µ)M1 0 S-Essential Spectrum and Measure of Non-Strict-Singularity 215 which implies that the operator ! 0 M1G(µ) ∈ I2(X ×Y), F(µ)M1 0 where g(·) is a measure of non-strict-singularity on the space L (X ×Y), given in (2.23). Then, we can deduce from Remark 7.2.2 (ii) and the fact that F(µ)M1G(µ), M2 and M3 are strictly singular that ! 0 M1G(µ) − M2 ∈ I2(X ×Y). F(µ)M1 − M3 F(µ)M1G(µ) Now, if we use Eq. (7.4) and apply Theorem 7.2.2, we can conclude that the operator λS−L is a Fredholm operator if, and only if, UV(λ)W is a Fredholm operator. Also observe that the operator U and W are bounded and have bounded inverse. Hence, the operator UV(λ)W is a Fredholm operator if, and only if, V(λ) has this property if, and only if, λM1 −A and

λM4 − S(µ) are Fredholm operators. By Theorem 2.2.19, we have

i(λS − L ) = i(U) + i(V(λ)) + i(W) = 0 + i(V(λ)) + 0. So,

i(λS − L ) = i(λM1 − A) + i(λM4 − S(µ)). (7.5) S Let λ 6∈ σe5,M1 (A) σe5,M4 (S(µ)) . The use of Proposition 3.4.1, we get λM1 − A and λM4 − S(µ) are Fredholm operators and

i(λM1 − A) = i(λM4 − S(µ)) = 0.

Then, λS − L is Fredholm and

i(λS − L ) = 0.

So,

λ 6∈ σe5,S(L ). This shows that [ σe5,S(L ) ⊆ σe5,M1 (A) σe5,M4 (S(µ)). 216 Linear Operators and Their Essential Pseudospectra

Now, let λ 6∈ σe5,S(L ). The use of Proposition 3.4.1, we get

λS − L is a Fredholm operator and

i(λS − L ) = 0.

Then, λM1 − A and λM4 − S(µ) are Fredholm operators. Since ΦA,M1 is connected and ρM1 (A) 6= /0(see, hypothesis (H1)), then by using Corollary 3.4.1, we get

i(λM1 − A) = 0. By, Eq. (7.5), we have

i(λM4 − S(µ)) = 0.

This shows that S σe5,S(L ) = σe5,M1 (A) σe5,M4 (S(µ)). Q.E.D. Chapter 8

S-Pseudospectra and Structured S-Pseudospectra

This chapter deals with the structured S-pseudospectra of a closed densely deﬁned linear operator on a Banach spaces. Precisely, we deﬁne the structured S-pseudospectra and we give some characterizations of this set. In particular, we establish a relationship between structured S-essential pseudospectra and S-essential spectra.

8.1 STUDY OF THE S-PSEUDOSPECTRA

Theorem 8.1.1 Let A ∈ C (X),S ∈ L (X) and ε > 0, then [ σS, ε(A) = σS(A + D). ♦ kDk<ε

Proof. • Let λ ∈ σS, ε(A), we try to ﬁnd D ∈ L (X) such that kDk < ε and

λ ∈ σS(A + D). 218 Linear Operators and Their Essential Pseudospectra

If λ ∈ σS(A), in this case we take D = 0. If λ ∈ σS, ε(A)\σS(A), then λ ∈ ρS(A) and 1 kR (λ,A)k > . S ε Since kRS(λ,A)wk kRS(λ,A)k = sup w6=0 kwk and according to the deﬁnition of the upper bound, we obtain that there exists w 6= 0 such that kwk kR (λ,A)wk > . S ε Therefore, we denote by v = RS(λ,A)w, then v ∈ D(A) and (λS − A)v = w. So, k(λS − A)vk kvk > . ε Since v 6= 0, then ε > k(λS − A)uk and v u = . kvk We have kuk = 1 so u 6= 0 and by using the Hahn-Banach theorem (see Theorem 2.1.4), we obtain the existence of ψ ∈ X∗ such that ψ(u) = kuk and kψk = 1. In this case, we denote by Dx = ψ(x)(λS − A)u. It is obviously clear that D is a linear operator. Furthermore, for x ∈ X, we have kDxk = kψ(x)(λS − A)uk, = |ψ(x)|k(λS − A)uk, ≤ kψkkxkk(λS − A)uk, < εkxk. S-Pseudospectra and Structured S-Pseudospectra 219

So, D is a bounded operator. We see immediately that

kDk < ε.

We have Du = ψ(u)(λS − A)u = (λS − A)u.

Then, for u 6= 0, we have

[λS − (A + D)]u = 0.

So, λS − (A + D) is not one-to-one. Consequently,

λ ∈ σS(A + D).

[ • Let λ ∈ σS(A+D), then there exists D ∈ L (X) such that kDk < ε kDk<ε and λ ∈ σS(A+D). We prove that λ ∈ σS, ε(A). We argue by contradiction. Suppose that λ ∈ ρS, ε(A), then

λS − A − D = (λS − A)[I − RS(λ,A)D].

Furthermore,

kRS(λ,A)Dk ≤ kRS(λ,A)kkDk < 1 this implies that I − RS(λ,A)D is invertible. Hence,

λ ∈ ρS(A + D) which is a contradiction. Q.E.D.

Corollary 8.1.1 Let A ∈ C (X),S ∈ L (X) and ε > 0 such that kSk ≤ 1, then

(i) σS(A)+B(0,ε) ⊆ σS, ε(A), where B(0,ε) is the open disk centered at O(0,0) and with radius ε. (ii) (A) + (0, ) ⊆ (A). ♦ σS, ε B δ σS, ε + δ 220 Linear Operators and Their Essential Pseudospectra

Proof. (i) Let λ ∈ σS(A)+B(0,ε), then there exists λ1 ∈ σS(A) and λ2 ∈ B(0,ε) such that λ = λ1 + λ2. Since λ1 ∈ σS(A), then

λ1 + λ2 ∈ σS(A + λ2S).

Furthermore,

kλ2Sk = |λ2|kSk < ε.

In this case, we denote by D := λ2S. So, D ∈ L (X), kDk < ε and

λ ∈ σS(A + D).

Consequently,

λ ∈ σS, ε(A).

(ii) Let λ ∈ σS, ε(A)+B(0,δ), then there exists λ1 ∈ σS, ε(A) and λ2 ∈ B(0,δ) such that λ = λ1 + λ2.

Since λ1 ∈ σS, ε(A), then there exists B ∈ L (X) such that kBk < ε and λ1 ∈ σS(A + B). Therefore,

λ = λ1 + λ2 ∈ σS(A + B + λ2S).

In addition, we have B + λ2S ∈ L (X) with

kB + λ2Sk ≤ kBk + kλ2Sk < ε + δ.

Consequently,

λ ∈ σS,ε+δ (A). Q.E.D.

Theorem 8.1.2 Let A ∈ C (X),S ∈ L (X) and ε > 0, then S σS, ε(A) = σS(A) n o λ ∈ C such that ∃ x ∈ D(A) and k(λS − A)xk < εkxk . ♦ S-Pseudospectra and Structured S-Pseudospectra 221

Proof. • Let λ ∈ σS, ε(A) this means that λ ∈ σS(A) or 1 kR (λ,A)k > . S ε

If λ ∈ σS, ε(A)\σS(A), then kR (λ,A)uk 1 sup S > . u6=0 kuk ε Using the deﬁnition of the upper bound, we obtain that there exists u 6= 0 such that kuk kR (λ,A)uk > . S ε Therefore, we denote by v := RS(λ,A)u this implies that

v ∈ D(A) and k(λS − A)vk < εkvk.

• Let λ ∈ C such that there is x ∈ D(A) and

k(λS − A)xk < εkxk or λ ∈ σS(A). If λ 6∈ σS(A) and we set

v = (λS − A)x this implies that

x = RS(λ,A)v. In this case, we obtain that

kvk < εkRS(λ,A)vk.

Since v 6= 0 it gives that 1 kR (λ,A)vk > kvk. S ε We see that 1 kR (λ,A)k > , S ε and so λ ∈ σS,ε (A). Q.E.D. 222 Linear Operators and Their Essential Pseudospectra

Theorem 8.1.3 Let A ∈ C (X),S ∈ L (X) and ε > 0, then S σS, ε(A) = σS(A) n o λ ∈ : ∃ xn ∈ D(A), kxnk = 1 and lim k(λS − A)xnk < ε . ♦ C n→+∞

Proof. • Let λ ∈ σS,ε (A)\σS(A), then λ ∈ ρS(A) and 1 kR (λ,A)k > . S ε On the one hand,

kRS(λ,A)k = sup kRS(λ,A)yk kyk=1

∗ means exactly that for every n ∈ N , there exists (yn)n such that kynk = 1 and 1 kR (λ,A)k − < kR (λ,A)y k ≤ kR (λ,A)k. S n S n S Consequently, 1 lim kRS(λ,A)ynk = kRS(λ,A)k > . n→+∞ ε On the other hand, we denote by

−1 xn = kRS(λ,A)ynk RS(λ,A)yn, this implies that kxnk = 1, xn ∈ D(A) and

−1 (λS − A)xn = kRS(λ,A)ynk yn.

Therefore,

−1 lim k(λS − A)xnk = lim kRS(λ,A)ynk , n→+∞ n→+∞ −1 = lim kRS(λ,A)ynk , n→+∞ < ε.

• Let λ ∈ C such that there is xn ∈ D(A), kxnk = 1 and

lim k(λS − A)xnk < ε n→+∞ S-Pseudospectra and Structured S-Pseudospectra 223 or λ ∈ σS(A). If λ 6∈ σS(A) and we denote by

−1 yn = k(λS − A)xnk (λS − A)xn, then kynk = 1, −1 RS(λ,A)yn = k(λS − A)xnk xn and −1 kRS(λ,A)ynk = k(λS − A)xnk . We know that

kRS(λ,A)k ≥ kRS(λ,A)ynk for kynk = 1, so

lim kRS(λ,A)k ≥ lim kRS(λ,A)ynk. n→+∞ n→+∞ Therefore, −1 kRS(λ,A)k ≥ lim k(λS − A)xnk . n→+∞ Thus, −1 kRS(λ,A)k ≥ lim k(λS − A)xnk , n→+∞ Consequently, 1 kRS(λ,A)k > ε . Q.E.D.

8.2 CHARACTERIZATION OF THE STRUCTURED S-PSEUDOSPECTRA

Our ﬁrst result is the following theorem.

Theorem 8.2.1 Let A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X), and ε > 0, then S 1 σS(A,B,C,ε) = σS(A) z ∈ C such that kBRS(z,A)Ck > . ♦ ε 224 Linear Operators and Their Essential Pseudospectra

Proof. We will discuss the following two cases: 1st case : If B or C = 0, then the result is immediately. 2nd case : If B 6= 0 and C 6= 0, then we can see immediately

σS(A) ⊆ σS(A,B,C,ε). 1 Let z 6∈ σ (A). If kBR (z,A)Ck ≤ , then for every kDk < ε, we have S S ε

kDBRS(z,A)Ck < 1.

Therefore,

I − DBRS(z,A)C is invertible. In view of Theorem 2.1.6, we obtain that for any kDk < ε

1 6∈ σ(DBRS(z,A)C) means exactly

1 6∈ σ(CDBRS(z,A)). Since we can write

zS − A −CDB = (I −CDBRS(z,A))(zS − A), this implies

z 6∈ σS(A +CDB) ∀ kDk < ε.

For the other inclusion, if z 6∈ σS(A), then 1 kBR (z,A)Ck > . S ε It follows 1 kR (z,A)k > . S εkBkkCk According to the deﬁnition of the upper bound, we have the existence of w 6= 0 such that kwk kR (z,A)wk > . S εkBkkCk S-Pseudospectra and Structured S-Pseudospectra 225

Therefore, set v = RS(z,A)w, then v ∈ D(A) and

(zS − A)v = w.

Which leads to

k(zS − A)uk < εkBkkCk for kuk = 1.

Using Hahn-Banach theorem (see Theorem 2.1.4), we obtain the existence of ψ ∈ X∗ such that ψ(u) = kuk and kψk = 1. Set CDBx = ψ(x)(zS − A)u.

So, kCDBxk = |ψ(x)|k(zS − A)uk ≤ kψkkxkk(zS − A)uk < εkxkkBkkCk. This implies that D ∈ L (Y,Z). We see immediately that kDk < ε. In addition, for u 6= 0, we have

[zS − (A +CDB)]u = 0.

Consequently,

z ∈ σS(A,B,C,ε). Q.E.D.

Proposition 8.2.1 Let A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X), and ε > 0 such that the operator B (resp. C) is not null. If B is one-to-one, then σS(A,B,C,ε) 6= /0. ♦ 226 Linear Operators and Their Essential Pseudospectra

Proof. Suppose that σS(A,B,C,ε) = /0,then

ρS(A) = C and 1 λ ∈ C such that kBRS(λ,A)Ck ≤ = C. ε Consider the function

ϕ : C −→ L (Z,Y) λ −→ BRS(λ,A)C.

Since ϕ is analytic on C and for every λ ∈ C, we have 1 kϕ(λ)k ≤ , ε then ϕ is an entire bounded function. Therefore, using Liouville theorem (see Theorem 2.1.5), we obtain that ϕ is a constant function. It follows that

BRS(λ,A)C is a null operator and so is C, which is impossible. Q.E.D.

Proposition 8.2.2 Let A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X), and ε > 0, then the structured S-pseudospectrum veriﬁes the following properties:

(i) σS(A,B,C,ε) 6= /0, (ii) the structured S-pseudospectra σS(A,B,C,ε) ε>0 are increasing sets in the sense of inclusion relative to the strictly positive parameter ε, and \ (iii) σS(A,B,C,ε) = σS(A). ♦ ε>0

Proof. (i) Suppose that σS(A,B,C,ε) = /0,then

ρS(A) = C and 1 λ ∈ C such that kBRS(λ,A)Ck ≤ = C. ε S-Pseudospectra and Structured S-Pseudospectra 227

Consider the function

ϕ : C −→ L (Z,Y) λ −→ BRS(λ,A)C.

RS(λ,A) is a null operator, which is impossible.

(ii) Let ε1, ε2 > 0 such that ε1 < ε2. Let λ ∈ σS(A,B,C,ε1), then 1 1 kBRS(λ,A)Ck > > . ε1 ε2 Therefore,

λ ∈ σS(A,B,C,ε2). (iii) We have \ σS(A,B,C,ε) ε>0 \ [ 1 = σS(A) λ ∈ C such that kBRS(λ,A)Ck > ε ε>0 ! S \ 1 = σS(A) λ ∈ C such that kBRS(λ,A)Ck > . ε ε>0 It sufﬁces to prove that \ 1 λ ∈ C such that kBRS(λ,A)Ck > = ε ε>0 {λ ∈ C such that kBRS(λ,A)Ck = +∞}.

For the inclusion in the direct sense: let λ ∈ C such that for all ε > 0 1 kBR (λ,A)Ck > . S ε This implies that

lim kBRS(λ,A)Ck = +∞. ε→0+ 228 Linear Operators and Their Essential Pseudospectra

Consequently,

kBRS(λ,A)Ck = +∞. For the other inclusion: let λ ∈ C such that 1 kBR (λ,A)Ck = +∞ > S ε for all ε > 0. Hence, \ Sn o σS(A,B,C,ε) = σS(A) λ ∈ C such that kBRS(λ,A)Ck = +∞ , ε>0 = σS(A), which completes the proof of proposition. Q.E.D.

Corollary 8.2.1 Let A ∈ C (X),S ∈ L (Y,Z),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0 such that kSk ≤ 1. If S1 = CSB, then

(i) σS1 (A) + B(0,ε) ⊆ σS1 (A,B,C,ε), where B(0,ε) is the open disk centered at O(0,0) and with radius ε.

(ii) σS1 (A,B,C,ε) + B(0,δ) ⊆ σS1 (A,B,C,ε + δ). ♦

Proof. (i) Let λ ∈ σS1 (A) + B(0,ε), then there exists λ1 ∈ σS1 (A) and λ2 ∈ B(0,ε) such that λ = λ1 + λ2.

Since λ1 ∈ σS1 (A), we have

λ1 + λ2 ∈ σS1 (A + λ2S1).

In this case, we take D := λ2S. Consequently,

λ ∈ σS1 (A,B,C,ε).

(ii) Let λ ∈ σS1 (A,B,C,ε) + B(0,δ), then there exists λ1 ∈

σS1 (A,B,C,ε) and λ2 ∈ B(0,δ) such that

λ = λ1 + λ2.

Since λ1 ∈ σS1 (A,B,C,ε), then there exists D ∈ L (Y,Z) such that kDk < ε and

λ1 ∈ σS1 (A +CDB).

In addition, λ = λ1 +λ2 ∈ σS1 (A+CD2B) with D2 := D+λ2S. Therefore, S-Pseudospectra and Structured S-Pseudospectra 229

λ ∈ σS1 (A,B,C,ε + δ). Q.E.D.

Theorem 8.2.2 Let ε > 0,A ∈ C (X),S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) such that S 6= A and S 6= 0, then S σS(A,B,C,ε) = σS(A) n o λ ∈ C : ∃ x ∈ D(A) and k(λS − A)xk < εkxkkBkkCk . ♦

Proof. We will discuss these two cases: st 1 case : If B or C = 0, then σS(A,B,C,ε) = σS(A,ε). nd 2 case : If B and C 6= 0. Let λ ∈ σS(A,B,C,ε)\σS(A), then 1 kBR (λ,A)Ck > . S ε Using the deﬁnition of the upper bound, we obtain that there exists u 6= 0 such that kuk kR (λ,A)uk > . S εkBkkCk

Therefore, we denote by v := RS(λ,A)u this implies that v ∈ D(A) and

k(λS − A)vk < εkvkkBkkCk.

• Let λ ∈ C such that there is x ∈ D(A) and

k(λS − A)xk < εkxkkBkkCk.

Therefore, k(λS − A)uk < εkBkkCk and kuk = 1. Using the Hahn-Banach theorem (see Theorem 2.1.4), we obtain the existence of ψ ∈ X∗ such that

ψ(u) = kuk and kψk = 1. Set CDBx = ψ(x)(λS − A)u. So,

kCDBxk = |ψ(x)|k(λS − A)uk, ≤ kψkkxkk(λS − A)uk, < εkxkkBkkCk. 230 Linear Operators and Their Essential Pseudospectra

This implies that D ∈ L (Y,Z). We see immediately that kDk < ε. Further- more, for u 6= 0, we have

[λS − (A +CDB)]u = 0.

Hence,

λ ∈ σS(A,B,C,ε). Q.E.D.

Theorem 8.2.3 Let A, S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X), and ε > 0 such that S 6= A +CDB for all D ∈ L (Y,Z) with kDk < ε. Then, S σap,S(A,B,C,ε) = σp,S(A,B,C,ε) [ n o λ ∈ C : R(λS − A −CDB) is not closed . ♦ kDk<ε Proof. • We shall ﬁrst show [ λ ∈ C : R(λS − A −CDB) is not closed ⊂ σap,S(A,B,C,ε). kDk<ε

Let λ 6∈ σap,S(A,B,C,ε), then for every kDk < ε, there exists α > 0 such that for any x ∈ X, we have

k(λS − A −CDB)xk ≥ α kxk. (8.1)

Let yn ∈ R(λS − A −CDB) such that yn → y as n → +∞. Hence, (yn)n is a Cauchy sequence with

yn = (λS − A −CDB)xn.

On the other hand, using Eq. (8.1), we obtain that (xn)n is a Cauchy sequence in X and this implies that it is convergent to x. Since λS − A − CDB is a bounded operator, then for any kDk < ε, the sequence

((λS − A −CDB)xn)n converges to (λS − A −CDB)x. By uniqueness of the limit, we obtain that for every kDk < ε, we have

y = (λS − A −CDB)x. S-Pseudospectra and Structured S-Pseudospectra 231

• Let λ ∈ σap,S(A,B,C,ε)\σp,S(A,B,C,ε). We assume that for any kDk < ε the range R(λS − A −CDB) is closed. Therefore, for any kDk < ε, we have λS − A −CDB is one-to-one and onto from X into R(λS − A −CDB) which is a Banach space. Then, using the Banach theorem, we get for every kDk < ε,

(λS − A −CDB)−1 is a bounded operator from R(λS −A−CDB) into X. This implies that for every kDk < ε, there exists M > 0 such that for any y ∈ R(λS−A−CDB), we have k(λS − A −CDB)−1yk ≤ Mkyk.

Set x := (λS − A −CDB)−1y, then for every kDk < ε and for any x ∈ X

kxk ≤ Mk(λS − A −CDB)xk. (8.2)

Since λ ∈ σap,S(A,B,C,ε), then there exists a sequence (xn)n of unit vectors in X and

lim k(λS − A −CDB)xnk = 0. n→+∞ The use of Eq. (8.2) leads to 1 ≤ 0 which is absurd. Q.E.D.

8.3 CHARACTERIZATION OF THE STRUCTURED S-ESSENTIAL PSEUDOSPECTRA

In all the sequel, we shall suppose that for every kDk < ε, we have S 6= CDB and S 6= A +CDB.

Theorem 8.3.1 Let A, S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, 232 Linear Operators and Their Essential Pseudospectra

b (i) If for every kDk < ε, we have ACDB ∈ F+(X) and SA = AS, then h [[ i σe1,S(A,B,C,ε)\ 0 ⊂ σe1,S(A) σe1,S(CDB) \ 0 . kDk<ε

b b If further, CDBA ∈ F+(X) for all kDk < ε and S ∈ Φ+(X), then h [[ i σe1,S(A,B,C,ε)\ 0 = σe1,S(A) σe1,S(CDB) \ 0 . kDk<ε

b (ii) If for every kDk < ε, we have ACDB ∈ F−(X) and SCDB =CDBS, then h [[ i σe2,S(A,B,C,ε)\ 0 ⊂ σe2,S(A) σe2,S(CDB) \ 0 . kDk<ε

b b If further, CDBA ∈ F−(X) for all kDk < ε and S ∈ Φ−(X), then h S [ i σe2,S(A,B,C,ε)\ 0 = σe2,S(A) σe2,S(CDB) \ 0 . ♦ kDk<ε

h S [ i Proof. • Let λ 6∈ σe1,S(A) σe1,S(CDB) or λ = 0. kDk<ε If λ 6= 0, then b λS − A ∈ Φ+(X) and for any kDk < ε, we have

b λS −CDB ∈ Φ+(X).

Using Theorem 2.2.5, we obtain

b (λS − A)(λS −CDB) ∈ Φ+(X)

b for every kDk < ε. Since ACDB ∈ F+(X) and applying both Eq. (3.16) and Lemma 2.3.1, we get

b S(λS − A −CDB) ∈ Φ+(X) for all kDk < ε. Furthermore, using Theorem 2.2.10, we obtain that

b λS − A −CDB ∈ Φ+(X) S-Pseudospectra and Structured S-Pseudospectra 233 for all kDk < ε. Consequently, λ 6∈ σe1,S(A,B,C,ε)\ 0 .

• Let λ 6∈ σe1,S(A,B,C,ε) or λ = 0. If λ 6= 0, then

b λS − A −CDB ∈ Φ+(X) for all kDk < ε. Since S ∈ Φ+(X), then for all kDk < ε, we have

b S(λS − A −CDB) ∈ Φ+(X) and b (λS − A −CDB)S ∈ Φ+(X). b b Since ACDB ∈ F+(X), CDBA ∈ F+(X), and applying Eqs. (3.16), (3.17) and Lemma 2.3.1, we obtain ∀ kDk < ε

b (λS − A)(λS −CDB) ∈ Φ+(X) and b (λS −CDB)(λS − A) ∈ Φ+(X). By using Theorem 2.2.10, it is clear that

b λS − A ∈ Φ+(X) and b λS −CDB ∈ Φ+(X) ∀ kDk < ε. Hence, h [[ i λ 6∈ σe1,S(A) σe1,S(CDB) \ 0 . kDk<ε In the same way we prove the item (ii). Q.E.D.

Theorem 8.3.2 Let A, S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. If for every kDk < ε, we have ACDB ∈ F b(X), α(S) < ∞ and SA = AS, then h [[ i σe4,S(A,B,C,ε)\ 0 ⊂ σe4,S(A) σe4,S(CDB) \ 0 . kDk<ε Further, if CDBA ∈ F b(X) for all kDk < ε and β(S) < ∞, then 234 Linear Operators and Their Essential Pseudospectra

h S [ i σe4,S(A,B,C,ε)\ 0 = σe4,S(A) σe4,S(CDB) \ 0 . ♦ kDk<ε

h S [ i Proof. • Let λ 6∈ σe4,S(A) σe4,S(CDB) or λ = 0. If λ 6= 0, then kDk<ε

λS − A ∈ Φb(X) and for any kDk < ε, we have

λS −CDB ∈ Φb(X).

