Spectral Theory and Applications of Linear Operators and Block Operator Matrices

Transcription

1 Spectral Theory and Applications of Linear Operators and Block Operator Matrices

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3 Aref Jeribi Spectral Theory and Applications of Linear Operators and Block Operator Matrices 123

4 Aref Jeribi Department of Mathematics University of Sfax Sfax, Tunisia ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer Cham Heidelberg New York Dordrecht London Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)

5 To my mother Sania, my father Ali, my wife Fadoua, my children Adam and Rahma, my brothers Sofien and Mohamed Amin, my sister Elhem, my mother-in-law Zineb, my father-in-law Ridha, and all members of my extended family

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7 Preface Several books have been devoted to the spectral theory and its applications. However, this volume is very special as compared with the previous ones. For example, the perturbation approach has been, for a long time, extensively studied and considered as one of the most useful methods used in order to study some mathematical and applied problems. The main idea is that, if we know something about the solution for an easier problem, lying close to the one we are studying, then we can say something about our problem, provided that the difference or the perturbation is sufficiently weak. In view of more advanced applications, especially the ones dealing with complicated evolutional problems in Physics, Chemistry, Technology, Biology, etc., where the natural setting doesn t involve single operators but operator matrices and polynomial operator pencils, the concept of compact perturbations is very often used, and it was shown that they were not sufficient for handling such problems. The main advantage of this book is the detailed description of the ways showing how the compactness condition can be relaxed, in a very general Banach space setting, so that the previously impossible problems become suddenly solvable. The method of extending results is not unique. That is why we have to devote a lot of space in order to describe the different extensions of the classical notions, and to demonstrate how they specifically work in different applications. More precisely, it is well known that the essential spectrum of an operator A consists of those points of the spectrum which cannot be removed from the spectrum by the addition to A of a compact operator. The most powerful result obtained in my thesis is that, in L 1 -spaces, the essential spectrum of an operator A is nothing else but the largest subset of the spectrum of A which remains invariant under weakly compact perturbations of A. This unexpected result has opened many prospects to develop innovative ways leading to a rigorous study of the Fredholm theory and in the whole book, we give an account of the recent research on the spectral theory by presenting a wide panorama of techniques including the weak topology, which vii

8 viii Preface contributes to an extra insight to the classical results and enables us to solve concrete problems from transport theory arising in their natural setting (L 1 -spaces). The main topics include: Riesz theory of polynomially compact operators. Time behavior of solutions for an abstract Cauchy problem on Banach spaces. Fredholm theory and characterization of essential spectra by means of measure of noncompactness, demicompact operator, measure of weak noncompactness, and graph measures. S-essential spectra and essential pseudospectra. Spectral theory of block operator matrices. Spectral graph theory. Applications in mathematical physics and biology. 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 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 probably 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. So, we are indebted to her. We would like to thank Nedra Moalla for the improvements she has made concerning the spectral mapping theorem. We would also like to thank Aymen Ammar for the improvements he has made throughout this book. So, we are very grateful to him. Concerning the chapter related to graph theory, we were fortunate to have the help of Hatem Baloudi, who assisted in the preparation of this chapter. So, we are indebted to him. We would like to thank Professor Sylvain Golénia for his generous permission to integrate, in this book, the results of Hatem Baloudi dealing with the graph theory. Moreover, we would like to mention that the thesis work results, performed under my direction, by my former students and presently colleagues Nedra Moalla, Afif Ben Amar, Faiçal Abdmouleh, Boulbeba Abdelmoumen, Salma Charfi, Ines Walha, Bilel Krichen, Omar Jedidi, Sonia Yengui, Aymen Ammar, Naouel Ben Ali, Rihab Moalla, Hatem Baloudi, Mohammed Zerai Dhahri, and Bilel Boukettaya, the obtained results have helped us in writing this book. Last but not least, we would like to thank Ridha Damak for improving the English of all chapters of this book. Finally, we apologize in case we have forgotten to quote any author who has contributed, directly or indirectly, to this work. Sfax, Tunisia June 2015 Aref Jeribi