Using Theorem 2.2.5, we obtain

(λS − A)(λS −CDB) ∈ Φb(X) ∀ kDk < ε.

Since for every kDk < ε, ACDB ∈ F b(X), applying Eq. (3.16), we have

S(λS − A −CDB) ∈ Φb(X) for all kDk < ε. In view of the nullity α(S) < ∞, together with Theorem 2.2.7, we conclude that

λS − A −CDB ∈ Φb(X) ∀ kDk < ε.

Consequently, λ 6∈ σe4,S(A,B,C,ε)\ 0 .

• For the inverse inclusion: Let λ 6∈ σe4,S(A +CDB) for every kDk < ε or λ = 0. If λ 6= 0, then for every kDk < ε, we have

λS − A −CDB ∈ Φb(X).

Since S ∈ Φb(X), then

S(λS − A −CDB) ∈ Φb(X) and (λS − A −CDB)S ∈ Φb(X). S-Pseudospectra and Structured S-Pseudospectra 235

Furthermore, we have for every kDk < ε,

ACDB ∈ F b(X) and CDBA ∈ F b(X), then by both Eqs. (3.16) and (3.17), we obtain for all kDk < ε

(λS − A)(λS −CDB) ∈ Φb(X) and (λS −CDB)(λS − A) ∈ Φb(X).

An immediate consequence of Lemma 2.2.3 that

λS − A ∈ Φb(X) and λS −CDB ∈ Φb(X) ∀ kDk < ε.

Thus, h S [ i λ 6∈ σe4,S(A) σe4,S(CDB) \ 0 . Q.E.D. kDk<ε

Lemma 8.3.1 Let A, S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, S σe5,S(A,B,C,ε) = σe4,S(A,B,C,ε) [ n o λ ∈ C such that i(λS − A −CDB) 6= 0 . ♦ kDk<ε Proof. Using Theorem 3.3.4, we obtain that for every kDk < ε

[n o σe5,S(A +CDB) = σe4,S(A +CDB) λ ∈ C : i(λS − A −CDB) 6= 0 .

Then, we can immediately deduce the result. Q.E.D. 236 Linear Operators and Their Essential Pseudospectra

Proposition 8.3.1 Let A, S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. If for every kDk < ε, we have C\σe4,S(A +CDB) is connected and ρS(A +CDB) 6= /0, then

σe5,S(A,B,C,ε) = σe4,S(A,B,C,ε). ♦

Proof. By virtue of Proposition 3.3.5, it is easy to get the following esti- mation for every kDk < ε

σe4,S(A +CDB) = σe5,S(A +CDB).

This equality allows us to reach the desired result. Q.E.D.

Theorem 8.3.3 Let A, S ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. If for every kDk < ε, ACDB ∈ F b(X), α(S) = β(S) < ∞ and SA = AS, then

h [[ i σe5,S(A,B,C,ε)\ 0 ⊂ σe5,S(A) σe5,S(CDB) \ 0 . kDk<ε

If further, C\σe4,S(A) is connected, ρS(A) 6= /0 and for all kDk < ε, we have CDBA ∈ F b(X), then h S [ i σe5,S(A,B,C,ε)\ 0 = σe5,S(A) σe5,S(CDB) \ 0 . ♦ kDk<ε

h S [ i Proof. • Let λ 6∈ σe5,S(A) σe5,S(CDB) \ 0 . Then, kDk<ε

λS − A ∈ Φb(X), and i(λS − A) = 0 and for every kDk < ε, we have

λS −CDB ∈ Φb(X) and i(λS −CDB) = 0. S-Pseudospectra and Structured S-Pseudospectra 237

Using Theorem 2.2.5, we infer that for all kDk < ε

(λS − A)(λS −CDB) ∈ Φb(X) and i(λS − A)(λS −CDB) = 0.

Moreover, since for any kDk < ε, ACDB ∈ F b(X), we can apply Eq. (3.16) and Lemma 2.3.1 to obtain that for all kDk < ε

S(λS − A −CDB) ∈ Φb(X) and iS(λS − A −CDB) = 0.

Then, for every kDk < ε

λS − A −CDB ∈ Φb(X) and i(λS − A −CDB) = 0.

Hence, λ 6∈ σe5,S(A,B,C,ε)\ 0 . • Let λ 6∈ σe5,S(A,B,C,ε)\ 0 , then for every kDk < ε,

λS − A −CDB ∈ Φb(X) and i(λS − A −CDB) = 0.

Since ACDB ∈ F b(X), CDBA ∈ F b(X) and α(S) = β(S) < ∞, it is easy to show that λS − A ∈ Φb(X) 238 Linear Operators and Their Essential Pseudospectra and for all kDk < ε, we have

λS −CDB ∈ Φb(X) and i(λS −CDB) = −i(λS − A).

Besides, C\σe4,S(A) is connected and ρS(A) 6= /0so we can use Lemma 3.3.2 to obtain the result. Q.E.D. Chapter 9

Structured Essential Pseudospectra

In this chapter, we study the structured essential pseudospectra of the closed, densely deﬁned linear operator on Banach space.

9.1 ON A CHARACTERIZATION OF THE STRUCTURED WOLF, AMMAR-JERIBI, AND BROWDER ESSENTIAL PSEUDOSPECTRA 9.1.1 Structured Ammar-Jeribi, and Browder Essential Pseudospectra

Theorem 9.1.1 Let A1 ∈ C (X),A2 ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Suppose that there exist a positive integer n and F ∈ F (X) n n such that A2 : D(A1) −→ D(A1) and for all kDk < ε, x ∈ D(A1), we have

(A1 +CDB)A2x = A2(A1 +CDB)x + Fx. Thereby, we obtain

(i) σe4(A1 + A2,B,C,ε) ⊆ σe4(A1,B,C,ε) + σe4(A2,B,C,ε). Further- more, if σe4(A1,B,C,ε) is empty set, then σe4(A1,B,C,ε)+σe4(A2,B,C,ε) is also empty set. 240 Linear Operators and Their Essential Pseudospectra

(ii) If, further, for every kDk < ε, we have C\σe4(A1 + CDB), C\σe4(A2 + CDB) and C\σe4(A1 + A2 + CDB) are connected such that ρ(A1 +CDB) 6= /0 and ρ(A1 + A2 +CDB) 6= /0, then

σe5(A1 + A2,B,C,ε) ⊆ σe5(A1,B,C,ε) + σe5(A2,B,C,ε).

(iii) Moreover, if for all kDk < ε, we assume that C\σe5(A1 +CDB), C\σe5(A2 +CDB) and C\σe5(A1 +A2 +CDB) are connected with ρ(A1 + CDB) 6= /0 and ρ(A1 + A2 +CDB) 6= /0, then

σe6(A1 + A2,B,C,ε) ⊆ σe6(A1,B,C,ε) + σe6(A2,B,C,ε). ♦

Proof. If σe4(A1,B,C,ε) + σe4(A2,B,C,ε) = C, then the theorem is triv- ially true. We, therefore, assume that σe4(A1,B,C,ε) + σe4(A2,B,C,ε) is not the entire plane. Let

γ 6∈ σe4(A1,B,C,ε) + σe4(A2,B,C,ε).

If λ ∈ σe4(A2,B,C,ε), then

γ − λ 6∈ σe4(A1,B,C,ε).

Since σe4(A2,B,C,ε) is compact, then there exists an open set

U ⊃ σe4(A2,B,C,ε) such that ∂U, the boundary of U, is bounded and λ ∈ U. Set A3 := γ −

A1 −CDB, we have for all kDk < ε, λ ∈ ΦA3 . Therefore,

σe4(A2,B,C,ε) ⊂ U ⊂ ΦA3 .

By Theorem 2.9.1, we obtain the existence of a bounded Cauchy domain

D1 such that

σe4(A2,B,C,ε) ⊂ D1 and

D1 ⊂ U. Structured Essential Pseudospectra 241

0 0 Since Φ (A3) (resp. Φ (A2)) does not accumulate in ΦA3 (resp. ΦA2 ), D1 0 0 can be chosen such that Rλ (A3) and Rλ (A2) are analytic on ∂D1. Deﬁne the operators N1 and N2 by Z 1 0 0 N1 = − Rλ (A3)Rλ (A2) dλ 2πi +∂D1 and Z 1 0 0 N2 = − Rλ (A2)Rλ (A3) dλ. 2πi +∂D1 0 Rλ (A3) is of the form TC(λ), where C(λ) is a bounded operator valued analytic function of λ and T is a ﬁxed bounded operator such that

T : X −→ D(A1).

Thereby, N1 : X −→ D(A1). We will now show that for all kDk < ε, there 0 0 exist two Fredholm perturbations F1 and F2 such that

0 (γ − A1 − A2 −CDB)N1 = I + F1 and 0 N2(γ − A1 − A2 −CDB) = I + F2 on D(A1). Hence, we can write

γ − A1 − A2 −CDB = −(λ − A3) + (λ − A2).

Thus,

(γ − A1 − A2 −CDB)N1 Z 1 0 0 = (λ − A3)Rλ (A3)Rλ (A2) dλ − 2πi +∂D1 Z 1 0 0 (λ − A2)Rλ (A3)Rλ (A2) dλ. (9.1) 2πi +∂D1 Since 0 (λ − A3)Rλ (A3) = I + K(λ), where K(λ) is a bounded ﬁnite rank operator depending analytically on λ, then 242 Linear Operators and Their Essential Pseudospectra

Z 1 0 0 (λ − A3)Rλ (A3)Rλ (A2) dλ = 2πi +∂D 1 Z Z 1 0 1 0 Rλ (A2)dλ + K(λ)Rλ (A2)dλ. 2πi +∂D1 2πi +∂D1 Using Theorem 2.9.5, we obtain Z 1 0 Rλ (A2)dλ is of the form I + K1, 2πi +∂D1 where K1 ∈ K (X). So, the ﬁrst integral of Eq. (9.1) is of the form I + K2,

K2 ∈ K (X). By virtue of Lemma 2.9.1, we get

0 0 0 0 Rλ (A3)Rλ (A2) = Rλ (A2)Rλ (A3) + F(λ).

The second integral of Eq. (9.1) is equal to Z 1 0 0 − (λ − A2)Rλ (A3)Rλ (A2) dλ 2πi +∂D1 Z Z 1 0 0 1 = − (λ − A2)Rλ (A2)Rλ (A3) dλ − (λ − A2)F(λ)dλ. 2πi +∂D1 2πi +∂D1 Z Z 1 0 1 = − (I + G(λ))Rλ (A3)dλ − (λ − A2)F(λ)dλ. 2πi +∂D1 2πi +∂D1 Since Z 1 0 Rλ (A3)dλ 2πi +∂D1 is compact (see Lemma 2.9.2) and 1 Z (λ − A2)F(λ)dλ 2πi +∂D1 is a Fredholm perturbation, then we obtain Z 1 0 0 − (λ − A2)Rλ (A3)Rλ (A2) dλ = F1, 2πi +∂D1 where F1 ∈ F (X). Consequently, for all kDk < ε, we have

0 (γ − A1 − A2 −CDB)N1 = I + F1,

0 where F1 ∈ F (X). By a similar argument, we get for all kDk < ε

0 N2(γ − A1 − A2 −CDB) = I + F2, Structured Essential Pseudospectra 243

0 where F2 ∈ F (X). So, for all kDk < ε, we have

γ − A1 − A2 −CDB ∈ Φ(X).

For the second result of (i), we argue by contradiction. We suppose that

σe4(A1,B,C,ε) + σe4(A2,B,C,ε) is not empty, then there exists

λ ∈ σe4(A1,B,C,ε) + σe4(A2,B,C,ε).

This implies that

λ = µ1 + µ2, where µ1 ∈ σe4(A1,B,C,ε) and µ2 ∈ σe4(A2,B,C,ε) which is absurd. Using Lemma 3.6.1, we obtain the items (ii) and (iii). Q.E.D.

Let A1 ∈ R(X), A2 ∈ R(X), B ∈ L (X,Y), C ∈ L (Z,X) and ε > 0 1 such that the operator B (resp. C) is not null. We assume that ε = kBkkCkkRk , where R ∈ L (X) satisfying

kRk > max{kA0k,kB0k} such that A0 (resp. B0) is a quasi-inverse operator of λ − A1 (resp. λ − A2) for all λ 6= 0. Now, we can see that for any D ∈ L (Y,Z) satisfying kDk < ε

A1 +CDB ∈ R(X) and

A2 +CDB ∈ R(X).

If there exists K ∈ K (X) such that for all kDk < ε, x ∈ X, we have

(A1 +CDB)A2x = A2(A1 +CDB)x + Kx, then we can deduce, from Theorem 2.3.2, that

A1 + A2 +CDB ∈ R(X). 244 Linear Operators and Their Essential Pseudospectra

This leads to

σe4(A1,B,C,ε) = {0},

σe4(A2,B,C,ε) = {0}, and

σe4(A1 + A2,B,C,ε) = {0}. Hence, the results of Theorem 9.1.1 are satisﬁed. In all the sequels we give some properties of the structured Ammar-Jeribi essential pseudospectrum.

Theorem 9.1.2 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe5(A,B,C,ε) = σ(A + K,B,C,ε). ♦ K∈K (X)

Proof. Let λ 6∈ σe5(A,B,C,ε), then for all kDk < ε, we have λ − A − CDB ∈ Φ(X) and i(λ − A −CDB) = 0.

Thus, there exists K ∈ K (X) such that for any D satisfying kDk < ε, we have λ − A − K −CDB ∈ Φ(X) and i(λ − A − K −CDB) = 0.

Therefore, we obtain

\ σe5(A,B,C,ε) = σe5(A + K,B,C,ε), K∈K (X) and so \ σe5(A,B,C,ε) ⊂ σ(A + K,B,C,ε). K∈K (X) Let \ λ ∈ σ(A + K,B,C,ε). K∈K (X) Structured Essential Pseudospectra 245

Without loss of generality, we may suppose that λ = 0. We argue by contradiction, we suppose that for all kDk < ε,

A +CDB ∈ Φ(X) and i(A +CDB) = 0.

0 0 Let d = α(A +CDB) = β(A +CDB) and {u1,··· ,ud} (resp. {z1,··· ,zd}) being the basis for N(A +CDB) (resp. N((A +CDB)∗)). By Lemma 2.1.1, 0 0 ∗ there are functionals u1,··· ,ud in X and elements z1,··· ,zd such that

0 u j(uk) = δ jk and 0 z j(zk) = δ jk, 1 ≤ j, k ≤ d, where ( 0 if j 6= k δ jk = 1 if j = k. Consider the operator K deﬁned by

d 0 Ku = ∑ ui(u) zi, u ∈ X. i=1 It is clear that K is a ﬁnite rank operator on X. Then, K is a compact operator on X. By virtue of Lemma 2.3.1 (i), we obtain that for all kDk < ε, A +CDB + K ∈ Φ(X) and α(A +CDB + K) = β(A +CDB + K). Obviously, if u ∈ N(A +CDB + K), then

(A +CDB)u = K(−u).

Since \ R(A +CDB) R(K) = {0} 246 Linear Operators and Their Essential Pseudospectra and \ N(A +CDB) N(K) = {0}, so we can deduce that u = 0. Thus, we have

α(A +CDB + K) = β(A +CDB + K) = 0.

Consequently, for any D satisfying kDk < ε, we have

0 ∈ ρ(A +CDB + K), which completes the proof. Q.E.D.

As a consequence of the previous theorem and Proposition 3.3.1 for S = I, we may state:

Corollary 9.1.1 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ (i) σe5(A,B,C,ε) = σe5(A), and ε>0 (ii) if ε1 < ε2, then

σe5(A,B,C,ε1) ⊂ σe5(A,B,C,ε2). ♦

Theorem 9.1.3 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe5(A,B,C,ε) = σ(A + F,B,C,ε). ♦ F∈F (X)

\ Proof. Let λ 6∈ σ(A+F,B,C,ε), then there exists F ∈ F (X) such F∈F (X) that for any D ∈ L (X) satisfying kDk < ε, we have

λ ∈ ρ(A + F +CDB).

By Lemma 2.3.1 (i), we can deduce that for all kDk < ε

λ − A −CDB ∈ Φ(X) Structured Essential Pseudospectra 247 and i(λ − A −CDB) = 0.

It is clear that

\ \ σ(A + F,B,C,ε) ⊂ σ(A + K,B,C,ε), F∈F (X) K∈K (X) because K (X) ⊂ F (X). Q.E.D.

Corollary 9.1.2 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. If E(X) be any subset of L (X) satisfying K (X) ⊂ E(X) ⊂ F (X), then \ σe5(A,B,C,ε) = σ(A + F,B,C,ε). ♦ F∈E(X)

Theorem 9.1.4 Let A1,A2 be operators in C (X),B ∈ L (X,Y),C ∈

L (Z,X) and ε > 0. If A2 is A1-compact, then

σe5(A1,B,C,ε) = σe5(A1 + A2,B,C,ε). ♦

Proof. Let us begin with the inclusion ⊂. Assume that

λ 6∈ σe5(A1 + A2,B,C,ε), then for all D ∈ L (Y,Z) such that kDk < ε, we have

λ − A1 − A2 −CDB ∈ Φ(X) and i(λ − A1 − A2 −CDB) = 0.

Since A2 is A1-compact, then for any D ∈ L (Y,Z) satisfying kDk < ε, we have A2 is (A1 +CDB)-compact. Thus, by Theorem 2.3.7, we get that A2 is

(A1 +A2 +CDB)-compact. So, we can see that A2 is (λ −A1 −A2 −CDB)- compact. Therefore, from Theorem 2.3.6, we can deduce that for all kDk < ε

λ − A1 −CDB ∈ Φ(X) and

i(λ − A1 −CDB) = 0. 248 Linear Operators and Their Essential Pseudospectra

Let us prove now the converse inclusion. Let λ 6∈ σe5(A1,B,C,ε) this means that for any D ∈ L (Y,Z) such that kDk < ε, we have

λ − A1 −CDB ∈ Φ(X) and

i(λ − A1 −CDB) = 0.

Using the fact that A2 is (A1 +CDB)-compact, then for all kDk < ε, we obtain

λ − A1 − A2 −CDB ∈ Φ(X) and

i(λ − A1 − A2 −CDB) = 0, which completes the proof. Q.E.D.

In addition, we have the following useful stability result for the structured Wolf and the structured Ammar-Jeribi essential pseudospectra.

Theorem 9.1.5 Let A1,A2 ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X), M (X) be a non zero two-sided ideal of L (X) satisfying K (X) ⊆ M (X) ⊆ F (X) and ε > 0. Assume that there are B1,B2 ∈ L (X) and J1,J2 ∈ M (X) such that

A1B1 = I − J1, (9.2)

A2B2 = I − J2. (9.3) T If 0 ∈ ΦA1 ΦA2 and B1 − B2 ∈ M (X), then

σe4(A1,B,C,ε) = σe4(A2,B,C,ε).

If, further, i(A1) = i(A2) = 0, then

σe5(A1,B,C,ε) = σe5(A2,B,C,ε). (9.4)

♦ Structured Essential Pseudospectra 249

Proof. By Eqs. (9.2) and (9.3), we have for any scalar λ and D ∈ L (Y,Z)

(λ −A1 −CDB)B1 −(λ −A2 −CDB)B2 = J1 −J2 +(λ −CDB)(B1 −B2). (9.5)

If λ 6∈ σe4(A2,B,C,ε), then for any D ∈ L (Y,Z) satisfying kDk < ε, we have

λ − A2 −CDB ∈ Φ(X).

Since A2 is closed, D(A2) endowed with the graph norm is a Banach space denoted by XA2 . Then, using Eq. (2.6), we infer that for all kDk < ε,

b λ − Ac2 −CDBd ∈ Φ (XA2 ,X).

Furthermore, since J2 ∈ M (X), it follows from Eq. (9.3) and Theorem b 2.2.7 that B2 ∈ Φ (X,XA2 ). Thereby, for every kDk < ε, we have

b (λ − Ac2 −CDBd )B2 ∈ Φ (X).

If B1 −B2 ∈ M (X), then Eq. (9.5) together with Lemma 2.3.1 (i) gives for all kDk < ε, b (λ − Ac1 −CDBd )B1 ∈ Φ (X) and h i h i i (λ − Ac1 −CDBd )B1 = i (λ − Ac2 −CDBd )B2 . (9.6)

Since A1 ∈ C (X), using Eq. (9.2) and arguing as above we conclude that b B1 ∈ Φ (X,XA1 ). Therefore, for all kDk < ε,

b λ − Ac1 −CDBd ∈ Φ (XA1 ,X).

This implies, for every kDk < ε, we have

λ − A1 −CDB ∈ Φ(X).

The opposite inclusion follows by symmetry. We now prove Eq. (9.4). If

λ 6∈ σe5(A2,B,C,ε), then for all kDk < ε, we have

λ − A2 −CDB ∈ Φ(X) and i(λ − A2 −CDB) = 0. 250 Linear Operators and Their Essential Pseudospectra

Since J1 and J2 belong to M (X) and

i(A1) = i(A2) = 0, applying Lemma 2.3.1 (i) to Eqs. (9.2) and (9.3) and using Atkinson theorem, we get

i(B1) = i(B2) = 0. This result together with Eq. (9.6), the Atkinson theorem and Eq. (2.6) show that for all kDk < ε

i(λ − A1 −CDB) = i(λ − A2 −CDB) = 0.

Consequently,

λ 6∈ σe5(A1,B,C,ε). The opposite inclusion follows by symmetry. Q.E.D.

We end this section by the following result which provides accurate characterization of the structured Ammar-Jeribi essential pseudospectrum in particular space.

Theorem 9.1.6 Let A ∈ C (X) such that ρ(A) 6= /0,B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, (i) If X has the Dunford-Pettis property, then \ σe5(A,B,C,ε) = σ(A + S,B,C,ε), S∈GA(X) where

n −1 o GA(X) = K ∈ L (X) such that (λ −A) K ∈ W (X) for some λ ∈ ρ(A) .

(ii) If X is isomorphic to one of the spaces Lp(Ω), p > 1, then \ σe5(A,B,C,ε) = σ(A + S,B,C,ε), 0 S∈GA(X) where 0 n −1 GA(X) = K ∈ L (X) such that (λ − A) K ∈ S (X) o for some λ ∈ ρ(A) . ♦ Structured Essential Pseudospectra 251

Proof. (i) Since K (X) ⊂ GA(X), then

\ σ(A + S,B,C,ε) ⊂ σe5(A,B,C,ε). S∈GA(X) \ Let λ 6∈ σ(A + S,B,C,ε), then there exists S ∈ GA(X) such that for S∈GA(X) any D ∈ L (Y,Z) satisfying kDk < ε, we have

λ ∈ ρ(A + S +CDB).

Thus, λ − A − S −CDB ∈ Φ(X) and i(λ − A − S −CDB) = 0.

Let µ ∈ ρ(A), we have

(λ −A−S−CDB)−1S = I + (λ − A − S −CDB)−1(µ − λ + S +CDB)×

(µ − A)−1S. (9.7)

Since S ∈ GA(X) and in view of Eq. (9.7), we deduce that for any kDk < ε, (λ − A − S −CDB)−1S is weakly compact on X. Using the fact that the composed of two weakly compact operators is compact, so we can deduce from Lemma 2.2.5 that I + (λ − A − S −CDB)−1S is a Fredholm operator with i(I + (λ − A − S −CDB)−1S) = 0.

Since we can write

λ − A −CDB = (λ − A − S −CDB)[I + (λ − A − S −CDB)−1S], then for all kDk < ε, λ − A −CDB ∈ Φ(X) and

i(λ − A −CDB) = 0.

The prove of the item (ii) follows by the same reasoning as (i). Q.E.D. 252 Linear Operators and Their Essential Pseudospectra

Remark 9.1.1 The Theorem 9.1.6 shows that for all K ∈ GA(X) or K ∈ 0 GA(X), we have

σe5(A + K,B,C,ε) = σe5(A,B,C,ε). ♦

Corollary 9.1.3 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe5(A,B,C,ε) = σ(A + K,B,C,ε), 00 K∈GA (X) where 00 n −1 GA (X) = K ∈ L (X) such that (λ − A) K ∈ F (X) o for some λ ∈ ρ(A) . ♦

9.1.2 A Characterization of the Structured Browder Essential Pseudospectrum

Theorem 9.1.7 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe6(A,B,C,ε) = σ(A + R,B,C,ε), R∈Rε (X) where n Rε (X) = R ∈ R(X) : ∀D ∈ L (Y,Z) satisfying kDk < ε, o we have R commutes with A +CDB . ♦

\ Proof. Suppose that λ 6∈ σ(A + R,B,C,ε), then there exists a

R∈Rε (X) Riesz operator R which commutes with A +CDB for every D ∈ L (Y,Z) satisfying kDk < ε, and

λ 6∈ σ(A + R,B,C,ε).

It follows from Lemma 2.4.3 that for all kDk < ε, R commutes with λ − A−R−CDB. Since for every kDk < ε, we have λ ∈ ρ(A+R+CDB), we can assert that

α(λ − A − R −CDB) = β(λ − A − R −CDB) = 0 Structured Essential Pseudospectra 253 and asc(λ − A − R −CDB) = 0.

By virtue of Theorem 2.4.5, we obtain for all kDk < ε

α(λ − A −CDB) = β(λ − A −CDB) < ∞ and asc(λ − A −CDB) = desc(λ − A −CDB) < ∞.

Consequently, for all kDk < ε,

λ 6∈ σe6(A +CDB).

Conversely, let λ 6∈ σe6(A,B,C,ε), then for any kDk < ε, we have

λ 6∈ σe6(A +CDB).

If λ ∈ σ(A + CDB)\σe6(A + CDB), then we may interchange T in Theorem 2.3.8 by A+CDB to get that λ is a pole of the resolvent operator, and Theorem 2.4.2 implies that for any kDk < ε, we have

asc(λ − A −CDB) < ∞.

Otherwise, for every kDk < ε

α(λ − A −CDB) = β(λ − A −CDB) = 0 and asc(λ − A −CDB) = 0, when \ λ ∈ ρ(A +CDB). kDk<ε Now, we see from Proposition 2.4.3 that there exists a ﬁnite dimensional range operator R which commutes with λ − A −CDB for all kDk < ε, and λ 6∈ σ(A + R +CDB). Q.E.D. 254 Linear Operators and Their Essential Pseudospectra

Remark 9.1.2 Theorem 9.1.7 remains true if “Riesz” is replaced by “compact”. More precisely, if we set n Kε (X) = K ∈ K (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε, K commutes o with A +CDB , (9.8) then \ σe6(A,B,C,ε) = σ(A + K,B,C,ε). K∈Kε (X) As a consequence of the previous identity that the structured Browder essential pseudospectrum, σe6(A,B,C,ε), remains invariant under perturbations of A by operators in Kε (X). ♦

Corollary 9.1.4 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe6(A,B,C,ε) = σ(A + F,B,C,ε), F∈Fε (X) where n Fε (X) = F ∈ F (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε, o K commutes with A +CDB . ♦

Remark 9.1.3 We conclude by Corollary 9.1.4 that for every F ∈ Fε (X), we have

σe6(A + F,B,C,ε) = σe6(A,B,C,ε). ♦

Theorem 9.1.8 Let A ∈ C (X),B ∈ L (X,Y) and C ∈ L (Z,X). Then, \ σe6(A,B,C,ε) = σe6(A). ♦ ε>0 Structured Essential Pseudospectra 255

Proof. We begin by observing that, for all ε > 0, [ σe6(A) ⊂ σe6(A +CDB). kDk<ε Hence, \ σe6(A) ⊂ σe6(A,B,C,ε). ε>0 \ On the other hand, let λ ∈ σe6(A,B,C,ε), then for all ε > 0 and for ε>0 every R ∈ Rε (X), we have λ ∈ σ(A+R,B,C,ε), where Rε (X) is given in Theorem 9.1.7. Since for all kDk < ε, we have R commutes with A+CDB this implies that Rx ∈ D(A +CDB) = D(A) for any x ∈ D(A) and

R(A +CDB)x = (A +CDB)Rx for all x ∈ D((A +CDB)2).

It follows that kRAx − ARxk ≤ 2 kCkkDkkBkkRkkxk < 2 εkCkkBkkRkkxk. Thus, lim kRAx − ARxk = 0, ε→0 and so for every x ∈ D(A), we have RAx = ARx. Furthermore, using Propo- sition 8.2.2 for S = I, we get \ σ(A + R,B,C,ε) = σ(A + R). ε>0 Whence \ λ ∈ σ(A + R), R∈Rc(X) where n o Rc(X) = R ∈ R(X) such that R commutes with A .

Using Theorem 3.1.3, we deduce that λ ∈ σe6(A). Q.E.D. Before continuing with the properties of the structured Browder essential pseudospectrum, we also show that we have a similar result as the previous theorem for the structured Wolf essential pseudospectrum. 256 Linear Operators and Their Essential Pseudospectra

Theorem 9.1.9 Let A ∈ C (X),B ∈ L (X,Y) and C ∈ L (Z,X). Then, \ σe4(A,B,C,ε) = σe4(A). ♦ ε>0 Proof. 1st case : If B = 0 or C = 0, then the proof follows immediately from Deﬁnition 3.6.1. nd 2 case : If B 6= 0 and C 6= 0, then for λ 6∈ σe4(A), we have λ − A ∈ 1 Φ(X). Set ε = , where A0 is a quasi-inverse of λ −A. Then, there kBkkCkkA0k exist K1, K2 ∈ K (X) such that

(λ − A)A0 = I − K1 whenever R(A0) ⊂ D(A), and

A0(λ − A) = I − K2.

Thus, for every kDk < ε, we have

h −1i (λ − A −CDB)A0 = I − K1(I −CDBA0) (I −CDBA0), whence

−1 −1 (λ − A −CDB)A0(I −CDBA0) = I − K1(I −CDBA0) .

In the same way, we can write that for any kDk < ε,

−1 −1 (I − A0CDB) A0(λ − A −CDB) = I − (I − A0CDB) K2.

−1 −1 Since K1(I −CDBA0) ∈ K (X) and (I − A0CDB) K2 ∈ K (X), then for any kDk < ε, we have λ − A −CDB ∈ Φ(X). Hence, \ λ 6∈ σe4(A,B,C,ε). ε>0 Conversely, we have [ σe4(A) ⊂ σe4(A +CDB), kDk<ε and so, \ σe4(A) ⊂ σe4(A,B,C,ε). Q.E.D. ε>0 Structured Essential Pseudospectra 257

Theorem 9.1.10 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe6(A,B,C,ε) = σ(A + K,B,C,ε), K∈Gε (X) where n Gε (X) = K ∈ L (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε,K commutes with −1 A +CDB and ∆ψ K(λ − A − K −CDB) < 1, o ∀ λ ∈ ρ(A + K +CDB) , and ∆ψ (·) is the measure of non-strict-singularity given in (2.11). ♦

Proof. We ﬁrst show that

\ σe6(A,B,C,ε) ⊂ σ(A + K,B,C,ε). K∈Gε (X) \ Let λ 6∈ σ(A + K,B,C,ε). There exists K ∈ Gε (X) such that for all K∈Gε (X) kDk < ε, we have λ ∈ ρ(A + K +CDB).

Therefore, for every kDk < ε, we have the following identity

λ −A−CDB = I + K(λ − A − K −CDB)−1(λ −A−K −CDB). (9.9)

Since −1 ∆ψ K(λ − A − K −CDB) < 1 = Γψ (I) (9.10) and by using Lemma 2.2.6, we can conclude that

−1 b I + K(λ − A − K −CDB) ∈ Φ+(X) and i(I + K(λ − A − K −CDB)−1) = i(I) = 0.

Therefore, for any kDk < ε

I + K(λ − A − K −CDB)−1 ∈ Φb(X) 258 Linear Operators and Their Essential Pseudospectra and i(I + K(λ − A − K −CDB)−1) = 0.

Furthermore, for all D ∈ L (Y,Z) satisfying kDk < ε, we have

λ − A − K −CDB ∈ Φ(X) and i(λ − A − K −CDB) = 0.

Consequently, for all kDk < ε

λ − A −CDB ∈ Φ(X) and i(λ − A −CDB) = 0.

It remains to show fo rall kDk < ε,

asc(λ − A −CDB) < ∞ and desc(λ − A −CDB) < ∞.

By using Eq. (9.9), we have n (λ − A −CDB)n = I + K(λ − A − K −CDB)−1 (λ − A − K −CDB)n

n for every n ∈ N. Since for any kDk < ε, we have (λ − A − K −CDB) is one-to-one, we infer N (λ − A −CDB)n ⊂ N (I + K(λ − A − K −CDB)−1)n .

Then, we can write for all kDk < ε, α (λ − A −CDB)n ≤ α (I + K(λ − A − K −CDB)−1)n and so, by Lemma 2.4.1, we obtain that α (λ − A −CDB)n ≤ Structured Essential Pseudospectra 259

asc I + K(λ − A − K −CDB)−1 α I + K(λ − A − K −CDB)−1 .

In addition, I ∈ B(X) and the use both Eq. (9.10) and Theorem 2.5.3 leads to for any kDk < ε,

I + K(λ − A − K −CDB)−1 ∈ B(X).

By Lemma 2.4.2, we can see that for every kDk < ε,

asc(λ − A −CDB) < ∞.

Likewise, Theorem 2.4.1 gives

desc(λ − A −CDB) = asc(λ − A −CDB) < ∞, whence

λ 6∈ σe6(A,B,C,ε). Conversely, by Theorem 2.1.8, we may easily observe that

Kε (X) ⊂ Gε (X), where Kε (X) is given in (9.8). Combining this result together with Remark 9.1.2, we obtain \ σ(A + K,B,C,ε) ⊂ σe6(A,B,C,ε). Q.E.D. K∈Gε (X)

Corollary 9.1.5 Let A ∈ C (X) such that ρ(A) 6= /0,B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then,

\ σe6(A,B,C,ε) = σ(A + K,B,C,ε), 0 K∈Gε (X) where 0 n Gε (X) = K ∈ L (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε,K commutes o with A +CDB and K(λ − A)−1 ∈ S (X), ∀ λ ∈ ρ(A) . ♦ 260 Linear Operators and Their Essential Pseudospectra

\ 0 Proof. Let λ 6∈ σ(A+K,B,C,ε), then there exists K ∈ Gε (X) such 0 K∈Gε (X) that for every kDk < ε, we have

λ ∈ ρ(A + K +CDB).

Let µ ∈ ρ(A), we adopt the following identity h i µ −A = I +(µ −λ +K +CDB)(λ −A−K −CDB)−1 (λ −A−K −CDB).

Hence,

K(λ − A − K −CDB)−1 = K(µ − A)−1× h i I + (µ − λ + K +CDB)(λ − A − K −CDB)−1 . (9.11)

Thus, we obtain from Theorem 2.1.8 together with Eq. (9.11) that K ∈

Gε (X), where Gε (X) is given in Theorem 9.1.10. Therefore,

\ \ σ(A + K,B,C,ε) ⊂ σ(A + K,B,C,ε). 0 K∈Gε (X) K∈Gε (X) On the other hand, we have

0 Kε (X) ⊂ Gε (X) and by Remark 9.1.2, we can deduce that \ σ(A + K,B,C,ε) ⊂ σe6(A,B,C,ε). Q.E.D. 0 K∈Gε (X) The following theorem gives a relationship between the structured essential pseudospectra of the sum of two bounded operators and the structured essential pseudospectra of one of them.

Theorem 9.1.11 Let A1,A2 ∈ L (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, Structured Essential Pseudospectra 261

(i) If for any D ∈ L (Y,Z) satisfying kDk < ε, we have A1(A2 + CDB) ∈ F b(X), then h [ i σei(A1 + A2,B,C,ε)\{0} ⊂ σei(A1) σei(A2,B,C,ε) \{0}, i = 4,5.

b If, further, (A2 +CDB)A1 ∈ F (X) whence h [ i σe4(A1 + A2,B,C,ε)\{0} = σe4(A1) σe4(A2,B,C,ε) \{0}.

Furthermore, if C\σe4(A1) is connected, then h [ i σe5(A1 + A2,B,C,ε)\{0} = σe5(A1) σe5(A2,B,C,ε) \{0}.

(ii) If the hypothesis of (i) is satisﬁed and if we have for any kDk < ε, C\σe5(A2 +CDB), C\σe5(A1 +A2 +CDB) and C\σe5(A1) are connected, then h S i σe6(A1 + A2,B,C,ε)\{0} = σe6(A1) σe6(A2,B,C,ε) \{0}. ♦

Proof. By virtue of Theorem 3.1.4, we can assert that for any kDk < ε, b we have if A1(A2 +CDB) ∈ F (X), then h [ i σei(A1 + A2 +CDB)\{0} ⊂ σei(A1) σei(A2 +CDB) \{0}, i = 4,5.

b If, further, (A2 +CDB)A1 ∈ F (X), then h [ i σe4(A1 + A2 +CDB)\{0} = σe4(A1) σe4(A2 +CDB) \{0}.

Furthermore, if C\σe4(A1) is connected, then h [ i σe5(A1 + A2 +CDB)\{0} = σe5(A1) σe5(A2 +CDB) \{0}.

Then, we can immediately deduce (i). Using Theorem 3.1.4 again, we obtain for any kDk < ε h [ i σe6(A1 + A2 +CDB)\{0} = σe6(A1) σe6(A2 +CDB) \{0}.

Therefore, h S i σe6(A1 +A2,B,C,ε)\{0} = σe6(A1) σe6(A2,B,C,ε) \{0}. Q.E.D. 262 Linear Operators and Their Essential Pseudospectra

9.2 SOME DESCRIPTION OF THE STRUCTURED ESSENTIAL PSEUDOSPECTRA 9.2.1 Relationship Between Structured Jeribi and Structured Ammar-Jeribi Essential Pseudospectra

We start our analysis by the following remark which gives some crucial properties of the structured Jeribi essential pseudospectrum.

Remark 9.2.1 We have the following properties:

(i) σ j(A,B,C,ε) ⊂ σe5(A,B,C,ε) ⊂ σ(A,B,C,ε). \ (ii) σ j(A,B,C,ε) = σ j(A). ε>0 (iii) If ε1 < ε2, then

σ j(A) ⊂ σ j(A,B,C,ε1) ⊂ σ j(A,B,C,ε2).

(iv) For all K ∈ W ∗(X), we have

σ j(A + K,B,C,ε) = σ j(A,B,C,ε). ♦

Theorem 9.2.1 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. If W ∗(X) = W (X) and X has the Dunford-Pettis property, then

σe5(A,B,C,ε) = σ j(A,B,C,ε). ♦

Proof. By virtue of Theorem 9.1.2, we have

\ σe5(A,B,C,ε) = σ(A + K,B,C,ε) K∈K (X) and since K (X) ⊂ W ∗(X), we obtain

σ j(A,B,C,ε) ⊂ σe5(A,B,C,ε).

∗ Let λ 6∈ σ j(A,B,C,ε), then there exists K ∈ W (X) such that for any D ∈ L (Y,Z) satisfying kDk < ε, we have

λ ∈ ρ(A + K +CDB). Structured Essential Pseudospectra 263

Thereby, λ − A − K −CDB ∈ Φ(X) and i(λ − A − K −CDB) = 0. Furthermore, we have for any kDk < ε,

(λ − A − K −CDB)−1K ∈ W ∗(X).

Then, by using Eq. (2.8), we get h i2 (λ − A − K −CDB)−1K ∈ K (X).

From Lemma 2.2.5, we can deduce that

I + (λ − A − K −CDB)−1K is a Fredholm operator and

i(I + (λ − A − S −CDB)−1K) = 0.

Thereby, we can write

λ − A −CDB = (λ − A − K −CDB)[I + (λ − A − K −CDB)−1K] and, by virtue of Theorem 2.2.19 for all kDk < ε,

λ − A −CDB ∈ Φ(X) and i(λ − A −CDB) = 0. Consequently,

λ 6∈ σe5(A,B,C,ε). Q.E.D.

Theorem 9.2.2 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. If ∗ W (X) = S (X) and X is isomorphic to one of the spaces Lp(Ω), p > 1, then

σe5(A,B,C,ε) = σ j(A,B,C,ε) ♦ 264 Linear Operators and Their Essential Pseudospectra

Proof. Since K (X) ⊂ S (X) together with Theorem 9.1.2, we get

σ j(A,B,C,ε) ⊂ σe5(A,B,C,ε).

It remains to shows that

σe5(A,B,C,ε) ⊂ σ j(A,B,C,ε).

For this, we reason in the same way as the proof of Theorem 9.2.1. Q.E.D.

Remark 9.2.2 Under the hypothesis of Theorem 9.2.1 or 9.2.2, we have for all K ∈ W ∗(X)

σe5(A + K,B,C,ε) = σe5(A,B,C,ε). ♦

9.2.2 A Characterization of the Structured Ammar-Jeribi Essential Pseudospectrum

We begin with the following theorem which gives a ﬁne description of the structured Ammar-Jeribi essential pseudospectrum.

Theorem 9.2.3 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then,

\ \ σe5(A,B,C,ε) = σ(A + K,B,C,ε) = σ(A + K,B,C,ε), 0 K∈Nn,ε (X) K∈Nn,ε (X) where n Nn,ε (X) = K ∈ L (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε, ∀λ ∈ ρ(A + K +CDB), − n ∗o f [(λ − A − K −CDB) 1K] < 1 for some n ∈ N ,

0 n Nn,ε (X) = K ∈ L (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε, ∀λ ∈ − n ∗o ρ(A + K +CDB), f [K(λ − A − K −CDB) 1] < 1 for some n ∈ N , and f (·) is the measure of non-strict-singularity given in (2.22). ♦ Structured Essential Pseudospectra 265

\ \ Proof. Let λ 6∈ σ(A + K,B,C,ε) (resp. λ 6∈ σ(A + 0 K∈Nn,ε (X) K∈Nn,ε (X) 0 K,B,C,ε)). There exists K ∈ Nn,ε (X) (resp. K ∈ Nn,ε (X)) such that for every kDk < ε, we have

λ ∈ ρ(A + K +CDB).

∗ Since for some n ∈ N , f [(λ − A − K −CDB)−1K]n < 1

(resp. f [K(λ − A − K −CDB)−1]n < 1), then the use of Proposition 2.6.4 leads to

I + (λ − A − K −CDB)−1K ∈ Φb(X)

(resp. I + K(λ − A − K −CDB)−1 ∈ Φb(X)) and i(I + (λ − A − K −CDB)−1K) = 0

(resp. i(I + K(λ − A − K −CDB)−1) = 0).

Using the following identity

λ − A −CDB = (λ − A − K −CDB)(I + (λ − A − K −CDB)−1K),

(resp. λ − A −CDB = (I + K(λ − A − K −CDB)−1)(λ − A − K −CDB)), we get for every D ∈ L (Y,Z) satisfying kDk < ε

λ − A −CDB ∈ Φ(X) and i(λ − A −CDB) = 0.

Consequently,

\ σe5(A,B,C,ε) ⊂ σ(A + K,B,C,ε) K∈Nn,ε (X) 266 Linear Operators and Their Essential Pseudospectra

\ (resp. σe5(A,B,C,ε) ⊂ σ(A + K,B,C,ε)). 0 K∈Nn,ε (X) For the inverse inclusion, we have K (X) ⊂ S (X) and by virtue of Propo- sition 2.6.3, we get

K (X) ⊂ Nn,ε (X) 0 (resp. K (X) ⊂ Nn,ε (X)). Furthermore, by using Theorem 9.1.2, we have \ σe5(A,B,C,ε) = σ(A + K,B,C,ε), K∈K (X) then we reach the desired result. Q.E.D.

Corollary 9.2.1 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X), ε > 0 and

Tε (X) be any subset of L (X) satisfying

K (X) ⊂ Tε (X) ⊂ Nn,ε (X) or 0 K (X) ⊂ Tε (X) ⊂ Nn,ε (X). Then, \ σe5(A,B,C,ε) = σ(A + K,B,C,ε). ♦ K∈Tε (X) We have not the stability of the structured Ammar-Jeribi essential pseu- 0 dospectra by perturbations by operators in Nn,ε (X) or Nn,ε (X). For this, ∗ we deﬁne the following subsets for n ∈ N

0 n In(X) = {K ∈ L (X) satisfying f ((BK) ) < 1 for all B ∈ L (X)},

n In(X) = {K ∈ L (X) satisfying f ((KB) ) < 1 for all B ∈ L (X)} and let 0 n Sn(X) = {B ∈ L (X) satisfying B ∈ S (X)}. It is proved in [122] that

0 S (X) ⊂ In(X) ⊂ Sn(X).

The following result is an immediate consequence of Theorem 3.1.1 (ii). Structured Essential Pseudospectra 267

Theorem 9.2.4 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, for all K ∈ In(X), we have

σe5(A + K,B,C,ε) = σe5(A,B,C,ε). ♦

Chapter 10

Structured Essential Pseudospectra and Measure of Noncompactness

This chapter deals with studying the structured Ammar-Jeribi and the structured Browder essential pseudospectra of closed densely deﬁned linear operators via the concept of the measure of noncompactness and the measure of non-strict-singularity.

10.1 NEW DESCRIPTION OF THE STRUCTURED ESSENTIAL PSEUDOSPECTRA 10.1.1 A Characterization of the Structured Ammar-Jeribi Essential Pseudospectrum by Kuratowski Measure of Noncompactness

∗ Theorem 10.1.1 Let n ∈ N ,A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe5(A,B,C,ε) = σ(A + K,B,C,ε), K∈Mn,ε (X) where 270 Linear Operators and Their Essential Pseudospectra

n Mn,ε (X) = K ∈ L (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε, o ∀λ ∈ ρ(A + K +CDB), γ [(λ − A − K −CDB)−1K]n < 1 , and γ(·) is the Kuratowski measure of noncompactness given in (2.20). ♦ \ Proof. Suppose that λ 6∈ σ(A + K,B,C,ε). Then, there exists

K∈Mn,ε (X) K ∈ Mn,ε (X) such that for every kDk < ε, we have

λ ∈ ρ(A + K +CDB).

Thereby, we can write h i λ − A −CDB = (λ − A − K −CDB) I + (λ − A − K −CDB)−1K .

Now, γ [(λ − A − K −CDB)−1K]n < 1, and hence k lim γ [(λ − A − K −CDB)−1K]n = 0. k→+∞ ∗ Therefore, there exists k0 ∈ N such that k0 1 γ [(λ − A − K −CDB)−1K]n < . 2 It follows from Lemma 2.6.1 (iii) that 1 γ [(λ − A − K −CDB)−1K]nk0 < . 2 We apply Theorem 2.6.1 (ii) for P(z) = znk0 and Q(z) = 1 − z, we obtain that I + (λ − A − K −CDB)−1K ∈ Φb(X). Next, for t ∈ [0,1], we have 1 γ [t(λ − A − K −CDB)−1K]nk0 < , 2 and so I +t(λ − A − K −CDB)−1K ∈ Φb(X).

Furthermore, for t ∈ [0,1], there is a β > 0 such that for all t1 ∈ [0,1] satisfying |t1 −t| < β, we have from Theorem 2.2.16 Structured Pseudospectra and Measure of Noncompactness 271

i I +t(λ − A − K −CDB)−1K

−1 −1 = i I +t1(λ − A − K −CDB) K + (t −t1)(λ − A − K −CDB) K

−1 = i I +t1(λ − A − K −CDB) K .

Using the Heine-Borel theorem, we get the existence of ﬁnite number of sets which cover [0,1]. Since the index is constant on each one of these sets which overlaps with at least one another, then we obtain i I + (λ − A − K −CDB)−1K = i(I) = 0.

Consequently, for all D ∈ L (Y,Z) such that kDk < ε, we have

λ − A −CDB ∈ Φ(X) and i(λ − A −CDB) = 0.

Conversely, since K (X) ⊂ Mn,ε (X), then

\ \ σ(A + K,B,C,ε) ⊂ σ(A + K,B,C,ε).

K∈Mn,ε (X) K∈K (X)

From Theorem 9.1.2, we have

\ σe5(A,B,C,ε) = σ(A + K,B,C,ε), K∈K (X) so we can deduce the result. Q.E.D.

∗ Corollary 10.1.1 Let n ∈ N ,A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X), M (X) be an arbitrary subspace of L (X) and ε > 0. If K (X) ⊂ M (X) ⊂

Mn,ε (X), then \ σe5(A,B,C,ε) = σ(A + K,B,C,ε). ♦ K∈M (X) 272 Linear Operators and Their Essential Pseudospectra

∗ Theorem 10.1.2 Let n ∈ N ,A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then,

[ σe5(A + K,B,C,ε) ⊂ σe5(A,B,C,ε), K∈Qε (X) n where Qε (X) = K ∈ L (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε,

the right quasi-inverse Nλ,ε of the operator λ − A −CDB n o satisﬁes γ (Nλ,ε K) < 1 . ♦

Proof. We assume that λ 6∈ σe5(A,B,C,ε), then for every kDk < ε, we have λ − A −CDB ∈ Φ(X) and i(λ − A −CDB) = 0.

Let K ∈ Qε (X), we shall prove that λ 6∈ σe5(A + K,B,C,ε). We have for any D ∈ L (Y,Z) satisfying kDk < ε the existence of a right quasi inverse

Nλ,ε of the operator λ − A −CDB, this implies that

R(Nλ,ε ) ⊂ D(A) and

(λ − A −CDB)Nλ,ε = I − K1, where K1 ∈ K (X). So, we can write h i λ − A − K −CDB = λ − A −CDB − (λ − A −CDB)Nλ,ε + K1 K

= (λ − A −CDB)[I − Nλ,ε K] − K1K.

Since n γ (Nλ,ε K) < 1, then by repeating the same argument used in the proof of Theorem 10.1.1, we obtain b I − Nλ,ε K ∈ Φ (X) and i(I − Nλ,ε K) = 0. Therefore, for every kDk < ε, we have

λ − A − K −CDB ∈ Φ(X) and i(λ − A − K −CDB) = 0. Q.E.D. Structured Pseudospectra and Measure of Noncompactness 273

We close this subsection by the following result which gives a characterization of the structured Ammar-Jeribi essential pseudospectrum by means of the measure of non-strict-singularity ∆ψ (·).

Theorem 10.1.3 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe5(A,B,C,ε) = σ(A + K,B,C,ε), K∈Hε (X) where n Hε (X) = K ∈ L (X) : ∀D ∈ L (Y,Z) with kDk < ε, −1 o ∆ψ (λ − A − K −CDB) K < 1 ∀ λ ∈ ρ(A + K +CDB) ,

and ∆ψ (·) is the measure of non-strict-singularity given in (2.11). ♦ \ Proof. Let λ 6∈ σ(A + K,B,C,ε), then there exists K ∈ Hε (X) K∈Hε (X) such that for all kDk < ε, we have

λ ∈ ρ(A + K +CDB).

Then, for any D ∈ L (Y,Z) satisfying kDk < ε, we can see that

λ − A − K −CDB ∈ Φ(X),

i(λ − A − K −CDB) = 0 and −1 ∆ψ (λ − A − K −CDB) K < 1 = Γψ (I), where Γψ (·) is given in (2.10). Using Lemma 2.2.6, we can conclude that

−1 b I + (λ − A − K −CDB) K ∈ Φ+(X) and i(I + (λ − A − K −CDB)−1K) = i(I) = 0.

Therefore, for any kDk < ε

I + (λ − A − K −CDB)−1K ∈ Φb(X) 274 Linear Operators and Their Essential Pseudospectra and i(I + (λ − A − K −CDB)−1K) = 0.

Since we can write

λ − A −CDB = (λ − A − K −CDB)(I + (λ − A − K −CDB)−1K), then we obtain that for all kDk < ε

λ − A −CDB ∈ Φ(X) and i(λ − A −CDB) = 0.

On the other hand, by Theorem 2.1.8, we can see that

K (X) ⊂ Hε (X).

Hence, \ σ(A + K,B,C,ε) ⊂ σe5(A,B,C,ε). Q.E.D. K∈Hε (X)

10.1.2 A Characterization of the Structured Browder Essential Pseudospectrum by Means of Measure of Non-Strict-Singularity

Theorem 10.1.4 Let A, K ∈ L (X) such that K commutes with A. Suppose that n n f (K ) < fM(A ) for some n ≥ 1, where fM(·) (resp. f (·)) is given in (2.21) (resp. (2.22)). Then, A ∈ B+(X) if, and only if, A + K ∈ B+(X). ♦ Structured Pseudospectra and Measure of Noncompactness 275

Proof. Let A ∈ B+(X), then A ∈ Φ+(X) and

asc(A) < ∞.

Using Lemma 2.6.4, we obtain

A + K ∈ Φ+(X).

Since N(Ap) is a closed subspace and asc(A) = p < ∞, then

N∞(A) = N∞(A).

By Proposition 2.5.1, we get

\ N∞(A) R∞(A) = {0}.

Set Aλ := A + λK, where λ ∈ [0,1]. Since

n n n n f ((λK) )) = λ f (K ) < fM(A ), then

Aλ ∈ Φ+(X). We can check that asc(A + K) < ∞ in the same way as that of the proof of Theorem 2.5.2. Conversely, let

A + K ∈ B+(X). Then, Lemma 2.6.4 proves that

A ∈ Φ+(X).

It remains to show that asc(A) < ∞.

To do this, we consider

(A + K)λ = A + K + λK = A + (λ + 1)K. where λ ∈ [−1,0] and we reason in the same way as above. Q.E.D. 276 Linear Operators and Their Essential Pseudospectra

Theorem 10.1.5 Let A, K ∈ L (X) such that K commutes with A. Suppose that n n f (K ) < fM(A ) for some n ≥ 1, where fM(·) (resp. f (·)) is given in (2.21) (resp. (2.22)). Then, A ∈ B(X) if, and only if, A + K ∈ B(X). ♦

Proof. Let A ∈ B(X), then A ∈ B+(X) and

desc(A) < ∞.

Using Theorem 10.1.4, we infer that A + K ∈ B+(X). So, the use of Theorem 2.4.1 leads to

desc(A + K) = asc(A + K) < ∞.

Conversely, we reason in the same way as above. Q.E.D. Now, we are ready to state the following result.

Theorem 10.1.6 Let A ∈ C (X),B ∈ L (X,Y),C ∈ L (Z,X) and ε > 0. Then, \ σe6(A,B,C,ε) = σ(A + K,B,C,ε), K∈Oε (X) where n Oε (X) = K ∈ L (X) : ∀D ∈ L (Y,Z) satisfying kDk < ε,K commutes with n A +CDB and f K(λ − A − K −CDB)−1 < 1, ∗o ∀ λ ∈ ρ(A + K +CDB) for some n ∈ N . ♦ \ Proof. Let λ 6∈ σ(A + K,B,C,ε). Then, there exists K ∈ Oε (X) K∈Oε (X) such that for all kDk < ε, we have

λ ∈ ρ(A + K +CDB).

Thereby, for every kDk < ε, we have the following equality

λ − A −CDB = I + K(λ − A − K −CDB)−1(λ − A − K −CDB). (10.1) Structured Pseudospectra and Measure of Noncompactness 277

Since I ∈ B(X) and n f K(λ − A − K −CDB)−1 < 1, then by Theorem 10.1.5, we obtain for every kDk < ε,

I + K(λ − A − K −CDB)−1 ∈ B(X).

Thus, I + K(λ − A − K −CDB)−1 ∈ Φ(X) and i(I + K(λ − A − K −CDB)−1) = 0.

In addition, for every kDk < ε, we have

λ − A − K −CDB ∈ Φ(X) and i(λ − A − K −CDB) = 0.

Therefore, for all kDk < ε

λ − A −CDB ∈ Φ(X) and i(λ − A −CDB) = 0.

By using Eq. (10.1), we have the following identity

n (λ − A −CDB)n = I + K(λ − A − K −CDB)−1 (λ − A − K −CDB)n for every n ∈ N. Since we have α (λ − A −CDB)n ≤ α (I + K(λ − A − K −CDB)−1)n for all kDk < ε and, by virtue of Lemma 2.4.1, we can conclude that α (λ − A −CDB)n ≤ asc I + K(λ − A − K −CDB)−1 α I + K(λ − A − K −CDB)−1 . 278 Linear Operators and Their Essential Pseudospectra

Lemma 2.4.2 proves that for every kDk < ε,

asc(λ − A −CDB) < ∞.

This implies by the use of Theorem 2.4.1 that

desc(λ − A −CDB) = asc(λ − A −CDB) < ∞.

Hence,

λ 6∈ σe6(A,B,C,ε). Conversely, by Remark 9.1.1, we have

\ σe6(A,B,C,ε) = σ(A + K,B,C,ε), K∈Kε (X) where Kε (X) is given in (9.8). Using Proposition 2.6.3 (i), we obtain

Kε (X) ⊂ Oε (X). Consequently, \ σ(A + K,B,C,ε) ⊂ σe6(A,B,C,ε). Q.E.D. K∈Oε (X) Chapter 11

A Characterization of the Essential Pseudospectra

In this chapter, we investigate some parts of the pseudospectrum of closed densely deﬁned operators on a Banach space from the viewpoint of Fred- holms theory by examining the different types of pseudospectra, such as approximation pseudospectrum, defect pseudospectrum, essential approximation pseudospectrum, essential defect pseudospectrum, essential structured approximation pseudospectrum and essential structured defect pseudospectrum.

11.1 APPROXIMATION OF ε-PSEUDOSPECTRUM

In order to obtain the strongest possible results we have to refine the notion of the ε-pseudospectrum of an operator (A,D(A)) defined on the Banach space X. For it turns out that the part σ(A)\σap(A) cannot always be approximated in the general case. For ε ≥ 0, we define the ε- approximate spectrum σε,ap(·) by

σε,ap(A) = {λ ∈ C such that αe(λ − A) ≤ ε}, 280 Linear Operators and Their Essential Pseudospectra where αe(·) is given in Remark 3.1.1 (iv). In particular σ0,ap(A) = σap(A) as well as

σε,ap(A) ⊂ Σε (A). The following results, given in this section, come from [161].

Theorem 11.1.1 Let A ∈ C (X). Then, σε,ap(A) is always closed. ♦

Proof. Let λ ∈/ σε,ap(A). Then,

αe(λ − A) > ε.

Now, let µ ∈ C satisfy

|λ − µ| < αe(λ − A) − ε.

Then,

αe(µ − A) = inf{k(µ − A)xk such that kxk = 1 and x ∈ D(A)} ≥ inf{|k(λ − A)xk − |λ − µ|kxk| such that kxk = 1 and x ∈ D(A)} > ε.

So, the complement of σε,ap(A) is open. Q.E.D.

Corollary 11.1.1 If A ∈ C (X) and ε > 0, then Σap,ε (A) is closed. ♦

Now, we present the following simple and useful result:

Proposition 11.1.1 Let A ∈ C (X) and ε > 0. Then,

(i) σap,ε (A) ⊂ σε (A), \ (ii) σap(A) = σap,ε (A), ε>0

(iii) if ε1 < ε2, then σap(A) ⊂ σap,ε1 (A) ⊂ σap,ε2 (A), (iv) if A ∈ L (X) and λ ∈ σap,ε (A), then |λ| < ε + kAk, (v) if α ∈ C and ε > 0, then σap,ε (A + α) = α + σap,ε (A), and (vi) if α ∈ C\{0} and ε > 0, then σap,|α|ε (αA) = α σap,ε (A). ♦ A Characterization of the Essential Pseudospectra 281

Proof. (i) If λ ∈/ σε (A), then 1 k(λ − A)−1k ≤ . ε Moreover, 1 kxk = sup inf k(λ − A)xk x∈D(A), kxk=1 k(λ − A)xk x∈D(A), kxk=1 kxk = sup 06=x∈D(A) k(λ − A)xk k(λ − A)−1yk = sup y∈X\{0} kyk = k(λ − A)−1k 1 ≤ . ε Hence, inf k(λ − A)xk) ≥ ε. x∈D(A), kxk=1 So,

λ ∈/ σap,ε (A).

(ii) It is clear that σap(A) ⊂ σap,ε (A), then

\ σap(A) ⊂ σap,ε (A). ε>0 \ Conversely, if λ ∈ σap,ε (A), then for all ε > 0, we have λ ∈ σap,ε (A). ε>0 Let n o λ ∈ λ ∈ C such that inf k(λ − A)xk < ε , x∈D(A), kxk=1 taking limits as ε → 0+, we get for all x ∈ D(A) such that kxk = 1,

inf k(λ − A)xk = 0. x∈D(T), kxk=1

We infer that λ ∈ σap(A). 282 Linear Operators and Their Essential Pseudospectra

(iii) Let λ ∈ σap,ε1 (A), then

inf k(λ − A)xk < ε1 < ε2. x∈D(A), kxk=1

Hence, λ ∈ σap,ε2 (A).

(iv) Let λ ∈ σap,ε (A), then inf k(λ − A)xk < ε, x∈D(A), kxk=1 and also

|λ| − kAxk < k(λ − A)xk. Hence, |λ| < ε + kAk.

(v) Let λ ∈ σap,ε (A + α), then inf k((λ − α) − A)xk < ε. x∈D(A), kxk=1 Hence,

λ − α ∈ σap,ε (A). This yields to

λ ∈ α + σap,ε (A). The second inclusion follows by using the same reasoning.

(vi) Let λ ∈ σap,|α|ε (αA), then λ inf k(λ − αA)xk = inf α − A x α 6= 0, x∈D(A), kxk=1 x∈D(A), kxk=1 α λ = |α| inf − A x x∈D(A), kxk=1 α < |α|ε.

Hence, λ ∈ σap (A). α ,ε So,

σap,|α|ε (αA) ⊆ α σap,ε (A). However, the reverse inclusion is similar. Q.E.D. A Characterization of the Essential Pseudospectra 283

Remark 11.1.1 P. H. Wolff shows that for all ε > 0 that Σap,ε (A) 6= Σε (A), (see [161]). ♦

Proposition 11.1.2 Let A ∈ C (X) and ε > 0. Then,

(i) Σap,ε (A) ⊂ Σε (A), \ (ii) Σap,ε (A) = σap(A), and ε>0

(iii) if ε1 < ε2, then σap(A) ⊂ Σap,ε1 (A) ⊂ Σap,ε2 (A). ♦

Proof. The proof of (i), (ii) and (iii) may be achieved in the same way as the proof of (i), (ii) and (iii) for Proposition 11.1.1. Q.E.D.

11.2 A CHARACTERIZATION OF APPROXIMATION PSEUDOSPECTRUM

In this section, we turn to the problem when the closure of σap,ε (A) is equal to Σap,ε (A) holds. We consider the following hypothesis for A :

(H6) There is no open set in ρap(A) := C\σap(A) on which the

λ −→ inf k(λ − A)xk x∈D(A), kxk=1 is constant. Our ﬁrst result is the following.

Theorem 11.2.1 Let A ∈ C (X) and ε > 0. If (H6) holds, then

σap,ε (A) = Σap,ε (A). ♦

Proof. Since σap,ε (A) ⊂ Σap,ε (A) and Σap,ε (A) is closed, then

σap,ε (A) ⊂ Σap,ε (A).

In order to prove the inverse inclusion, we take λ ∈ Σap,ε (A). We notice the existence of two cases: st 1 case : If λ ∈ σap,ε (A), then λ ∈ σap,ε (A). 284 Linear Operators and Their Essential Pseudospectra

nd 2 case : If λ ∈ Σap,ε (A)\σap,ε (A), then

inf k(λ − A)xk = ε. x∈D(A), kxk=1,

By using Hypothesis (H6), there exists a sequence λn ∈ ρap(A) such that

λn → λ, and

inf k(λn − A)xk < inf k(λ − A)xk = ε. x∈D(A), kxk=1 x∈D(A), kxk=1

We deduce that λn ∈ σap,ε (A) and hence, λ ∈ σap,ε (A). So,

Σap,ε (A) ⊂ σap,ε (A). Q.E.D.

Theorem 11.2.2 Let A ∈ C (X) and ε > 0. Then, the following properties are equivalent:

(i) λ ∈ σap,ε (A). (ii) There exists a bounded operator D ∈ L (X) such that kDk < ε and λ ∈ σap(A + D). ♦

Proof. (i) ⇒ (ii) Let λ ∈ σap,ε (A). Then, there exists x0 ∈ X such that kx0k = 1 and

k(λ − A)x0k < ε. (11.1) By using the Hahn Banach theorem (see Theorem 2.1.4), there exists x0 ∈ X∗ (dual of X) such that kx0k = 1 and

0 x (x0) = kx0k.

Consider the operator D deﬁned by the formula

0 D : X −→ X, x −→ Dx := x (x)(λ − A)x0.

Then, D is a linear operator everywhere deﬁned on X. It is bounded, since

0 kDxk = kx (x)(λ − A)x0k 0 ≤ kx kkxkk(λ − A)x0k, A Characterization of the Essential Pseudospectra 285 for x 6= 0. Therefore, kDxk ≤ k(λ − A)x k. kxk 0 Hence, by using Eq. (11.1), we have

kDk < ε.

Let x0 ∈ X, then

inf k(λ − A − D)xk ≤ k(λ − A − D)x0k x∈D(A), kxk=1 0 ≤ k(λ − A)x0 − x (x0)(λ − A)x0k = 0.

This prove that inf k(λ − A − D)xk = 0. x∈D(A), kxk=1 (ii) ⇒ (i) We assume that there exists a bounded operator D ∈ L (X) such that kDk < ε and λ ∈ σap(A + D), which means that

inf k(λ − A − D)xk = 0. x∈D(A), kxk=1 In order to prove that

inf k(λ − A)xk < ε, x∈D(A), kxk=1 we can write

k(λ − A)x0k = k(λ − A − D + D)x0k

≤ k(λ − A − D)x0k + kDx0k.

Then, inf k(λ − A)xk < ε. Q.E.D. x∈D(A), kxk=1 We can derive from Theorem 11.2.2 the following result.

Corollary 11.2.1 Let A ∈ C (X) and ε > 0. Then, [ σap,ε (A) = σap(A + D). ♦ kDk<ε 286 Linear Operators and Their Essential Pseudospectra

Theorem 11.2.3 Let A ∈ C (X) and ε > 0. Then,

[ σap,ε (A) = σap(A + D), D∈Θε (X) where n o Θε (X) := D ∈ L (X) such that kDk < ε and dimR(D) ≤ 1 . ♦

Proof. Let λ ∈ σap,ε (A), then there exists x0 ∈ X such that kx0k = 1 and

k(λ − A)x0k < ε.

−1 Putting kx0k = k(λ − A) (λ − A)x0k implies that 1 k(λ − A)−1k > . ε

Then, we can ﬁnd y0 ∈ X such that ky0k = 1 and 1 k(λ − A)−1y k > . 0 ε Hence, 1 k(λ − A)−1y k = , 0 δ where δ < ε. By using the Hahn Banach theorem (see Theorem 2.1.4), there exists x0 ∈ X∗ such that kx0k = 1 and

1 x0 (λ − A)−1y = k(λ − A)−1y k = . 0 0 δ Now, we can deﬁne the rank-one operator by,

0 D : X −→ X, x −→ Dx := δx (x)y0.

Clearly, D is a linear operator everywhere deﬁned on X. It is bounded, since

0 kDxk = kδx (x)y0k 0 ≤ δkx kky0kkxk. A Characterization of the Essential Pseudospectra 287

Then, kDk ≤ δ < ε. Furthermore,

−1 0 −1 D (λ − A) y0 = δx (λ − A) y0 y0 1 = δ y δ 0 = y0.

−1 Putting x = (λ − A) y0, we will discuss these two cases: st 1 case : If x = x0, we obtain

inf k(λ − A − D)xk ≤ k(λ − A − D)x0k x∈D(A), kxk=1

≤ k(λ − A)x0 − Dx0k

= ky0 − y0k = 0.

nd 2 case : If x 6= x0. First, let x = 0, then

−1 (λ − A) y0 = 0, which is a contradiction with 1 k(λ − A)−1y k = . 0 δ Second, let x 6= 0, then

Dx = y0 = (λ − A)x.

Hence, inf k(λ − A − D)xk = 0. x∈D(A), kxk=1

We deduce that λ ∈ σap(A + D) and D ∈ Θε (X). The second inclusion is clear. Q.E.D.

Theorem 11.2.4 Let A ∈ C (X) and ε > 0. Let E ∈ L (X) such that kEk < ε. Then,

σap,ε−kEk(A) ⊆ σap,ε (A + E) ⊆ σap,ε+kEk(A). ♦ 288 Linear Operators and Their Essential Pseudospectra

Proof. Let λ ∈ σap,ε−kEk(A). Then, by using Theorem 11.2.2, there exists a bounded operator D ∈ L (X) with kDk < ε − kEk such that λ ∈ σap(A + D) = σap (A + E) + (D − E) .

The fact that kD − Ek ≤ kDk + kEk < ε allows us to deduce that λ ∈ σap,ε (A+E). Using a similar reasoning to the

ﬁrst inclusion, we deduce that λ ∈ σap,ε+kEk(A). Q.E.D.

Theorem 11.2.5 Let A ∈ L (X) and V ∈ L (X) be invertible. If B = V −1AV, then

σap(B) = σap(A), and for ε > 0 and k = kV −1kkVk, we have

σ ε (A) ⊆ σap,ε (B) ⊆ σap,k (A), and (11.2) ap, k ε Σ ε (A) ⊆ Σap,ε (B) ⊆ Σap,k (A). (11.3) ap, k ε

♦

Proof. We can write

kλ − Bk = kV −1(λ − A)Vk ≤ kkλ − Ak, (11.4) kλ − Ak = kV(λ − B)V −1k ≤ kkλ − Bk. (11.5)

Let λ ∈ σap(A), which implies that

inf k(λ − A)xk = 0. kxk=1

By using relation (11.4), it follows that

inf k(λ − B)xk = 0. kxk=1

Hence, λ ∈ σap(B). The converse is similar, it is sufﬁcient to use relation (11.5). For the second result, if λ ∈ σ ε (A), then using (11.4), we obtain ap, k A Characterization of the Essential Pseudospectra 289

λ ∈ σap,ε (B). If λ ∈ σap,ε (B), then using relation (11.5), we obtain λ ∈

σap,kε (B). The second formula in (11.2) holds and the proof is similar for the relation (11.3). Q.E.D.

The closure of σap,ε (A) is always contained in Σap,ε (A), but equality holds if, and only if, A does not have constant inﬁmum norm on any open set. The present part addresses the question on whether or not a similar equality holds in the case of non-strict inequalities: ? [ Σap,ε (A) = σap(A + D). kDk≤ε

Theorem 11.2.6 Let A ∈ C (X) and ε > 0. Then, [ σap(A + D) ⊂ Σap,ε (A). ♦ kDk≤ε

Proof. Let λ ∈/ Σap,ε (A), then

inf k(λ − A)xk > ε. x∈D(A), kxk=1 It is easy to see inf k(λ − A − D)xk > 0 x∈D(A), kxk=1 for all D ∈ L (X) such that kDk ≤ ε. So, [ λ ∈/ σap(A + D), kDk≤ε which completes the proof of theorem. Q.E.D.

We first consider the following example: Let l1(N) the space defined by ( +∞ ) l1(N) = (x j) j≥1 such that x j ∈ C and ∑ |x j| < ∞ j=1 which equipped with the following norm ∞ kxk := ∑ |x j|. j=1 Define the operator A by 290 Linear Operators and Their Essential Pseudospectra

A : l1(N) −→ l1(N), x −→ Ax, where

∞ ! Ax = (1 + 2ε)x1 − ∑ x j,−ε2x2,··· ,−εnxn,··· , j=2 x = (x1,x2,··· ,xn,···) ∈ l1(N), ε > 0, and εn, where n = 2,3,···, is a sequence of positive numbers monotonically decreasing to 0. Then, we have

∞ ! (A − 2ε)x = x1 − ∑ x j,−(2ε + ε2)x2,··· ,−(2ε + εn)xn,··· , j=2 and ∞ ∞ ∞

k(A − 2ε)xk = x1 − ∑ x j + 2ε ∑ |x j| + ∑ ε j|x j|. j=2 j=2 j=2 Suppose kxk = 1. Then, we have

∞ ∞

k(A − 2ε)xk = x1 − ∑ x j + 2ε(1 − |x1|) + ∑ ε j|x j|. j=2 j=2

Let ∞

m(x) := x1 − ∑ x j + 2ε(1 − |x1|). j=2 1 If |x1| ≥ 2 , then we have ∞ m(x) ≥ |x1| − ∑ |x j| + 2ε(1 − |x1|) j=2 = 2(1 − ε)|x1| + 2ε − 1 1 ≥ 2(1 − ε) + 2ε − 1 2 = ε

1 If |x1| ≤ 2 , then we have

m(x) ≥ 2ε(1 − |x1|) ≥ ε. A Characterization of the Essential Pseudospectra 291

Hence, m(x) ≥ ε, kxk = 1,

1 where the equality is achieved if, and only if, |x1| = 2 and

∞ ∑ x j = x1. j=2

Consequently,

∞ k(A − 2ε)xk = m(x) + ∑ ε j|x j| > ε, kxk = 1, (11.6) j=2 and k(A − 2ε)x(k)k → ε as k → ∞, where (k) 1 1 x = ,0,··· ,0, ,0,··· , k ∈ N. 2 | {z } 2 k zeros

Thus,

inf k(A − 2ε)xk = ε. (11.7) kxk=1 It is easy to see that A − 2ε is invertible. It follows from (11.7) that 1 k(A − 2ε)−1k = . ε Hence, 2ε ∈ Σε (A). Let D ∈ L (X) such that kDk ≤ ε. Then, D − 2ε is invertible and A + D − 2ε is a Fredholm operator of index 0. Suppose that its kernel is not trivial. Then, there exists x ∈ l1(N) such that kxk = 1 and

(A − 2ε)x = −Dx.

It follows from (11.6) that the left-hand side of the last equality has a norm strictly larger than ε. The norm of the right-hand side, on the other hand, is less than or equal to ε. This contradiction implies that the kernel of A+D− 292 Linear Operators and Their Essential Pseudospectra

2ε is trivial. A+D−2ε is therefore invertible and we have 2ε ∈ ρ(A+D).

It follows that 2ε ∈ Σap,ε (A) and 2ε ∈/ σap(A + D) for all kDk ≤ ε. Then,

[ 2ε ∈/ σap(A + D). kDk≤ε

Hence, [ σap(T + D) Σap,ε (A). ♦ kDk≤ε

Theorem 11.2.7 Let A ∈ C (X) and ε > 0. If (H6) holds, then

[ Σap,ε (A) = σap(A + D). (11.8) kDk≤ε

♦

Proof. It follows from both Theorems 11.2.6 and 11.2.2 that

[ σap,ε (A) = σap(A + D) ⊆ Σap,ε (A). kDk≤ε

So, by using the hypothesis (H6), we have

σap,ε (A) = Σap,ε (A).

Hence,

[ σap(A + D) = Σap,ε (A). kDk≤ε

It follows from both Theorems 11.2.6 and 11.2.2 that Eq. (11.8) is an equality if, and only if, the level set n o λ ∈ C such that inf k(λ − A)xk = ε x∈D(A), kxk=1 is a subset of [ σap(A + D). Q.E.D. kDk=ε A Characterization of the Essential Pseudospectra 293

11.3 ESSENTIAL APPROXIMATION PSEUDOSPECTRUM

In this section, we have the following useful stability result for the essential approximation pseudospectrum.

Proposition 11.3.1 Let A ∈ C (X) and ε > 0. Then, \ (i) σeap,ε (A) = σe7(A), ε>0

(ii) if ε1 < ε2, then σe7(A) ⊂ σeap,ε1 (A) ⊂ σeap,ε2 (A), and (iii) σeap,ε (A + F) = σeap,ε (A) for all F ∈ K (X). ♦

Proof. (i) σe7(A) ⊂ σeap,ε (A). Indeed, Let λ ∈/ σeap,ε (A). Then, there exists K ∈ K (X) such that

inf k(λ − A − K)xk > ε > 0. x∈D(X), kxk=1

Hence, λ ∈/ σe7(A). So,

\ σe7(A) ⊂ σeap,ε (A). ε>0 Conversely, let \ λ ∈ σeap,ε (A). ε>0

Then, for all ε > 0, we have λ ∈ σeap,ε (A). Hence, for every K ∈ K (X), we obtain λ ∈ σap,ε (A + K). This implies that

inf k(λ − A − K)xk < ε. x∈D(X), kxk=1

+ Taking limits as ε → 0 , we infer that λ ∈ σe7(A).

(ii) Let λ ∈ σeap,ε1 (A), then there exists K ∈ K (X), such that

inf k(λ − A − K)xk < ε1 < ε2. x∈D(X), kxk=1

So, λ ∈ σeap,ε2 (A). 294 Linear Operators and Their Essential Pseudospectra

(iii) It follows immediately from the deﬁnition of σeap,ε (A) that

σeap,ε (A + F) = σeap,ε (A) for all F ∈ K (X). Q.E.D.

Theorem 11.3.1 Let A ∈ C (X) and ε > 0. Then, the following properties are equivalent:

(i) λ ∈/ σeap,ε (A). (ii) For all D ∈ L (X) such that kDk < ε, we have λ −A−D ∈ Φ+(X) and i(λ − A − D) ≤ 0. ♦

Proof. (i) ⇒ (ii) Let λ ∈/ σeap,ε (A). Then, there exists a compact operator

K on X such that λ ∈/ σap,ε (A + K). By using Theorem 11.2.2, we notice that λ ∈/ σap(A + D + K), for all D ∈ L (X) such that kDk < ε. So,

λ − A − D − K ∈ Φ+(X) and i(λ − A − D − K) ≤ 0, for all D ∈ L (X) such that kDk < ε. Using Lemma 2.3.1, we get for all D ∈ L (X) such that kDk < ε,

λ − A − D ∈ Φ+(X) and i(λ − A − D) ≤ 0.

(ii) ⇒ (i) We assume that for all D ∈ L (X) such that kDk < ε, we have

λ − A − D ∈ Φ+(X) and i(λ − A − D) ≤ 0.

Based on Lemma 2.2.7, λ − A − D can be expressed in the form

λ − A − D = S + K, where K ∈ K (X) and S ∈ C (X) is an operator with closed range and α(S) = 0. So,

λ − T − D − K = S and α(λ − A − D − K) = 0. A Characterization of the Essential Pseudospectra 295

By using Theorem 2.1.3, there exists a constant c > 0 such that

k(λ − A − D − K)xk ≥ ckxk, for all x ∈ D(A).

This proves that

inf k(λ − A − D − K)xk ≥ c > 0. x∈D(T), kxk=1

Thus, λ ∈/ σap(A + D + K), and therefore λ ∈/ σeap,ε (A). Q.E.D.

Remark 11.3.1 It follows immediately from Theorem 11.3.1 that λ ∈/

σeap,ε (A) if, and only if, for all D ∈ L (X) such that kDk < ε, we obtain

λ − A − D ∈ Φ+(X) and i(λ − A − D) ≤ 0.

This is equivalent to [ σeap,ε (A) = σe7(A + D). ♦ kDk<ε

From the deﬁnition of σeap,ε (A) and Proposition 11.3.1, we get the following corollary:

Corollary 11.3.1 Let A ∈ C (X) and ε > 0. Then,   \ \ [ σe7(T) = lim σap,ε (A + K) =  σe7(A + D). ♦ ε→0 K∈K (X) ε>0 kDk<ε

Theorem 11.3.2 Let A ∈ C (X) and ε > 0. Then, \ σeap,ε (A) = σap,ε (A + F). ♦ F∈F+(X) 296 Linear Operators and Their Essential Pseudospectra

\ Proof. Let λ ∈/ σap,ε (A + F), then there exists F ∈ F+(X) such F∈F +(X) that

λ ∈/ σap,ε (A + F). In view of Theorem 11.2.2, we have

λ ∈/ σap(A + F + D), for all D ∈ L (X) such that kDk < ε. Therefore,

λ − A − F − D ∈ Φ+(X) and i(λ − A − F − D) ≤ 0.

Using Lemma 2.3.1, we conclude that for all D ∈ L (X) such that kDk < ε,

λ − A − D ∈ Φ+(X) and i(λ − A − D) ≤ 0.

Finally, Theorem 11.3.1 shows that λ ∈/ σeap,ε (A). For the second inclusion, it is clear that

\ \ σap,ε (A + F) ⊂ σap,ε (A + F) =: σeap,ε (A), F∈F+(X) F∈K (X) because K (X) ⊂ F+(X), which completes the proof. Q.E.D.

Remark 11.3.2 Let A ∈ C (X) and ε > 0.

(i) Using Theorem 11.3.2, we infer that σeap,ε (A + F) = σeap,ε (A) for all F ∈ F+(X). (ii) Let I(X) be any subset of L (X). If K (X) ⊂ I(X) ⊂ F+(X), then

\ σeap,ε (A) = σap,ε (A + M) M∈I(X) and

σeap,ε (A + J) = σeap,ε (A) for all J ∈ I(X). ♦ A Characterization of the Essential Pseudospectra 297

Lemma 11.3.1 Let ε > 0, A and B be two elements of C (X). Assume that for a bounded operator D such that kDk < ε, the operator B is (A + D)- compact, then

σeap,ε (A) = σeap,ε (A + B). ♦

Proof. Let λ ∈/ σeap,ε (A), then for all D ∈ L (X) such that kDk < ε, we have λ − A − D ∈ Φ+(X) and

i(λ − A − D) ≤ 0.

Since B is (A + D)-compact and applying Theorem 2.3.5, we get

λ − A − B − D ∈ Φ+(X) and i(λ − A − B − D) ≤ 0.

Therefore,

λ ∈/ σeap,ε (T + D). We conclude that

σeap,ε (A + B) ⊂ σeap,ε (A).

Conversely, let λ ∈/ σeap,ε (A+B). Then, for all D ∈ L (X) such that kDk < ε, we have

λ − A − B − D ∈ Φ+(X) and i(λ − A − B − D) ≤ 0.

On the other hand, B is (A+D)-compact. Using Theorem 2.3.7, we deduce that B is (A + B + D)-compact. Hence,

λ − A − D ∈ Φ+(X) and i(λ − A − D) ≤ 0.

Therefore, λ ∈/ σeap,ε (A). This proves that

σeap,ε (A) ⊂ σeap,ε (A + B). Q.E.D. 298 Linear Operators and Their Essential Pseudospectra

11.4 PROPERTIES OF ESSENTIAL PSEUDOSPECTRA

In this section, we give the results of stability and sum for the essential pseudospectra.

Theorem 11.4.1 Let X be a Banach space, ε > 0 and consider A ∈ C (X). Then, (i) If J ∈ F (X), then

σei,ε (A) = σei,ε (A + J), i = 4,5.

(ii) If J ∈ F+(X), then

σei,ε (A) = σei,ε (A + J), i = 1,ap.

(iii) If J ∈ F−(X), then

σei,ε (A) = σei,ε (A + J), i = 2,δ.

T (iv) If J ∈ F+(X) F−(X), then

σe3,ε (A) = σe3,ε (A + J). ♦

Proof. (i) Let λ ∈/ σe5,ε (A), then λ − A − D ∈ Φ(X) and

i(λ − A − D) = 0 for all ||D|| < ε. Hence, using Lemma 2.3.1, we have λ −A−D−J ∈ Φ(X) and i(λ − A − D − J) = 0 for ||D|| < ε. Therefore, λ ∈/ σe5,ε (A + J), i.e.,

σe5,ε (A + J) ⊂ σe5,ε (A).

The opposite inclusion is obtained by symmetry:

σe5,ε (A) = σ5,ε (A + J − J) ⊂ σe5,ε (A + J). A Characterization of the Essential Pseudospectra 299

(ii) Let λ ∈/ σeap,ε (A), then λ − A − D ∈ Φ+(X) and i(λ − A − D) ≤ 0 for all ||D|| < ε. Hence, using Lemma 2.3.1, we have

λ − A − D − J ∈ Φ+(X) and i(λ − A − D − J) ≤ 0 for all ||D|| < ε. Therefore,

λ ∈/ σeap,ε (A + J), i.e.,

σeap,ε (A + J) ⊂ σeap,ε (A). Analogously, by using Lemma 2.3.1 and arguing as above, we can deriving easily the opposite inclusion

σeap,ε (A) ⊂ σeap,ε (A + J).

So,

σei,ε (A + J) = σei,ε (A), i = 1,ap. Statements (iii), (iv) can be checked in the same way (ii). Q.E.D.

Theorem 11.4.2 Let X be a Banach space, ε > 0 and consider A, B ∈ b C (X). Assume that there are A0,B0 ∈ L (X) and F1,F2 ∈ F (X) such that

AA0 = I − F1, (11.9)

BB0 = I − F2. (11.10)

T b (i) If 0 ∈ ΦA ΦB,A0 − B0 ∈ F+(X) and i(A) = i(B), then

σeap,ε (A) = σeap,ε (B). (11.11)

T (ii) If 0 ∈ ΦA ΦB,A0 − B0 ∈ F−(X) and i(A) = i(B), then

σeδ,ε (A) = σeδ,ε (B). ♦ 300 Linear Operators and Their Essential Pseudospectra

Proof. Let λ be a complex number, Eqs. (11.9) and (11.10) imply that

(λ − A − D)A0 − (λ − B − D)B0 = F1 − F2 + (λ − D)(A0 − B0). (11.12)

(i) Let λ ∈/ σeap,ε (B), then

λ − B − D ∈ Φ+(X) and i(λ − B − D) ≤ 0 for all D ∈ L (X) such that ||D|| < ε. Since B + D is closed and D(B + D) = D(B) endowed with the graph norm is a Banach space denoted by XB+D. We can regard B + D as operator from XB+D into X. This will be denoted by B\+ D. Using Eq. (2.6), we can show that

b λ − (B\+ D) ∈ Φ+(X,XB+D) and i(λ − (B\+ D)) ≤ 0.

b Moreover, since F2 ∈ F (X), Eq. (11.10), Lemma 2.3.1, and Theorem b 2.2.22, we ﬁnd that B0 ∈ Φ (X,XB+D) and consequently,

b (λ − (B\+ D))B0 ∈ Φ+(X).

b Since A0 − B0 ∈ F+(X), and by using both Eq. (11.12) and Lemma 2.3.1, we can prove that b (λ − A − D)A0 ∈ Φ+(X) and

i[(λ − A − D)A0] = i[(λ − B − D)B0]. (11.13) A similar reasoning show that

b A0 ∈ Φ (X,XA+D), where XA+D = (D(A),||.||A+D). By Theorem 2.2.3, we can write

A0S = I − F on XA+D, (11.14) A Characterization of the Essential Pseudospectra 301 where S ∈ L (XA+D,X) and F ∈ K (XA). Hence, by Eq. (11.14), we have

(λ − (A\+ D))A0S = (λ − (A\+ D)) − (λ − (A\+ D))F. (11.15)

b Since S ∈ Φ (XA+D,X), using Theorem 2.2.5, we show that

b (λ − (A\+ D))A0S ∈ Φ+(X,XA+D). By using Eq. (11.15) and, by applying Lemma 2.3.1, we prove that

b λ − (A\+ D) ∈ Φ+(X,XA+D), and in view of Eq. (2.6), we have

λ − A − D ∈ Φ+(X).

It follows that ε λ − A ∈ Φ+(X). (11.16) b Since, F1, F2 ∈ F (X), Eqs. (11.9), (11.10) and Lemma 2.3.1 give

i(A) + i(A0) = i(I − F1) = 0 and

i(B) + i(B0) = i(I − F2) = 0. Since i(A) = i(B), then

i(A0) = i(B0). Using both Eq. (11.13) and Theorem 2.2.5, we can write

i(λ − A − D) + i(A0) = i(λ − B − D) + i(B0).

Therefore, i(λ − A − D) ≤ 0. (11.17) Using both Eqs. (11.16) and (11.17), we conclude that

λ ∈/ σeap,ε (A).

Therefore,

σeap,ε (A) ⊂ σeap,ε (B). The opposite inclusion follows by symmetry and we obtain (11.11). 302 Linear Operators and Their Essential Pseudospectra

(ii) Similarly, we can prove the statement

σeδ,ε (A) = σeδ,ε (B). Q.E.D.

Theorem 11.4.3 Let X be a Banach space, ε > 0 and consider A, B ∈ L (X). Then, (i) If for all bounded operator D such that ||D|| < ε and A(B + D) ∈ F b(X), then [ σe4,ε (A + B)\{0} ⊂ [σe4(A) σe4,ε (B)]\{0}.

If, further, (B + D)A ∈ F b(X), then [ σe4,ε (A + B)\{0} = [σe4(A) σe4,ε (B)]\{0}.

(ii) If for all bounded operator D such that ||D|| < ε and A(B + D) ∈ b F+(X), then [ σe1,ε (A + B)\{0} ⊂ [σe1(A) σe1,ε (B)]\{0}.

b If, further, (B + D)A ∈ F+(X), then [ σe1,ε (A + B)\{0} = [σe1(A) σe1,ε (B)]\{0}. (11.18)

(iii) If for all bounded operator D such that ||D|| < ε and A(B + D) ∈ b F−(X), then [ σe2,ε (A + B)\{0} ⊂ [σei(A) σe2,ε (B)]\{0}.

b If, further, (B + D)A ∈ F−(X), then [ σe2,ε (A + B)\{0} = [σe2(A) σe2,ε (B)]\{0}. (11.19)

(iv) If for all bounded operator D such that ||D|| < ε and A(B + D) ∈ b T b F+(X) F−(X), then

σe3,ε (A + B)\{0} ⊂ S S T S T [(σe3(A) σe3,ε (B)) (σe3(A) σe2,ε (B)) (σe2(A) σe1,ε (B))]\{0}. b T b Moreover, if (B + D)A ∈ F+(X) F−(X), then

σe3,ε (A + B)\{0} = S S T S T [(σe3(A) σe3,ε (B)) (σe3(A) σe2,ε (B)) (σe2(A) σe1,ε (B))]\{0}. ♦ A Characterization of the Essential Pseudospectra 303

Proof. (i) For λ ∈ C, we can write

(λ − A)(λ − B − D) = A(B + D) + λ(λ − A − B − D) (11.20) and

(λ − B − D)(λ − A) = (B + D)A + λ(λ − A − B − D). (11.21)

S S b Let λ 6∈ σe4(A) σe4,ε (B) {0}. Then, λ −A ∈ Φ (X) and for all kDk < ε, λ − B − D ∈ Φb(X). It follows from Theorem 2.2.5 that

(λ − A)(λ − B − D) ∈ Φb(X).

Since A(B+D) ∈ F b(X), applying Eq. (11.20), we have λ −A−B−D ∈ b Φ (X), then λ 6∈ σe4,ε (A + B). Therefore, [ σe4,ε (A + B)\{0} ⊆ σe4(A) σe4,ε (B) \{0}. (11.22)

Now, we prove the inverse inclusion of Eq. (11.22). Suppose λ 6∈

σe4,ε (A + B)\{0}, then for all D ∈ L (X) such that kDk < ε, we have λ − A − B − D ∈ Φb(X). In view of A(B + D) ∈ F b(X) and (B + D)A ∈ F b(X), and by using both Eqs. (11.20) and (11.21), we have

(λ − A)(λ − B − D) ∈ Φb(X) and (λ − B − D)(λ − A) ∈ Φb(X).

Applying Lemma 2.2.3, it is clear that λ −A ∈ Φb(X) and for all kDk < ε, b S we have λ −B−D ∈ Φ (X). Therefore, λ 6∈ σe4(A) σe4,ε (B). This proves that [ σe4,ε (A + B)\{0} = σe4(A) σe4,ε (B) \{0}. S S (ii) Suppose that λ ∈/ σe1(A) σe1,ε (B) {0}, then

b λ − A ∈ Φ+(X) and ε λ − B ∈ Φ+(X). 304 Linear Operators and Their Essential Pseudospectra

b b Hence, λ − A ∈ Φ+(X) and λ − B − D ∈ Φ+(X) for all ||D|| < ε. Using Theorem 2.2.5, we have b (λ − A)(λ − B − D) ∈ Φ+(X). b Since A(B + D) ∈ F+(X), and by applying Eq. (11.20), we infer that λ − b ε A − B − D ∈ Φ+(X). Hence, λ − A − B ∈ Φ+(X). So, λ ∈/ σe1,ε (A + B). Therefore, [ σe1,ε (A + B)\{0} ⊂ [σe1(A) σe1,ε (B)]\{0}. S ε Suppose λ ∈/ σe1,ε (A+B)] {0}, then λ −A−B ∈ Φ+(X), so λ −A−B− b D ∈ Φ+(X) for all bounded operator D such that ||D|| < ε. Since A(B + b b D) ∈ F+(X) and (B + D)A ∈ F+(X), then by Eqs. (11.20), (11.21) and Lemma 2.3.1, we have b b (λ −A)(λ −B−D) ∈ Φ+(X) and (λ −B−D)(λ −A) ∈ Φ+(X). (11.23) b By using (11.23) and Theorem 2.2.10, it is clear that λ − A ∈ Φ+(X) and ε S λ − B ∈ Φ+(X). Hence, λ ∈/ σe1(A) σe1,ε (B). Therefore, [ σe1(A) σe1,ε (B) ⊂ σe1,ε (A + B). This prove that [ σe1(A) σe1,ε (B) = σe1,ε (A + B). (iii) can be checked in the same way as (ii). (iv) Since the equalities \ σe3(A) = σe1(A) σe2(A), \ σe3,ε (B) = σe1,ε (B) σe2,ε (B), and \ σe3,ε (A + B) = σe1,ε (A + B) σe2,ε (A + B) b T b and, in view of both A(B + D) ∈ F+ F−(X) and (B + D)A ∈ b T b F+(X) F−(X), one can infer from both Eqs. (11.18) and (11.19) that

σe3,ε (A + B)\{0} = S S T S T [(σe3(A) σe3,ε (B)) (σe3(A) σe2,ε (B)) (σe2(A) σe1,ε (B))]\{0}. Q.E.D. A Characterization of the Essential Pseudospectra 305

11.5 PSEUDOSPECTRUM OF BLOCK OPERATOR MATRICES

Let X1 and X2 be two Banach spaces and consider the 2 × 2 block operator matrices deﬁned on X1 × X2 by ! AB L := , CD where A ∈ L (X1), B ∈ L (X2 × X1), C ∈ L (X1 × X2), and D ∈ L (X2). Let ! T 0 T = 1 , 0 T2 where Ti ∈ L (Xi), i = 1,2. Deﬁning the norm of operator matrix T as ||T|| = max kT1k,kT2k .

Lemma 11.5.1 Let A ∈ L (X1),B ∈ L (X2) and consider the 2 × 2 block operator matrices ! AC MC := , 0 B where C ∈ L (X2,X1). Then, b b b (i) if A ∈ Φ (X1) and B ∈ Φ (X2), then MC ∈ Φ (X1 × X2) for every

C ∈ L (X2,X1), b b b (ii) if A ∈ Φ+(X1) and B ∈ Φ+(X2), then MC ∈ Φ+(X1 × X2) for every C ∈ L (X2,X1), and b b b (iii) if A ∈ Φ−(X1) and B ∈ Φ−(X2), then MC ∈ Φ−(X1 ×X2) for every C ∈ L (X2,X1). ♦

Proof. (i) Let us write MC in the form ! ! ! I 0 IC A 0 MC = . (11.24) 0 B 0 I 0 I 306 Linear Operators and Their Essential Pseudospectra

b b Since A ∈ Φ (X1) and B ∈ Φ (X2), then ! A 0 0 I and ! I 0 0 B are both Fredholm operators. So, MC is a Fredholm operator since ! IC 0 I is invertible for every C ∈ L (X2,X1). (ii) and (iii) can be checked in the same way as (i). Q.E.D.

Remark 11.5.1 Using the same reasoning as in the proof of Lemma 11.5.1, we can show that: b b (i) if A ∈ Φ (X1) and B ∈ Φ (X2), then ! A 0 MD := DB is a Fredholm operator on X1 × X2 for every D ∈ L (X1,X2), b b b (ii) if A ∈ Φ+(X1) and B ∈ Φ+(X2), then MD ∈ Φ+(X1 ×X2) for every D ∈ L (X1,X2), and b b b (iii) if A ∈ Φ−(X1) and B ∈ Φ−(X2), then MD ∈ Φ−(X1 ×X2) for every D ∈ L (X1,X2). ♦

Lemma 11.5.2 Let A ∈ L (X1),B ∈ L (X2) and consider the 2 × 2 block operator matrices ! AC MC := , 0 B where C ∈ L (X2,X1). Then, b b (i) if MC ∈ Φ+(X1 × X2), then A ∈ Φ+(X1), and b b (ii) if MC ∈ Φ−(X1 × X2), then B ∈ Φ−(X2). ♦ A Characterization of the Essential Pseudospectra 307

Proof. The result follows immediately from Eq. (11.24). Q.E.D.

b Remark 11.5.2 (i) It follows from Lemma 11.5.2 that, if MC ∈ Φ (X1 × b b X2), then A ∈ Φ+(X1) and B ∈ Φ−(X2). (ii) Using the same reasoning as in the proof of Lemma 11.5.1, we can show that, if the operator ! A 0 b is in Φ (X1 × X2) DB

b b for some D ∈ L (X1,X2), then A ∈ Φ−(X1) and B ∈ Φ+(X2). ♦

Theorem 11.5.1 (A. Jeribi, N. Moalla, and S. Yengui [102]) Let ! F F F := 11 12 , F21 F22

b where Fi j ∈ L (Xj,Xi), with i, j = 1,2. Then, F ∈ F (X1 ×X2) if, and only b if, Fi j ∈ F (Xj,Xi), with i, j = 1,2. ♦

Proof. In order to prove the second implication, let us consider the following decomposition: ! ! ! ! F 0 0 F 0 0 0 0 F = 11 + 12 + + . 0 0 0 0 F21 0 0 F22

b It is sufﬁcient to prove that if Fi j ∈ F (Xj,Xi), with i, j = 1,2, then each operator in the right side of the previous equality is a Fredholm perturbation on X1 ×X2. For example, we will prove the result for the ﬁrst operator. The proofs for the other operators will be similarly achieved. Consider ! AB b L = ∈ Φ (X1 × X2) CD and let us denote ! F11 0 Fe := . 0 0 308 Linear Operators and Their Essential Pseudospectra

From Theorem 2.2.3, it follows that there exist ! A0 B0 L0 = ∈ L (X1 × X2) C0 D0 and ! K11 K12 K = ∈ K (X1 × X2), K21 K22 such that

LL0 = I − K on X1 × X2. Then, ! I − K11 + F11A0 −K12 + F11B0 (L + Fe)L0 = I − K + FLe 0 = . −K21 I − K22

b Since F11 ∈ F (X1), and using Theorem 2.3.1, we will have

b I − K11 + F11A0 ∈ Φ (X1).

b This, together with the fact that I − K22 ∈ Φ (X2), allows us to deduce, from Lemma 11.5.1 (i), that ! 0 0 (L + Fe)L0 − −K21 0 is a Fredholm operator on X1 × X2. The fact that K21 is a compact operator b b and L0 ∈ Φ (X1 × X2) leads, by Theorem 2.2.6, to L + Fe ∈ Φ (X1 × X2). b Conversely, assume that F ∈ F (X1 × X2). We will prove that F11 ∈ b b F (X1). Let A ∈ Φ (X1) and let us deﬁne the operator ! A −F12 L1 := . 0 I

From Lemma 11.5.1 (i), it follows that

b L1 ∈ Φ (X1 × X2). A Characterization of the Essential Pseudospectra 309

Hence, ! A + F11 0 b F + L1 = ∈ Φ (X1 × X2). F21 I + F22 The use of Remark 11.5.2 (ii) leads to

b A + F11 ∈ Φ−(X1). (11.25) In the same way, we may consider the Fredholm operator ! A 0 . −F21 I Using Remarks 11.5.1 (i) and 11.5.2 (i), it is easy to deduce that

b A + F11 ∈ Φ+(X1). (11.26) b From Eqs. (11.25) and (11.26), it follows that F11 ∈ F (X1). In the same way, we can prove that

b F22 ∈ F (X2).

b b Now, we have to prove that F12 ∈ F (X2,X1) and F21 ∈ F (X1,X2). For b b this, let us consider A ∈ Φ (X2,X1) and B ∈ Φ (X1,X2). Then, ! 0 A b ∈ Φ (X1 × X2). B 0

b b Using the facts that F11 ∈ F (X1), that F22 ∈ F (X2), as well as the result of the second implication, we can deduce that ! −F11 0 b F + ∈ F (X1 × X2). 0 −F22 Hence, ! 0 A + F12 b ∈ Φ (X1 × X2). B + F21 0 So, b A + F12 ∈ Φ (X2,X1) and

b B + F21 ∈ Φ (X1,X2). Q.E.D. 310 Linear Operators and Their Essential Pseudospectra

Lemma 11.5.3 Let A ∈ L (X1),B ∈ L (X2 × X1), ε > 0 and consider the block operator matrices ! AC MC = , 0 B where C ∈ L (X2 × X1). Then, S (i) σei,ε (MC) ⊂ σei,ε (A) σei,ε (B), i = 4,5, S (ii) σei,ε (MC) ⊂ σei,ε (A) σe,ε (B) i = 1,ap, and S (iii) σei,ε (MC) ⊂ σei,ε (A) σei,ε (B) i = 2,δ. ♦

Proof. Let T1 ∈ L (X1), T2 ∈ L (X2) and consider ! T 0 T = 1 0 T2 such that ||T|| < ε. The operator λ − MC − T can be written as the following form ! ! ! I 0 I −C λ − A − T1 0 λ − MC − T = . (11.27) 0 λ − B − T2 0 I 0 I S If λ ∈/ σe4,ε (A) σe4,ε (B), then

ε λ − A ∈ Φ (X1) and ε λ − B ∈ Φ (X2). Since ||T|| < ε, then

||T1|| < ε and

||T2|| < ε.

So, λ − A − T1 and λ − B − T2 are Fredholm operators. Therefore, ! λ − A − T1 0 b ∈ Φ (X1 × X2) 0 I A Characterization of the Essential Pseudospectra 311 and ! I 0 b ∈ Φ (X1 × X2). 0 λ − B − T2 Since ! I −C 0 I is invertible, then b λ − MC − T ∈ Φ (X1 × X2) for all ||T|| < ε. Therefore,

ε λ − MC ∈ Φ (X1 × X2).

Hence, [ σe4,ε (MC) ⊂ σe4,ε (A) σe4,ε (B). On the other hand, Eq. (11.27), the invertibility of ! I −C 0 I and Theorem 2.2.19 leads to " !# " !# I 0 λ − A − T1 0 i(λ − MC − T) = i + i . 0 λ − B − T2 0 I So,

i(λ − MC − T) = i(λ − A − T1) + i(λ − B − T2). Hence, [ σe5,ε (MC) ⊂ σe5,ε (A) σe5,ε (B). (ii) and (iii) can be checked in the same way as (i). Q.E.D. In all that follows we will make the following assumptions: b b (H7) CB ∈ F (X2) and C(A + T1) ∈ F (X1 × X2),

(H8) CB ∈ K (X2) and C(A + T1) ∈ K (X1 × X2), and

(H9) BC ∈ K (X2) and (D + T2)C ∈ K (X1 × X2). 312 Linear Operators and Their Essential Pseudospectra

Theorem 11.5.2 (i) If (H7) holds, then [ σei,ε (L )\{0} ⊆ [σei,ε (A) σei,ε (D)]\{0}, 4,5.

(ii) If (H8) holds, then [ σe1,ε (L )\{0} ⊆ [σe1,ε (A) σe1,ε (D)]\{0}.

(iii) If (H9) holds, then S σe2,ε (L )\{0} ⊆ [σe2,ε (A) σe2,ε (D)]\{0}. ♦

Proof. Let T1 ∈ L (X1), T2 ∈ L (X2) and consider ! T 0 T = 1 0 T2 such that ||T|| < ε. Then, for all λ ∈ C\{0}, we have λ − L − T =   ! I 0 ! 1 0 0 λ − A − T1 −B +  C  . λ −C(A + T1) −CB − I 0 λ − D − T2 λ (11.28) S S (i) Suppose that λ ∈/ σe5,ε (A) σe5,ε (D) {0}, then by Lemma 11.5.3, ! λ − A − T1 −B b ∈ Φ (X1 × X2) 0 λ − D − T2 and " !# λ − A − T −B i 1 = 0. 0 λ − D − T2 Since   I 0  C  − I λ is invertible, using Theorem 2.2.19   I 0 ! λ − A − T1 −B  C  − I 0 λ − D − T2 λ is a Fredholm operator and A Characterization of the Essential Pseudospectra 313

   I 0 ! λ − A − T1 −B i C   = − I 0 λ − D − T2 λ   I 0 " !# λ − A − T1 −B i C  + i = 0. − I 0 λ − D − T2 λ On the other hand, it follows from the hypothesis (H7) and Theorem 11.5.1 that ! 0 0 b ∈ F (X1 × X2). −C(A + T1) −CB ε So, using Eq. (11.28), λ − L ∈ Φ (X1 × X2) and

i(λ − L − T) = 0 for all ||T|| < ε. Thus,

λ ∈/ σe5,ε (L ). Hence,

[ σei,ε (L )\{0} ⊆ [σei,ε (A) σei,ε (D)]\{0}, i = 4,5.

S S (ii) If λ ∈/ σe1,ε (A) σe1,ε (D) {0}, then

ε λ − A ∈ Φ+(X1) and ε λ − D ∈ Φ+(X2). Using Lemma 11.5.3, we have ! λ − A − T1 −B b ∈ Φ+(X1 × X2) 0 λ − D − T2 for all ||T|| < ε. Since   I 0  C  − I λ 314 Linear Operators and Their Essential Pseudospectra is invertible, then   I 0 ! λ − A − T1 −B b  C  ∈ Φ+(X1 × X2). − I 0 λ − D − T2 λ Hence, it follows from both hypothesis (H8) and Eq. (11.28), that

b λ − L − T ∈ Φ+(X1 × X2) for all ||T|| < ε. Thus,

λ ∈/ σe1,ε (L ). Hence, [ σe1,ε (L )\{0} ⊆ [σe1,ε (A) σe1,ε (D)]\{0}. (iii) The proof of (iii) may be checked in the same way as the proof of the item (ii). Q.E.D. Chapter 12

Conditional Pseudospectra

Our aim in this chapter is to show some properties of condition pseudospectra of a linear operator A in Banach spaces and reveal the relation between their condition pseudospectrum.

12.1 SOME PROPERTIES OF Σε (A)

In the next lemma, we give some properties of the condition pseudospectral radius rε (·).

Lemma 12.1.1 Let A ∈ L (X) and 0 < ε < 1. Then, 1 + ε (i) r (A) ≤ r (A) ≤ kAk. σ ε 1 − ε k 1 (ii) If kA k ≤ δ < ε for all k ≥ 0, then 1 + δ 2ε r (A) ≤ . ♦ ε 1 − δε Proof. (i) Since σ(A) ⊆ Σε (A) for 0 < ε < 1, we obtain that

rσ (A) ≤ rε (A). 316 Linear Operators and Their Essential Pseudospectra

Now, we prove that 1 + ε r (A) ≤ kAk. ε 1 − ε Let λ ∈ Σε (A). We will discuss these two cases: 1st case : If |λ| ≤ kAk, then it is clear that

1 + ε r (A) ≤ kAk. ε 1 − ε

2nd case : If |λ| > kAk, then λ − A is an invertible operator and 1 k(λ − A)−1k ≤ . |λ| − kAk

So, |λ| + kAk 1 < εk(λ − A)−1kkλ − Ak ≤ ε. |λ| − kAk After a simple computation, we have 1 + ε |λ| ≤ kAk. 1 − ε Hence, 1 + ε r (A) ≤ kAk. ε 1 − ε (ii) Since λ ∈ Σε (A) and

k 1 rσ (A) = lim kA k k k→∞ 1 = lim δ k k→∞ = 1, then there are two possible cases: Conditional Pseudospectra 317

1st case : If |λ| ≤ 1, then it is clear that 1 + δ 2ε |λ| ≤ . 1 − δε 2nd case : If |λ| > 1, then

λ ∈/ σ(A).

We get λ − A is an invertible operator. Therefore, ∞ k −1 1 kA k k(λ − A) k ≤ ∑ k |λ| k=0 |λ| δ ∞ 1 ≤ ∑ k |λ| k=0 |λ| δ = . |λ| − 1 Hence, δ 1 < εk(λ − A)−1kkλ − Ak ≤ ε(|λ| + δ) , |λ| − 1 which implies that 1 + δ 2ε |λ| ≤ . 1 − δε So, 1 + δ 2ε r (A) ≤ . Q.E.D. ε 1 − δε The following theorem establish the relationship between condition pseudospectrum and pseudospectrum of a bounded linear operator A ∈ L (X). We set n o δA := inf kλ − Ak : λ ∈ C . Theorem 12.1.1 Let A ∈ L (X) such that A 6= I and 0 < ε < 1. Then,

ε υε Σ (A) ⊆ σγε (A) ⊆ Σ (A), where 2εkAk γ = ∈]0,1[, ε 1 − ε 2εkAk υε = ∈]0,1[ (1 − ε)δA and σγε (A) is the pseudospectrum of A. ♦ 318 Linear Operators and Their Essential Pseudospectra

Proof. Let λ ∈ Σε (A), then

1 + ε |λ| ≤ kAk 1 − ε and 1 kλ − Akk(λ − A)−1k > . ε Thus, 1 k(λ − A)−1k > εkλ − Ak 1 > ε(|λ| + kAk) 1 − ε ≥ . 2εkAk Hence,

λ ∈ σγε (A).

For the second inclusion, let λ ∈ σγε (A). Then, 1 − ε k(λ − A)−1k > . 2εkAk

Also, we have n o kλ − Ak ≥ inf kλ − Ak : λ ∈ C := δA.

Hence, 1 − ε kλ − Akk(λ − A)−1k > δ . A 2εkAk Therefore, λ ∈ Συε (A). Q.E.D.

In the following we gives a precise information about the condition pseudospectrum of bounded linear operator under linear transformation.

Proposition 12.1.1 Let A ∈ L (X) and 0 < ε < 1. Then, \ (i) σ(A) = Σε (A). 0<ε<1 Conditional Pseudospectra 319

(ii) If 0 < ε1 < ε2 < 1, then

σ(A) ⊂ Σε1 (A) ⊂ Σε2 (A).

(iii) If α ∈ C, then

Σε (A + αI) = α + Σε (A).

(iv) If α ∈ C\{0}, then Σε (αA) = αΣε (A). ♦

Proof. (i) It is clear that

σ(A) ⊂ Σε (A) for all 0 < ε < 1, then \ σ(A) ⊂ Σε (A). 0<ε<1 Conversely, if \ λ ∈ Σε (A), 0<ε<1 then for all 0 < ε < 1, we have

λ ∈ Σε (A).

We will discuss these two cases: 1st case : If λ ∈ σ(A), then we get the desired result. nd −1 1 2 case : If λ ∈ λ ∈ C : kλ − Akk(λ − A) k > , then taking ε limits as ε → 0+, we get

kλ − Akk(λ − A)−1k = ∞.

Thus, λ ∈ σ(A).

(ii) Let λ ∈ Σε1 (A), then 1 1 kλ − Akk(λ − A)−1k > > . ε1 ε2 320 Linear Operators and Their Essential Pseudospectra

Hence, λ ∈ Σε2 (A). (iii) Let λ ∈ Σε (A + αI), then 1 k(λ − α)I − Akk((λ − α)I − A)−1k > . ε Hence, λ − α ∈ Σε (A). This yields to λ ∈ α + Σε (A). The opposite inclusion follows by symmetry. (iv) Let λ ∈ Σε (αA) such that α 6= 0, then

1 −1 −1 λ −1 λ < kλ − αAk k(λ − αA) k = α − A α − A , ε α α

−1 λ λ = − A − A . α α Hence, λ ∈ Σε (A), α so Σε (αA) ⊆ αΣε (A). However, the opposite inclusion follows by symmetry. Q.E.D.

Theorem 12.1.2 Let A ∈ L (X), k = kAkkA−1k and 0 < ε < 1. Then, (i) λ ∈ Σε (A) if, and only if, λ ∈ Σε (A∗). (ii) If λ ∈ Σε (A−1)\{0}, then 1 ∈ Σεk(A)\{0}. ♦ λ Proof. (i) Using the identity

kλ − Akk(λ − A)−1k = kλ − A∗kk(λ − A∗)−1k, it is easy to see that the condition pseudospectrum of A∗ is given by the mirror image of Σε (A) with respect to the real axis. Conditional Pseudospectra 321

(ii) Let λ ∈ Σε (A−1)\{0}, then 1 < kλ − A−1kk(λ − A−1)−1k ε and

kλ − A−1kk(λ − A−1)−1k

1 1 −1 −1 −1 = −λ − A A −λ A − A , λ λ

1 1 −1 −1 ≤ kAkkA k − A − A . λ λ Hence, 1 ∈ Σεk(A)\{0}. Q.E.D. λ Theorem 12.1.3 Let A ∈ L (X) and V ∈ L (X) be invertible. Let B = V −1AV. Then, for all 0 < ε < 1 and k = kV −1kkVk such that 0 < k2ε < 1, we have ε 2 Σ k2 (A) ⊆ Σε (B) ⊆ Σk ε (A). ♦ Proof. We can write kλ − Bkk(λ − B)−1k = kV −1(λ − A)VkkV −1(λ − A)−1Vk 2 ≤ kVkkV −1k kλ − Akk(λ − A)−1k ≤ k2kλ − Akk(λ − A)−1k, (12.1) and kλ − Akk(λ − A)−1k = kV(λ − B)V −1kkV(λ − B)−1V −1k 2 ≤ kVkkV −1k kλ − Bkk(λ − B)−1k ≤ k2kλ − Bkk(λ − B)−1k. (12.2) ε Let λ ∈ Σ k2 (A), then using relation (12.2), we obtain λ ∈ Σε (B). If λ ∈ Σε (B), then using relation (12.1), we obtain

2 λ ∈ Σk ε (A). Q.E.D. 322 Linear Operators and Their Essential Pseudospectra

12.2 CHARACTERIZATION OF CONDITION PSEUDOSPECTRUM

In this section, we give a characterization of the condition pseudospectrum of linear operators on a Banach space. Our ﬁrst result is the following.

Lemma 12.2.1 Let A ∈ L (X) and 0 < ε < 1. Then, λ ∈ Σε (A)\σ(A) if, and only if, there exists x ∈ X, such that k(λ − A)xk < εkλ − Akkxk. ♦

Proof. Let λ ∈ Σε (A)\σ(A), then 1 kλ − Akk(λ − A)−1k > . ε Thus, 1 k(λ − A)−1k > . εkλ − Ak Moreover, k(λ − A)−1yk 1 sup > . y∈X\{0} kyk εkλ − Ak Then, there exists a nonzero y ∈ X, such that kyk k(λ − A)−1yk > . εkλ − Ak

Putting x = (λ − A)−1y. We have the result. Conversely, we assume that there exists x ∈ X such that

k(λ − A)xk < εkλ − Akkxk.

Let λ ∈/ σ(A) and x = (λ − A)−1y, then

kxk ≤ k(λ − A)−1kkyk.

Moreover, 1 < εkλ − Akk(λ − A)−1k. So, Conditional Pseudospectra 323

λ ∈ Σε (A)\σ(A). Q.E.D. In the following theorem, we investigate the relation between the condition pseudospectrum and the usual spectrum in a complex Banach space.

Theorem 12.2.1 Let A ∈ L (X), λ ∈ C, and 0 < ε < 1. If there is D ∈ L (X) such that kDk ≤ εkλ − Ak and λ ∈ σ(A + D), then λ ∈ Σε (A). ♦

Proof. We assume that there exists D such that kDk < εkλ − Ak and λ ∈ σ(A + D). If λ ∈/ Σε (A), then 1 kλ − Akk(λ − A)−1k ≤ . ε Now, we deﬁne the operator S : X −→ X by

∞ n S := ∑ (λ − A)−1 D(λ − A)−1 . n=0 Since kD(λ − A)−1k < 1, we can write −1 S = (λ − A)−1 I − D(λ − A)−1 . Then, there exists y ∈ X such that −1 S I − D(λ − A)−1 y = (λ − A)−1y.

Let x = (λ − A)−1y. Then,

S(λ − A − D)x = x for every x ∈ X. Similarly, we can prove that

(λ − A − D)Sy = y for all y ∈ X. Hence, λ − A − D is invertible, so

λ ∈ Σε (A). Q.E.D. 324 Linear Operators and Their Essential Pseudospectra

Theorem 12.2.2 Let X be a complex Banach space satisfying the following property: for all bounded operator A with 0 ∈ ρ(A), there exists a non invertible bounded operator D such that 1 kA − Dk = . kA−1k

If λ ∈ Σε (A), then there exists D ∈ L (X) such that kDk ≤ εkλ − Ak and λ ∈ σ(A + D). ♦

Proof. Suppose λ ∈ Σε (A). We will discuss these two cases: 1st case : If λ ∈ σ(A), then it is sufﬁcient to take D = 0. 2nd case : If λ ∈ Σε (A)\σ(A), then by assumption, there exists an element B ∈ L (X) such that 1 kλ − A − Bk = . k(λ − A)−1k

Let D = λ − A − B. Then 1 kDk = ≤ εkλ − Ak. k(λ − A)−1k

Also B = λ − (A + D), is not invertible. So, λ ∈ σ(A + D). Q.E.D.

The next corollary is an immediate consequence of Theorems 12.2.1 and 12.2.2.

Corollary 12.2.1 Let X be a complex Banach space satisfying the hypothesis of Theorem 12.2.2. Then, λ ∈ Σε (A) if, and only if, there exists D ∈ L (X) such that kDk ≤ εkλ − Ak and λ ∈ σ(A + D). ♦

In the sequel of this section, we consider the Hilbert space X, we use the notation conv(Σε (A)), the convex hull in C of a set Σε (A), B(a,r) the open Conditional Pseudospectra 325 ball with center at a and radius r and we deﬁne the distance between two nonempty subsets U and V by the formula

inf{ku − vk : u ∈ U, v ∈ V}.

The next theorem gives a relation between the condition pseudospectrum and the numerical range of A, given by

W(A) := {hAx,xi such that x ∈ SX }, where SX := {x ∈ X such that kxk = 1} is the unit sphere in X. It is well known that W(A) is a convex set whose closure contains the spectrum σ(A) of A.

Theorem 12.2.3 Let A ∈ L (X) and 0 < ε < 1. Then, ε 2ε conv(Σ (A)) ⊆ W(A) + B 0, kAk . ♦ 1 − ε

Proof. Let λ ∈ C such that

dist(λ,W(A)) > 0, then λ − A is invertible and, we have 1 k(λ − A)−1k ≤ . dist(λ,W(A))

Assume that λ ∈ Σε (A). There are two possible cases: 1st case : If λ ∈ W(A), then the result is trivial. 2nd case : If λ ∈ Σε (A)\W(A). Then,

1 kλ − Ak < kλ − Akk(λ − A)−1k ≤ . ε dist(λ,W(A)) 326 Linear Operators and Their Essential Pseudospectra

It follows that

dist(λ,W(A)) < εkλ − Ak ≤ ε (|λ| + kAk).

Using Lemma 12.1.1, we obtain that

1 + ε dist(λ,W(A)) ≤ ε kAk + kAk 1 − ε 2ε = kAk. 1 − ε Hence, ε 2ε Σ (A) ⊆ W(A) + B 0, kAk . 1 − ε The result follows from the fact that W(A) is convex set. Q.E.D.

Corollary 12.2.2 Let A ∈ L (X) and 0 < ε < 1. Then, n o ε λ ∈ C such that dist(λ,W(A)) < εkλ − Ak ⊆ Σ (A). ♦

Proof. Let λ ∈/ Σε (A). By using Theorem 12.2.3, we have 1 ≤ k(λ − A)−1k dist(λ,W(A)) 1 ≤ . εkλ − Ak Hence, dist(λ,W(A)) ≥ εkλ − Ak. Q.E.D. Bibliography

1. Jeribi, A., (2015). Spectral Theory and Applications of Linear Oper- ators and Block Operator Matrices, Springer-Verlag, New York.

2. Jeribi, A., (2018). Spectral Theory of Linear Non-Self-Adjoint Oper- ators in Hilbert and Banach Spaces: Denseness and Bases with Applications, preprint.

3. Jeribi, A., (2018). Denseness, Bases and Frames in Banach Spaces and Applications, de Gruyter, Berlin.

4. Jeribi, A., & Krichen, B., (2016). Nonlinear Functional Analysis in Banach Spaces and Banach Algebras. Fixed point theory under weak topology for nonlinear operators and block operator matrices with applications. (English) Monographs and Research Notes in Mathe- matics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-3388-5/hbk; 978-1-4987-3389-2/ebook). xvi, 355 p.

5. Jeribi, A., Hammami, M. A., & Masmoudi, A., (eds.), (2015). Applied Mathematics in Tunisia. International Conference on Advances in Applied Mathematics (ICAAM), Hammamet, Tunisia, December 1619, 2013. Springer proceedings in mathematics and statistics 131. Cham: Springer (ISBN 978-3-319-18040-3/hbk; 978- 3-319-18041-0/ebook). xix, 397 p.

6. Abdelmoumen, B., Dehici, A., Jeribi, A., & Mnif, M., (2008). Some new properties of fredholm theory, Schechter essential spectrum, and application to transport theory, J. Inequal. Appl., Art. ID 852676, 1–14. 328 Bibliography

7. Abdelmoumen, B., Jeribi, A., & Mnif, M., (2011). Invariance of the Schechter essential spectrum under polynomially compact operators perturabation, Extr. Math., 26(1), 61–73.

8. Abdelmoumen, B., Jeribi, A., & Mnif, M., (2012). On graph measures in banch spaces and description of essential spectra of multidimensional transport equation, Acta Math. Sci. Ser. B Engl. Ed., 32(5), 2050–2064.

9. Abdelmoumen, B., Jeribi, A., & Mnif, M., (2012). Measure of weak noncompactness, some new properties in fredholm theory, characterization of the Schechter essential spectrum and application to transport operators, Ricerche Mat., 61, 321–340.

10. Abdmouleh, F., Ammar, A., & Jeribi, A., (2013). Stability of the S-Essential Spectra on a Banach Space, Math. Slovaca, 63(2), 299– 320.

11. Abdmouleh, F., Ammar, A., & Jeribi, A., A (2015). Characteriza- tion of the pseudo-browder essential spectra of linear operators and application to a transport equation, J. Comput. Theor. Transp., 44(3), 141–153.

12. Abdmouleh, F., Charﬁ, S., & Jeribi, A., (2012). On a characterization of the essential spectra of the sum and the product of two operators, J. Math. Anal. Appl., 386(1), 83–90.

13. Abdmouleh, F., & Jeribi, A., (2010). Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators, J. Math. Anal. Appl., 364, 414–223.

14. Abdmouleh, F., & Jeribi, A., (2011). Gustafson, Weidmann, Kato, Wolf, Schechter, Browder, Rakocevi˘ c´ and Schmoger essential spectra of the sum of two bounded operators and application to a transport operator, Math. Nachr., 284(2–3), 166–176. Bibliography 329

15. Aiena, P., (1989). Riesz Operators and Perturbation Ideals, Note Mat., 9(1), 1–27.

16. Aiena, P., (2007). Semi-Fredholm Operators, Perturbation Theory and Localized SVEP, Caracas, Venezuela.

17. Akhmerov, R. R., Kamenskii, M., I., Potapov, A. S., Rodkina, A., E., & Sadovskii, B. N., (1992). Measures of Noncompactness and Condensing Operators, Birkhauser¨ Verlag, Basel Boston Berlin.

18. Alvarez, T., (1998). Perturbation and coperturbation functions characterizing semi-Fredholm type operators, Math. Proc. R. Ir. Acad., 98 A(1), 41–26.

19. Ammar, A., Boukattaya, B., & Jeribi, A., (2014). Stability of the S-left and S-right essential spectra of a linear operator, Acta Math. Scie., Ser. B Engl. Ed., 34(5), 1–13.

20. Ammar, A., Boukettaya, B., & Jeribi, A., (2014). Stability of the S-left and S-right essential spectra of a linear operator, Acta Math. Scie. Ser. B Engl. Ed., 34(6), 1922–1934.

21. Ammar, A., Boukettaya, B., & Jeribi, A., (2016). A note on the essential pseudospectra and application, Linear and Multilinear Algebra, 64(8), 1474–1483.

22. Ammar, A., Bouzayeni, F., & Jeribi, A., (2017). Perturbation of unbounded linear operators by γ-relative boundedness, Ricerche Mat., DOI 10.1007/s11587-017-0341-0, 1–13.

23. Ammar, A., Dhahri, M. Z., & Jeribi, A., (2015). Stability of essential approximate point spectrum and essential defect spectrum of linear operator, Filomat, 29(9), 1983–1994.

24. Ammar, A., Dhahri, M. Z., & Jeribi, A., (2015). Some properties of the M-essential spectra of closed linear operator on a Banach space, Functional Analysis, Approximation and Computation, 7(1), 15–28. 330 Bibliography

25. Ammar, A., Dhahri, M. Z., & Jeribi, A., (2015). A Characterization of S-essential spectrum by mean of measure of non-strict-singularity and application, Azerbaijan J. Math., V. 5(1), 2218–6816.

26. Ammar, A., Dhahri, M. Z., & Jeribi, A., (2015). Some properties of upper triangular 3 × 3-block matrices of linear relations, Boll. Unione Mat. Ital., 8(3), 189–204.

27. Ammar, A., Djaidja, N., & Jeribi, A., (2017). The essential spectrum of a sequence of linear operators in Banach spaces, International J. Anal. Appl., 1–7.

28. Ammar, A., Heraiz, T., & Jeribi, A., (2017). Essential approximate point spectrum and essential defect spectrum of a sequence of linear operators in Banach spaces, preprint.

29. Ammar, A., & Jeribi, A., (2013). A characterization of the essential pseudospectra on a Banach space, Arab. J. Math., 2(2), 139–145.

30. Ammar, A., & Jeribi, A., (2013). A characterization of the essential pseudospectra and application to a transport equation, Extr. Math., 28, 95–112.

31. Ammar, A., & Jeribi, A., (2014). Measures of noncompactness and essential pseudo-spectra on Banach Space, Math. Methods Appli. Sci., 37(3), 447–452.

32. Ammar, A., & Jeribi, A., (2016). The Weyl essential spectrum of a sequence of linear operators in Banach spaces, Indag. Math. (N.S.), 27(1), 282–295.

33. Ammar, A., Jeribi, A., & Mahfoudhi, K., (2017). A characterization of the essential approximation pseudospectrum on a Banach space, Filomat 31(11), 3599–3610.

34. Ammar, A., Jeribi, A., & Mahfoudhi, K., (2017). Global bifurcation from the real leading eigenvalue of the transport operator, J. Comput. Theor. Transp., 46(4), 229–241. Bibliography 331

35. Ammar, A., Jeribi, A., & Mahfoudhi, K., (2017). A Characterization of the Condition Pseudospectrum, preprint.

36. Ammar, A., Jeribi, A., & Moalla, N., (2013). On a characterization of the essential spectra of a 3 × 3 operator matrix and application to three-group transport operators, Ann. Funct. Anal. AFA 4(2), 153– 170, electronic only.

37. Ammar, A., Jeribi, A., & Saadaoui, B., (2017). Frobenius- Schur factorization for multivalued 2 × 2 matrices linear operator, Mediterr. J. Math., 14(1), Paper No. 29, 29 p.

38. Astala, K., (1980). On measure of noncompactness and ideal varia- tions in Banach spaces, Ann. Acad. Sci. Fenn. Ser. A.I. Math. Diss., 29.

39. Atkinson, F. V., Langer, H., Mennicken, R., & Shkalikov, A. A., (1994). The essential spectrum of some matrix operators, Math. Nachr., 167, 5–20.

40. Baloudi, H., & Jeribi, A., (2014). Left-Right Fredholm and Weyl spectra of the sum of two bounded operators and application, Mediterr. J. Math., 11, 939–953.

41. Baloudi, H., & Jeribi, A., (2016). Holomorphically Weyl- decomposably regular. (English) Funct. Anal. Approx. Comput., 8(2), 13–22.

42. Banas,´ J., & Geobel, K., (1980). Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, Volume 60, Marcel Dekker, New York, 259–262.

43. Ben Ali, N., Jeribi, A., & Moalla, N., (2010). Essential spectra of some matrix operators, Math. Nachr., 283(9), 1245–1256.

44. Ben Amar, A., Jeribi, A., & Mnif, M., (2010). Some applications of the regularity and irreducibility on transport theory, Acta Appl. Math., 110, 431–448. 332 Bibliography

45. Ben Amar, A., Jeribi, A., & Krichen, B., (2010). Essential spectra of a 3 × 3 operator matrix and application to three-group transport equation, Inte. Equa. Oper. Theo., 68, 1–21.

46. Ben Amar, A., Jeribi, A., & Mnif, M., (2014). Some results on Fred- holm and semi-Fredholm operators, Arab. J. Math. (Springer) 3(3), 313–323.

47. Benharrat, M., Ammar, A., Jeribi, A., & Messirdi, B., (2014). On the Kato, semi-regular and essentially semi-regular spectra, Functional Analysis, Approximation and Computation 6(2), 9–22.

48. Browder, F. E., (1961). On the spectral theory of elliptic differential operators, I, Math. Ann., 142, 22–130.

49. Caradus, S. R., (1966). Operators of Riesz type, Paciﬁc J. Math., 18, 61–71.

50. Caradus, S. R., (1966). Operators with ﬁnite ascent and descent, Paciﬁc J. Math., 18, 437–449.

51. Chaker, W., Jeribi, A., & Krichen, B., (2015). Demicompact linear operators, essential spectrum and some perturbation results, Math. Nachr., 288(13), 1476–1486.

52. Charﬁ, S., & Jeribi, A., (2009). On a characterization of the essential spectra of some matrix operators and applications to two-group transport operators, Math. Z., 262(4), 775–794.

53. Charﬁ, S., Jeribi, A., & Moalla, N., (2013). Time asymptotic behavior of the solution of an abstract Cauchy problem given by a one-velocity transport operator with Maxwell boundary condition, Collectanea Mathematica, 64, 97–109.

54. Charﬁ, S., Jeribi, A., & Moalla, R., (2014). Essential spectra of operator matrices and applications, Methods Appli. Sci. 37(4), 597–608. Bibliography 333

55. Charﬁ, S., Jeribi, A., & Walha, I., (2010). Essential Spectra, Matrix Operator and Applications, Acta Appl. Math., 111, (3), 319–337.

56. Damak, M., & Jeribi, A., (2007). On the essential spectra of some matrix operators and applications, Electron. J. Differential Equa- tions 11, 1–16.

57. Danes, J., (1972). On the istratescu measure of noncompactness, Bull. Math. Soc. R.S. Roumanie, 16(64), 403–406.

58. Davies, E. B., (2007). Linear Operators and Iheir Spectra, Cambridge University Press, New York.

59. Dehici, A., Latrach, K., & Jeribi, A., (2002). On a transport operator arising in growing cell populations I. Spectral analysis, Advances in Mathematics Research, pp. 149–175.

60. Dehici, A., Jeribi, A., & Latrach, K., Spectral analysis of a transport operator arising in growing cell populations, Acta Appl. Math., 92(1), 37–62 (2006).

61. Diagana, T., (2015). Perturbations of unbounded Fredholm linear operators in Banach spaces, in: Handbook on Operator Theory, 875– 880, Springer Basel.

62. Diestel, J., (1980). A survey of results related of the Dunford-Pettis property, Amer. Math. Soc. of Conf. on integration, Topology and Geometric in Linear Spaces, 15–60.

63. Dominguez Benavides, T., (1986). Some properties of the set and ball measures of noncompactness and applications, J. London Math. Soc., 34(2), 120–128.

64. Dunford, N., & Pettis, B. J., (1940). Linear operators on summable functions, Trans. Amer. Math. Soc., 47, 323–392.

65. Dunford, N., & Schwartz, J. T., (1971). Linear Operators, Part III, Wiley-Interscience, New York. 334 Bibliography

66. Edmum, D. E., & Evans, W. D., (1987). Spectral Theory and Differ- ential Operators, Oxford Science Publications.

67. Elleuch, A., & Jeribi, A., (2016). On a characterization of the structured Wolf, Schechter and Browder essential pseudospectra, Indag. Math., 27, 212–224.

68. Elleuch, A., & Jeribi, A., (2016). New description of the structured essential pseudospectra, Indag. Math., 27, 368–382.

69. Faierman, M., Mennicken, R., & Muller, M., (1995). A boundary eigenvalue problem for a system of partial differential operators occurring in magnetohydrodynamics, Math. Nachr., 173, 141–167.

70. Fakhfakh, F., & Mnif, M., (2009). Browder and semi-Browder operators and perturbation function, Extr. Math., 24(3), 219–241.

71. Gethner, R. M., & Shapiro, J. H., (1987). Universal vectors for operators on space of holomorphic functions, Proc. Am. Math. Soc., 100, 281–288.

72. Gohberg, I. C., & Krein, G., (1960). Fundamental theorems on deﬁ- ciency numbers, root numbers and indices of linear operators, Amer. Math. Soc. Transl., Ser. 2, 13, 185–264.

73. Gohberg, I. C., & Krein, M. G., (1969). Introduction to the theory of linear non-self-adjoint operators in Hilbert space, Amer. Math. Soc., Providence.

74. Gohberg, I., Markus, A., ad Feldman, I. A., (1967). Normally solv- able operators and ideals associated with them, Proc. Amer. Math. Soc. Tran. Ser. 2(61), 63–84.

75. Goldberg, S., (1966). Unbounded Linear Operators, New York: McGraw-Hill. Bibliography 335

76. Goldberg, S., (1974). Perturbations of semi-Fredholm operators by operators converging to zero compactly, Proc. Amer. Math. Soc., 45, 93–98.

77. Gonzalez, M., & Onieva, M. O., (1985).On Atkinson operators in locally convex spaces, Math. Z., 190, 505–517.

78. Grothendieck, B., (1953). Sur les applications lineaires´ faiblement compactes d’espaces du type C(K), Canad. J. Math., 5, 129–173.

79. Gustafson, K., & Weidmann, J., (1969). On the essential spectrum, J. Math. Anal. Appl., 25, 121–127.

80. Harrabi, A., (1998). Pseudospectre d’une suite d’operateurs´ bornes,´ RAIRO Model.´ Math. Anal. Numer.,´ 32, 671–680.

81. Hinrichsen, D., & Kelb, B., (1993). Spectral value sets: a graphical tool for robustness analysis, Systems Control Lett., 21(2), 127–136.

82. Jeribi, A., (1997). Quelques remarques sur les oprateurs de Fred- holm et application l’quation de transport, C. R. Acad. Sci., Paris, Srie, I 325, 43–48.

83. Jeribi, A., (1998). Quelques remarques sur le spectre de Weyl et applications, C. R. Acad. Sci., Paris Sr. I Math., 327(5), 485–490.

84. Jeribi, A., (1998). Developpement´ de certaines propriet´ es´ ﬁnes de la theorie´ spectrale et applications a` des modeles` monocinetiques´ et a` des modeles` de Reggeons, Thesis of Mathematics. University of Corsica, Frensh (16 Janvier 1998).

85. Jeribi, A., (2000). Une nouvelle caractrisation du spectre essentiel et application, C. R. Acad. Sci. Paris, t. 331, Srie I, pp. 525–530.

86. Jeribi, A., (2002). A characterization of the Schechter essential spectrum on Banach spaces and applications, J. Math. Anal. Appl., 271(2), 343–358. 336 Bibliography

87. Jeribi, A., (2002). Some remarks on the Schechter essential spectrum and applications to transport equations, J. Math. Anal. Appl., 275(1), 222–237.

88. Jeribi, A., (2002). A characterization of the essential spectrum and applications, Boll. dell. Unio. Mate. Ital., 8 B-5, 805–825.

89. Jeribi, A., (2002). On the Schechter essential spectrum on Banach spaces and applications, Facta Univ. Ser. Math. Inform., 17, 35–55.

90. Jeribi, A., (2003). Time asymptotic behavior for unbounded linear operator arising in growing cell populations, Nonl. Anal. Real Worl. Appli., 4, 667–688.

91. Jeribi, A., (2004). Fredholm operators and essential spectra, Arch. Inequal. Appl., 2(2–3), 123–140.

92. Jeribi, A., Krichen, B., & Zarai Dhahri, M. (2016). Essential spectra of some of matrix operators involving γ-relatively bounded entries and an applications, Linear and Multilinear Algebra, 1654–1668.

93. Jeribi, A., & Latrach, K., (1996). Quelques remarques sur le spectre essentiel et application l’quation de transport, C. R. Acad. Sci. Paris, t. 323, Srie I, 469–474.

94. Jeribi, A., Latrach, K., & Megdiche, H., (2005). Time asymptotic behavior of the solution to a Cauchy problem governed by a transport operator, J. Inte. Equa. Appl., Volume 17, Number 2, 121–139.

95. Jeribi, A., Megdiche, H., & Moalla, N., (2005). On a transport operator arising in growing cell populations II. Cauchy problem, Math. Meth. Appl. Scie., 28, 127–145.

96. Jeribi, A., & Mnif, M., (2005). Fredholm operators, essential spectra and application to transport equations, Acta Appl. Math., 89(1–3), 155–176. Bibliography 337

97. Jeribi, A., & Moalla, N., (2006). Fredholm operators and Riesz theory for polynomially compact operators, Acta Appl. Math., 90(3), 227–245.

98. Jeribi, A., & Moalla, N., (2009). A characterization of some subsets of Schechter’s essential spectrum and application to singular transport equation, J. Math. Anal. Appl., 358, 434–444.

99. Jeribi, A., Ould Ahmed Mahmoud, S. A., & Sfaxi, R., (2007). Time asymptotic behavior for a one-velocity transport operator with Maxwell boundary condition, Acta Appl. Math., 3, 163–179.

100. Jeribi, A., Moalla, N., & Walha, I., (2009). Spectra of some block operator matrices and application to transport operators, J. Math. Anal. Appl., 351(1), 315–325.

101. Jeribi, A., Moalla, N., & Yengui, S., (2014). S-essential spectra and application to an example of transport operators, Math. Methods Appl. Sci., 37(16), 2341–2353.

102. Jeribi, A., Moalla, N., & Yengui, S., (2017). Some results on perturbation theory of matrix operators, M-essential spectra of matrix operators and application to an example of transport operators, preprint.

103. Jeribi, A., & Walha, I., (2011). Gustafson, Weidmann, Kato, Wolf, Schechter and Browder Essential Spectra of Some Matrix Operator and Application to Two-Group Transport Equation, Math. Nachr., 284(1), 67–86.

104. Kaashoek, M. A., (1967). Ascent, descent, nullity and defect, a note on a paper by A.E. Taylor, Math. Ann., 172, 105–115.

105. Kaashoek, M. A., & Lay, D. C., (1972). Ascent, descent and commuting perturbations, Trans. Amer. Math. Soc., 169, 35–47. 338 Bibliography

106. Karow, M., (2006). Eigenvalue condition numbers and a formula of Burke, Lewis and Overton, Electron. J. Linear Algebra 15, 143–153.

107. Kato, T., (1966). Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York.

108. Kato, T., (1980). Perturbation Theory for Linear Operators, Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin Heidelberg, New York.

109. Kulkarni, S. H., & Sukumar, D., (2008). The condition spectrum, Acta Sci. Math. (Szeged), 74, 625–641.

110. Kumar, G. K., & Lui, S. H., (2015). Pseudospectrum and condition spectrum, Operators and Matrices, 1, 121–145.

111. Langer, H., Markus, A., Matsaev, V., & Tretter, C., (2003). Self- adjoint block operator matrices with non-separated diagonal entries and their Schur complements, J. Funct. Anal., 199(2), 427–451.

112. Latrach, K., & Dehici, A., (2001). Fredholm, semi-Fredholm perturbations, and essential spectra, J. Math. Anal. Appl., 259(1), 277–301.

113. Latrach, K., & Jeribi, A., (1996). On the essential spectrum of trans-

port operators on L1-spaces, J. Math. Phys., 37(12), 6486–6494.

114. Latrach, K., & Jeribi, A., (1997). Sur une quation de transport inter- venant en dynamique des populations, C. R. Acad. Sci. Paris, t. 325, Serie´ I, pp. 1087–1090.

115. Latrach, K., & Jeribi, A., (1998). Some results on Fredholm operators, essential spectra and application, J. Math. Anal. Appl., 225(2), 461–485.

116. Latrach, K., Megdiche, H., & Jeribi, A., (2005). Time asymptotic behavior of the solution to a Cauchy problem governed by a transport operator, J. Inte. Equa. Appl. 17(2), 121–140. Bibliography 339

117. Lebow, A., & Schechter, M., (1971). Semigroups of operators and measures of noncompactness, J. Func. Anal., 7, 1–26.

118. Leon-Saavedra, F., & Muller,¨ V., (2004). Rotations of hypercyclic operators, Inte. Equa. Oper. Theo., 50(3), 385–391.

119. Markus, A. S., (1988). Introduction to the spectral theory of polynomial operator pencils, Amer. Math. Soc., Providence, RI . iv+250 pp. ISBN: 0-8218-4523-3.

120. Mennicken, R., & Motovilov, A. K., (1999). Operator interpretation of resonances arising in spectral problems for 2 × 2 operator matrices, Math. Nachr., 201, 117–181.

121. M. M. Milovanovic-Arandjelovi´ c,´ (2001). Measures of noncompactness on uniform spaces—the axiomatic approach, IMC Filomat 2001, Nis,ˇ 221–225.

122. Moalla, N., (2012). A characterization of Schechter’s essential spectrum by mean of measure of non-strict-singularity and application to matrix operator, Acta Math. Sci. Ser. B Engl. Ed., 32(6), 2329–2340.

123. Moalla, N., Damak, M., & Jeribi, A., (2006). Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions, J. Math. Anal. Appl., 323(2), 1071–1090.

124. Muller,¨ V., (2003). Spectral Theory of Linear Operator and Spectral Systems in Banach Algebras, Birkhauser¨ Verlag.

125. Muller,¨ V., (2007). Spectral theory of linear operators and spectral system in Banach algebras, Second edition, Operator Theory Advance and Application, vol. 139, Birkhauser¨ Verlag, Basel.

126. Murray, F. J., (1937). On complementary manifolds and projections

in spaces Lp and lp. Trans. Amer. Math. Soc., 41, 138–152. 340 Bibliography

127. Pelczynski, A., (1965). On strictly singular and strictly cosingular operators. II. Strictly singular and strictly cosingular operators in L(µ)−spaces, Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys., 13, 37–41.

128. Pelczynski, A., (1965). On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C (Ω)-spaces, Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys., 13, 31–36.

129. Rakocevi˘ c,´ V., (1981). On one subset of M. Schester’s essential spectrum, Math. Vesnik., 5(18), 33(4), 389–391.

130. Rakoc ˘evic ´, V. R., (1983). Measures of non-strict-singularity of operators, Math. Vesnik., 35, 79–82.

131. Rakocevi˘ c,´ V., (1984). On the essential approximate point spectrum, ll, Math. Vesnik., 36, 89–97.

132. Rakocevi˘ c,´ V., (1995). Semi-Fredholm operators with ﬁnite ascent or descent and perturbations, Amer. Math. Soci., 123(12), December.

133. Rakocevi˘ c,´ V., (1998). Measures of noncompactness and some applications, Filomat, 12(2), 87–120.

134. Ransford, T. J., (1984). Generalized spectra and analytic multivalued functions, J. London Math. Soc., 29, 306–322.

135. Reddy, S. C., & Trefethen, L. N., (1990). Lax-stability of fully discrete spectral methods via stability regions and pseudo- eigenvalues, Comput. Meth. Appl. Mech. and Engrg., 80(1), 147– 164.

136. Riesz, F., & Nagy, B. S., (1952). Lec¸ons d’analyse fonctionnelle, Acadmie des Sciences de Hongrie, Akadmiai Kiado,´ Budapest.

137. Schechter, M., (1966). On the essential spectrum of an arbitrary operator, I. J. Math. Anal. Appl., 13, 205–215. Bibliography 341

138. Schechter, M., (1967). Basic theory of Fredholm operators, Anna. Scuola Norm. Sup., Pisa 21(3), 261–280.

139. Schechter, M., (1968). Riesz operators and Fredholm perturbations, Bull. Amer. Math. Soc., 74, 1139–1144.

140. Schechter, M., (1971). Spectra of Partial Differential Operators, North-Holland, Amsterdam.

141. Schechter, M., (1971). Principles of Functional Analysis, New Work: Academic Press.

142. Schechter, M., (1972). Quantities related to strictly singular operators, Ind. Univ. Math. J., 21(11), 1061–1071.

143. Schmoeger, C., (1997). The spectral mapping theorem for the essential approximate point spectrum, Colloq. Math. 74(2) 167–176.

144. Shapiro, J., & Snow, M., (1974). The Fredholm spectrum of the sum and product of two operators, Trans. Amer. Math. Soc., 191, 387– 393.

145. Shargorodsky, E., (2009). On the deﬁnition of pseudospectra, Bull. London Math. Soc., 41, 524–534.

146. Shkalikov, A. A., (1995). On the essential spectrum of some matrix operators, Math. Notes, 58(5–6), 1359–1362.

147. Shkalikov, A. A., & Tretter, C., (1996). Spectral analysis for linear pencils N −λP of ordinary differential operators, Math. Nachr. 179, 275–305.

148. Taylor, A. E., (1951). Spectral theory of closed distributive operators, Acta Math., 84, MR 12, 717, 189–224.

149. Taylor, A. E., (1966). Theorems on ascent, descent, nullity, and defect of linear operators, Math. Annalen, 163, 18–49. 342 Bibliography

150. Trefethen, L. N., (1990). Approximation theory and numerical linear algebra, Algorithms for Approximation II, Mason, J. C., & Cox, M. G., eds., Chapman and Hall, London, 336–360.

151. Trefethen, L. N., (1992). Pseudospectra of matrices, numerical analysis. In: Pitman Research notes in Mathematics Series, Longman Science and Technology, Harlow, 260, 234–266.

152. Trefethen, L. N., (1997). Pseudospectra of linear operators, SAIM rev., 39(3), 383–406.

153. Trefethen, L. N., & Embree, M., (2005). Spectra and Pseudospectra (The behavior of non normal matrices and operators), Princeton University Press.

154. Tretter, C., (2008). Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London.

155. Varah, J. M., (1967). The computation of bounds for the invariant subspaces of a general matrix operator, PhD thesis, Stanford Univer- sity Pro. Quest LLC, Ann. Arbor.

156. Vladimirskii, Ju. I., (1967). Strictly cosingular operators, Soviet. Math. Dokl. 8, 739–740.

157. West, T. T., (1987). A Riesz-Schauder theorem for semi-Fredholm operators, Proc. Roy. Irish Acad. Sect. A, 87(2), 137–146.

158. Weyl, H., (1909). Uber beschrankte quadratische Formen, deren differenz vollstelig ist., Rend. Circ. Mat. Palermo 27, 373–392.

159. Wilkinson, J. H., (1986). Sensitivity of eigenvalues II, Util. Math., 30, 243–286.

160. Wolf, F., (1959). On the invariance of the essential spectrum under a change of boundary conditions of partial differential boundary operators, Indag. Math., 21, 142–147. Bibliography 343

161. Wolff, M. P. H., (2001). Discrete approximation of unbounded operators and approximation of their spectra, J. Approx. Theo., 113, 229– 244.

162. Yood, B., (1951). Properties of linear transformations preserved under addition of a completely continuous transformation, Duke Math. J., 18, 599–612.

163. Zivkoviˇ c-Zlatanovi´ c,´ S. C.,ˇ Djordjevic,´ D. S., & Harte, R. E., (2011). On left and right Browder operators, J. Korean Math. Soc., 48, 1053–1063.

164. Zivkoviˇ c-Zlatanovi´ c,´ S. C.,ˇ Djordjevic,´ D. S., & Harte, R. E., (2011). Left-right Browder and left-right Fredholm operators, Inte. Equa. Oper. Theo., 69, 347–363.

165. Zivkoviˇ c-Zlatanovi´ c,´ S. C.,ˇ Djordjevic,´ D. S., Harte, R. E., & Duggal, B. P., (2014). On polynomially Riesz operators, Filomat, 28, 197–205.

Index

(H1), 211–213, 216 σe2,S (A), 6, 104, 105, 187, 190 (H2), 211 σe2,S (A, B, C, ε), 128, 232 (H3), 211, 212 σe2,ε (A), 10, 122, 123, 312 (H4), 211, 213 σe3(A), 3, 86, 87, 302, 304 (H5), 212–214 σe3,S (A), 6, 104, 191 (H6), 283, 284, 292 σe3,S (A, B,C, ε), 128 (H7), 311–313 σe3,ε (A), 10, 122, 123, 298 (H8), 311, 312, 314 σe4(A), 3, 86, 87, 89, 90, 256, 302, 303 (H9), 311, 312 σe4(A, B, C, ε), 13, 126, 256 α(A), 19, 27, 34, 35, 43, 55, 58, 62, 63, 88 σe4,S (A), 6, 104–108, 236, 238 β(A), 19, 27, 34, 35, 37, 58, 62, 63 σe4,S (A, B, C, ε), 128, 233–236 γ(A), 44, 69, 70, 72 σe4,ε (A), 10, 122, 123, 310, 311 γ(T), 44, 70, 71, 78, 133, 134 σe5(A), 3, 86–90, 143, 144, 246 γ-diagonally dominant, 75, 76, 137 σe5(A, B, C, ε), 13, 126, 244, 246, 247, γ-relative boundedness, 15, 129 250–252, 262–267, 269, 271–274

γ-relatively bounded, 73 σe5,S (A), 6, 104–108, 110, 112, 183, 184, γ1(A), 32, 44 186, 187, 191–194, 200, 207–210 δ(M, N), 76, 77 σe5,S (A, B, C, ε), 128, 235–237 δ(T, S), 77, 78 σe5,ε (A), 10, 119, 121–123, 298, 311, 312 ε-pseudospectrum, 16 σe6(A), 3, 10, 86, 87, 90, 91, 254, 255 σ(A), 3, 8, 9, 14, 25, 26, 35, 56, 58, 87, σe6(A, B, C, ε), 13, 126, 252–255, 257, 113–115, 123–125, 227, 279, 315, 259, 260, 276, 278

317–319, 322–325 σe6,S (A), 100, 104, 108, 191, 192, 204 σ(A, B, C, ε), 12, 124, 125, 262 σe7(A), 3, 86, 87, 89, 156, 158–161, 293 σa p(A), 3, 11, 86, 88, 279–281, 283, 288 σe8(A), 3, 86, 87, 89, 156, 158, 159 σa p,S (A), 99 σea p,S (A), 6, 104, 183, 187, 188, σa p,S (A, B, C, ε), 127, 128, 230, 231 190–193 Σap,ε(A), 11, 122, 280, 283, 284, 289, 292 σea p,ε (A), 10, 11, 122, 123, 293–297, σa p,ε (A), 11, 122, 280–286, 289, 292 299, 301 σbl,S (A), 6, 104, 105, 202–204 Σea p,ε (A), 11, 122 σbr,S (A), 6, 104, 105, 202, 204 σel (A), 86 σc(A), 26, 87, 144 σel,S (A), 6, 104, 105, 197–199 σc,S (A), 99 σer (A), 86 σc,S (A, B, C, ε), 127 σer,S (A), 6, 104, 105, 198, 199 σdS (A), 100 σewl (A), 3, 4, 86, 91, 143, 144, 146,

σe1(A), 3, 86, 87, 89, 302–304 148–150, 152–154 σe1,S (A), 6, 104, 105, 187, 188, 190, 191 σewl,S (A), 6, 104, 105, 198–201 σe1,S (A, B, C, ε), 128, 232, 233 σewr (A), 3, 4, 86, 91, 143, 144, 146, 148, σe1,ε (A), 10, 122, 123, 312–314 149, 151–153 σe2(A), 3, 86, 87, 302, 304 σewr,S (A), 6, 104, 105, 198–200 346 Index

σeδ,S (A), 6, 104, 183, 187, 190, 193 Φ−A, 35, 104, 145, 156, 195 σeδ,ε (A), 10, 122, 123, 299, 302 Φ−A,S, 35, 104, 195

σj(A), 4, 86, 87, 91, 155, 262 ∆ψ(A), 33 σj(A, B, C, ε), 126, 262–264 ⊥ R, 23 σl (A), 3, 86 σl(A), 4, 91, 150, 152, 153, 155 A A σl,S ( ), 98 A-g-bounded, 73 A σp( ), 26 A-bounded, 27–29, 54, 111 A σp,S ( ), 99 A-closed, 20 A, B, C, ε σp,S ( ), 127, 230, 231 A-compact, 27, 28, 54, 55, 111 A σR( ), 26 operators, 27 A σr ( ), 3, 86 perturbations, 55 A σr( ), 4, 91, 151, 152 A-convergent, 20 A σr,S ( ), 98 A-defined, 27, 31, 32, 54 A σR,S ( ), 99 A-Fredholm perturbation, 54 A, B,C, ε σR,S ( ), 127 AF (X,Y), 54 σS(A), 98–100, 116–118, 218–224, F A +(X,Y), 54 226–229 AF (X,Y), 54 A, B,C, ε − σS( ), 126, 127, 223–230 AK (X,Y), 28 A σS, ε( ), 116–118, 217–222 Algebaric A σδ( ), 3, 86 complement, 24 σδ,S (A), 183 ε sum, 135, 136 Σ (A), 14, 123, 291, 315–320, 322–326 Ammar-Jeribi essential pseudospectrum, A σε ( ), 8, 9, 113–116, 123, 124, 280, 281 119, 269 A Σε( ), 9, 14, 113, 114, 123, 280, 283, Analytic function, 22, 79, 82, 241 291, 315–320, 322–326 Annihilator, 23, 109, 184 A σε,ap( ), 279, 280 Approximate point spectrum, 3, 5, 15, Φ(X,Y), 34–36, 43, 45, 46, 48–50 16, 87, 89, 159, 177, 178 X,Y Φ+( ), 34, 35, 49, 50, 53 A-pseudo-compact, 29 Φ+A, 35, 104, 145, 156, 187, 188, 191, Arbitrary 195 neighborhood, 171, 180, 182 Φ+A,S, 35, 104, 187, 188, 191, 195 0 operator, 82, 83 Φ (A), 81–83 positive measure space, 150, 151 ΦA, 35, 46, 47, 55, 79–83, 86, 88, 89, subspace of L (X), 271 107, 110, 121, 145, 159, 185, 187, AS (X,Y), 31 188, 190–192, 195, 213, 216, 241, 299 S C b A (X,Y), 32 Φ (X,Y), 35, 37–39, 51, 52 asc(A), 57–59, 62–64, 275 A Φi( ), 80, 83 Ascent, 57, 62, 63, 65, 66 X,Y Φl ( ), 35, 36, 48 descent operators, 57 ΦlA, 35, 86, 148, 149, 153, 155 A-strictly X,Y Φr ( ), 35, 36, 48 cosingular, 32, 54 ΦrA, 35, 86, 144, 149, 155 singular, 31, 54 X,Y Φ±( ), 34, 35, 50, 134 Asymptotic behavior, 1, 8, 14 Φε (X), 10, 122, 301, 303, 304 ε Atkinson theorem, 250 Φ A, 10, 37, 122, 301, 303, 304, 310, A-weakly compact, 29, 54 311, 313 operators, 29 Φ−(X,Y), 34, 35, 49, 50, 53 Index 347

A-γ-bound, 73, 76, 140 C0-semigroup, 33, 34 Aλ,S, 100–102 Canonical injection, 32 Cauchy B domain, 81–83, 240 B (X), 6, 64–66, 91, 104, 259, 276, 277 sequence, 131, 230 B Characterization of essential spectra, 88 −(X), 63, 64 B Closable, 20, 21, 25, 27, 75, 78, (λ0, ε), 93 B 129–131, 137, 211, 212 +(X), 63, 64, 274–276 Backward unilateral shifts, 199 linear operator, 25, 78 Banach Closed, 1, 2, 4, 5, 7–10, 12, 13, 16, 20, algebras, 14, 15 21, 23–27, 29–32, 34–37, 40–42, space, 2, 4, 5, 7–10, 12–14, 16, 21, 44–46, 48, 49, 51, 53, 55, 56, 58, 61, 24, 25, 27–29, 31, 33, 34, 39, 43, 62, 70, 73, 75, 77–81, 84, 85, 87, 44, 52, 53, 57, 63, 64, 68, 70–72, 90–92, 94, 95, 99, 105, 111, 113, 119, 75, 79, 85, 88, 90–92, 105, 119, 122, 124, 125, 129, 133, 135–137, 140, 122, 131, 133, 137, 143, 147, 153, 143, 144, 150–153, 159, 168, 169, 177, 156, 159, 160, 162, 177, 183, 195, 182, 183, 205, 211, 217, 230, 231, 239, 200, 201, 210, 231, 239, 249, 279, 249, 269, 275, 279, 280, 283, 294, 300 298–300, 302, 322–324 closable operators, 20 graph theorem, 21, 26, 95, 211 setting, 4 linear operator, 5, 9, 10, 16, 20, 21, theorem, 231, 284 44, 46, 55, 56, 77, 90, 94, 95, 113, B l(X), 6, 64, 104, 202, 203 122, 168, 177 Block operator matrices, 7, 16, 129, 137, subspace, 23–25, 30, 31, 37, 44, 80, 275 139, 210, 305, 306, 310 Closed subspace of X, 23, 31, 44 Bounded Codimension, 19, 31, 42 finite rank operators, 80 Commutativity assumption, 203 Fredholm operators, 37 Commutes, 28, 40, 60, 62, 64, 73, 95, linear operator, 5, 6, 14, 20, 21, 26, 96, 252–255, 257, 259, 274, 276 46, 56, 57, 62, 89, 94, 95, 99, 166, Compact, 2–4, 13, 16, 26–31, 40, 49, 177, 195, 201, 317, 318 50, 53–56, 62, 68–70, 74, 83, 89, operator, 5, 7, 12, 20–22, 27, 39, 44, 108–111, 119–121, 136, 147, 178, 45, 56, 57, 72, 78–80, 94, 99, 114, 185, 196, 201, 240, 242, 245, 247, 116, 125, 170, 172, 177, 178, 188, 248, 251, 254, 294, 297, 308 190, 210, 214, 219, 230, 231, 241, linear operator, 62 284, 285, 288, 297, 302, 304, 324 operator, 2, 4, 29, 30, 49, 55, 108– subset, 68 110, 119–121, 185, 196, 201, 245, B r(X), 6, 64, 104 251, 294, 308 Browder Complemented, 24, 25, 34, 35, 144, 145, essential spectrum, 3, 10, 91 195 operators, 6, 63, 64 Complex B , 70 X analysis, 2 C number, 35, 37, 214, 300 valued, 79 C (X,Y), 21, 27, 31, 32, 34, 35, 43–46, Condensing, 70 48, 50, 54, 77, 78, 92–94 Condition pseudospectrum, 123, 315, c0, 25, 30 322 348 Index

Constant infimum norm, 289 spectrum, 2–5, 9, 10, 12, 16, 85, 87, Continuous spectrum, 2, 85, 87 88, 91, 108, 125, 146, 149, 151, conv(A), 68 205, 210 Converge in the generalized sense to T, 78 F Convergence, 1, 5, 8, 14–16, 77, 78, 83, F (A), 82 162, 181 F (X,Y), 49–51 compactly, 56, 162 F ∗(A), 82, 83 Convergent subsequence, 28, 29, 43, 110 F b(X,Y), 49–51 F Convex −(X,Y), 49, 50 F hull, 68, 324 +(X,Y), 49, 50 F set, 325, 326 0(X), 183, 184, 186 Core of A, 88 Finite algebraic multiplicity, 2, 85 D dimensional range operator R, 253 D (A), 3, 11, 19–21, 23, 27–29, 32–34, rank, 27, 56, 80, 100–103, 109, 120, 44, 45, 48, 53, 55, 62, 73, 75, 79–81, 183, 185, 196, 241, 245 86, 88, 92, 94, 95, 99, 110, 115, 122, operator, 101, 103, 109, 120, 185, 127, 137, 139, 147, 160, 210, 211, 241, 245 218, 220–222, 225, 229, 255, 256, Finite-dimensional subspace of X, 120 272, 279–285, 287, 289, 292, 295, 300 Finitely meromorphic, 100 n D (A ), 28, 81, 239 Forward unilateral shift, 203 desc(A), 57–59, 62–64, 276 Fredholm, 1, 3, 6, 7, 10, 13, 15, 16, Differential/integral equations, 2 34–39, 43–49, 51, 53–56, 61, 63, Direct sum, 24 65–67, 71, 78, 81, 86, 88, 108, 133, dist(x, N(A)), 44 134, 144, 148, 152, 156–158, 195, Dunford-Pettis property, 4, 29, 152, 153, 196, 200, 215, 216, 241, 242, 251, 250, 262 263, 291, 306–310, 312 Dynamical systems, 1, 8, 14 operator, 1, 3, 5, 6, 10, 34–39, 43, 45, 46, 47, 53, 55, 61, 63, 65–67, 71, E 86, 88, 89, 108, 133, 134, 148, 200, Eigenvalues, 2, 8, 10, 85 215, 216, 263, 291, 306, 308, 309, Endomorphisms, 24 310, 312 Essential perturbation, 7, 13, 16, 48, 49, 51, 54, approximation pseudospectrum, 10, 81, 196, 241, 242, 307 16, 279, 293 semi-Fredholm perturbations, 48 defect pseudospectrum, 279 theory, 279 effect spectrum, 4, 5, 16, 89, 156, Function theory, 2 159, 167, 177 Functional analysis, 67 numerical range, 9 pseudospectra, 1, 9, 10, 12–14, 16, G 122, 125, 128, 217, 239, 248, 260, G(A), 20, 21 266, 269, 298 Gap spectra, 1–10, 12–16, 85, 91, 104, between S and T, 77 105, 122, 143, 146, 153, 156, 162, between T and S, 78 183, 200, 205, 217 metric, 77 Index 349

topology, 15, 76, 77 Jeribi essential spectrum, 4, 86, 151 Generalization of convergence, 5, 78 Jordan curves, 81 Generalized range, 63 G l (X), 35, 98, 201 K G X r ( ), 35, 98 K (X,Y), 26, 38, 43, 49 Graph, 20, 21, 26, 27, 95, 147, 160, 211, k-ball-contraction, 70 249, 300 Ker(µ), 68, 69 Gustafson/Weidman’s essential Iernel, 40, 41, 63, 68, 100, 291 spectra, 86 k-set-contraction, 70 Kuratowski’s measure of H noncompactness, 69, 135, 269, 270 Hahn-Banach theorem, 21, 218, 225, 229, 286 L Hausdorff L (X,Y), 12, 13, 21, 22, 25, 26, 29–31, metric, 84 33, 35, 38, 39, 43, 48–50, 52, 71, 72, space, 30, 31, 50 78, 79, 92–94, 124–128, 133, 223, Heine-Borel theorem, 47, 271 225, 226, 228–231, 233, 235, 236, Hilbert space, 2, 30, 85, 87, 324 239, 243, 244, 246–248, 250, 252, Holomorphic, 93, 97 254, 256, 257, 259, 260, 262–264, Homeomorphism, 30 266, 267, 269, 271–273, 276 l (N), 289–291 I 1 l∞, 25 I (X), 32 Left and right I-essential spectra, 210 Jeribi essential spectra, 15 i(A), 3, 35, 36, 38, 39, 43–46, 48, 50, 54, Weyl 55, 57, 58, 73–75, 88, 145, 147, 148, essential spectra, 153 154, 159, 161, 195, 212, 299, 301 operator sets, 4 Idempotent operator, 24 spectrum, 152 iM , 32, 74 Left Immediate consequence, 51, 192, 194, Fredholm inverse of A, 53 201, 204, 235, 266, 324 invertible, 3, 34, 35, 86, 144, 201 I n(X), 88, 89, 212, 266, 267 Weyl operators, 6, 36 Index of A, 3, 35 Lim Infinite dimensional inf En, 84 Banach spaces, 14 sup En, 84 subspace, 30 Limit inferior/superior, 83, 84 Infinitesimal generator, 34 Linear operators, 1–5, 7–9, 12, 14–16, Integro-differential operator, 7 19, 20, 21, 26, 28, 29, 56, 57, 73, 77, Intertwining operator B, 75, 137, 210 78, 85, 90, 91, 102, 105, 119, 123, Invariance, 108, 159 129, 139, 143, 162, 168, 169, 177, Invertible operator, 39, 127, 195–197, 182, 205, 269, 322 316, 317 Liouville theorem, 22, 117, 226, 227 Isomorphic, 31, 50, 145, 195, 250, 263 Lower semi-Browder operator, 63 J semi-Fredholm, 3, 5, 34, 38, 49, J (X), 50 53, 89 350 Index

M 286, 289, 294, 297, 300, 305–308, 310, 323 Matrix matrix T, 305 operator, 16, 205, 210, 213 valued function T, 33 theory, 2 Opposite inclusion, 131, 148, 161, 192, Maximal gap, 76 199, 202, 249, 250, 298, 299, 301, Measure of noncompactness, 5, 13, 15, 320 16, 67–70, 73, 135, 156, 269 Measure of non-strict-singularity, 33, 71, 274 P M P F X,68, 69 (X), 65, 67, 159–161 P R (X), 61, 146, 147 N Perturbation function, 32, 33, 64 N(A), 19, 21, 27, 32, 34, 35, 37, 40, 42, operators, 7, 196 44, 45 ⊥ result, 73 N , 23 structure, 12, 125 N∞(A), 63, 275 Projections, 24, 138, 141 Neumann series, 174, 176 Pseudo-Browder essential spectrum, 9, 10 Newton’s binomial formula, 40, 60 Pseudospectra, 8, 217, 239, 262, 269, Nonnegative integer n, 62 279 Non-strict A, 9, 113 inequalities, 289 Pseudospectral radius, 123, 315 singularity, 7, 8, 13, 16, 33, 44, 72, Pseudospectrum, 1, 8–17, 113, 116, 119, 88, 205, 207, 209, 212, 215, 257, 123–126, 226, 254, 255, 262, 279, 264, 269, 273 315, 317, 318, 320, 322, 323, 325 Nonzero complex polynomial, 51, 59, P , 100–103 61, 65, 67 λ,S P , 100 Norm operators, 129 λ0,S Normed spaces, 21, 32, 33, 77 Q vector space X, 23, 24 q(A), 32, 33 Null Quasi-inverse, 15, 79, 80, 243, 256, 272 operator, 226, 227 closed operator A, 79 space, 19 λ − A, 80, 256 Nullity, 19, 234 Quasi-inverse operator, 15, 79, 243

NX, 68, 69 R O rε (A), 123, 315–317 Operator, 4, 5, 7–12, 14, 16, 19–23, rσ(A), 123, 315, 316 25, 27, 29–31, 33, 34, 36–38, 45, 46, R ′(A), 80–83 51–55, 62, 65, 67, 69, 70, 73, 75, 76, R(A), 19, 21, 27, 34, 35, 37, 40, 42, 44, 78–80, 89, 95, 99, 100, 103, 106, 45, 55, 62 108, 109, 114, 116, 120, 125, 129, R (X), 7, 52, 53, 59–61, 64, 146, 197, 135–137, 139, 145, 147, 149, 155, 198, 200, 203, 204, 243, 252, 255 ′ 157–160, 180–182, 184, 190, 194, R∞ (A), 79, 82 195, 198, 210–213, 215, 225, 241, R∞(A), 63, 275

243, 245, 251, 253, 272, 279, 284, Rb,S (A, λ), 101, 102 Index 351

RS (λ, A), 92–98, 116–118, 218, 219, value sets, 12, 124 221–223, 227 Spectrum, 1–3, 9–11, 14, 15, 25, 26, 85, Reduced minimum modulus, 44 87, 90, 91, 100, 114, 146, 195, 279, Regular, 69 323, 325 Relative bound, 28 S-point spectrum, 99 Relatively S-pseudospectra, 13, 16, 117, 127, 217 bounded with respect to A, 28, 73 S-pseudospectrum, 115 compact with respect to A, 28 S-residual spectrum of A, S, 99 weakly compact with respect S-resolvent set, 15, 92, 93 to A, 29 S-Riesz integral for A, 100 Resolvent S-Riesz projector, 100 operator, 92, 253 S-right spectrum of A, 98 set, 3, 25, 211 S-Schechter essential spectrum, 207 Resolvent set and spectrum, 25 S-spectra, 15, 92, 98 Riesz S-spectrum, 98, 99 operator, 4, 7, 52, 53, 60, 62, 90, 146, Stability of essential spectra, 156 196, 203, 252 Standard abbreviation, 33 Schauder operators, 6, 64 Strictly Right Browder operators, 6 cosingular, 31, 32, 49, 54 Right Fredholm operators, 31, 49 inverse of A, 53 positive parameter, 226 inverse of λ − A, 158 singular, 4, 30, 31, 33, 49, 54, 206, operators, 6 210, 213, 215 invertible, 3, 34, 35, 86 operators, 4, 30, 49, 213 quasi inverse, 272 Structured Weyl operators, 6, 36 Ammar-Jeribi essential pseudospectrum, 13, 16, 126, 244, S 250, 264, 273 Browder essential pseudospectrum, S (X,Y), 30, 72 13, 16 S-Browder’s resolvent, 100, 101 essential pseudospectra, 1, 125, 260 S-structured essential pseudospectra, 1 Jeribi essential pseudospectrum, 126 S-approximate point spectrum, 99 pseudospectra, 12, 15, 124, 125 S-continuous spectrum of A, 99 S-discrete spectrum of A, 100 S-approximate point Self-adjoint operator, 2, 85 pseudospectrum, 127 Semi S-continuous pseudospectrum, 127 Browder operators, 63 S-essential pseudospectra, 128 Fredholm operator, 38, 56 S-point pseudospectrum, 127 S-essential spectra, 1, 7, 15, 92, 104, S-pseudospectra, 16, 126, 217, 226 108, 183, 188, 205 S-residual pseudospectrum, 127 S-left and S-right Browder spectra, 201, Wolf essential pseudospectrum, 203 126, 255 S-left and S-right spectrum, 7 Sublinear, 68, 69 S-left spectrum, 98 Surjection modulus, 32 S-left spectrum of A, 98 Surjective, 3, 31, 86, 144, 183 Spectral S , 325 radius, 123 X 352 Index

T W c W Tn→T, 56 (A), 325, 326 c W Tn→0, 56 (X,Y), 29 g W Tn→T, 78, 79, 178 *(X), 4, 86, 91, 126, 262–264 Topological Weakly compact, 29, 49, 54 complement, 25 operators, 29, 49 problems, 67 Weyl operators, 36 Whole complex plane, 9, 98 W U l (X), 6, 36, 53, 104, 198, 199, 201 nbounded Wolf’s essential spectrum, 3, 87 W X Fredholm operators, 43 r ( ), 6, 36, 53, 104 operators, 62, 178 pper X A-Fredholm perturbation, 54 XA, 27, 31, 32, 54, 55, 147, 148, 160, A-semi-Fredholm perturbation, 54 161, 188, 189, 194, 249, 300, 301 Fredholm perturbation, 48 pseudo semi-Fredholm operator, 37 semi-Browder operators, 63 Y semi-Fredholm, 3, 5, 34, 38, 45, 49, 89 Y*, 22, 23