9 Contents 1 Introduction Spectral Theory and Cauchy Problem Time Behavior of Solutions to an Abstract Cauchy Problem on Banach Spaces Fredholm Theory and Essential Spectra S-Essential Spectra and Essential Pseudospectra Spectral Theory of Block Operator Matrices Spectral Graph Theory Applications in Mathematical Physics and Biology Outline of Contents Fundamentals Basic Properties Closed and Closable Operators Adjoint Operator Elementary Results Fredholm Operators Spectrum Relatively Boundedness and Relatively Compactness Sum of Closed Operators Strictly Singular and Strictly Cosingular Operators Fredholm and Semi-Fredholm Perturbations Dunford Pettis Property Basic Notions Basics on Bounded Fredholm Operators Gap Topology Semi-Regular and Essentially Semi-Regular Operators Basics on Unbounded Fredholm Operators ix

10 x Contents Quasi-Inverse Operator Basics on Unbounded Browder Operators Positive Operator Positive Operator on L p -Spaces Positive Operator on Banach Lattice Integral Operator Integral Operator on L p -Spaces Integral Operator on L 1 -Spaces Cauchy s Type Integral Semigroup Theory Strongly Continuous Semigroup The Hille-Yosida Theorem The Essential Spectral Radius Borel Mappings Baire Measurable Functions Banach Module Measure of Noncompactness Measure of Noncompactness of a Bounded Subset Measure of Noncompactness of an Operator Measure of Weak Noncompactness Graph Measures Graph Measure of Noncompactness Graph Measure of Weak Noncompactness Seminorm Related to Upper Semi-Fredholm Perturbations Quadratic Forms Schur Test Generalities about graphs Unoriented Graph Tree Bipartite Graph Subgraph Fredholm Operators and Riesz Theory for Polynomially Compact Operators Riesz Theory Some Results on Polynomially Compact Operators Generalized Riesz Operator First and Second Kind Operator Equation Spectral Analysis Localization of Eigenvalues of Polynomially Compact Operators Polynomially Riesz Operators Some Results on Polynomially Fredholm Perturbation

11 Contents xi 4 Time-Asymptotic Description of the Solution for an Abstract Cauchy Problem Abstract Cauchy Problem Compactness Results The Remainder Term of the Dyson Philips Expansion Time Behavior of Solutions for an Abstract Cauchy Problem (4.0.1) on Banach Spaces Time Behavior of Solutions for an Abstract Cauchy Problem (4.0.1) on L p -Spaces (1 <p<1) Fredholm Theory Related to Some Measures Fredholm Operators Fredholm Theory by Means of Noncompactness Measures Fredholm Theory by Means of Non-strict Singularity Measures Fredholm Theory by Means of Demicompact Operator Demicompactness S-Demicompactness Fredholm Theory by Means of Weak Noncompactness Measures Fredholm Theory with Finite Ascent and Descent Stability of Semi-Browder Operators Convergence to Zero Compactly Perturbation Results Definitions and Notations Fredholm and Semi-Fredholm Operators Semi-Fredholm Perturbations Fredholm Inverse Operator Fredholm Perturbations Some Perturbation Results for Matrix Operators Some Fredholm Theory Results for Matrix Operators Essential Spectra of Linear Operators Definitions and Notations The Jeribi Essential Spectrum Relationship Between Jeribi and Schechter Essential Spectra on L 1 -Spaces Relationship Between Jeribi and Schechter Essential Spectra on Banach Space Satisfying the Dunford Pettis Property Other Characterization of the Schechter Essential Spectrum by the Jeribi Essential Spectrum on L p -Spaces

12 xii Contents 7.3 Auxiliary Results Essential Spectra of the Sum of Two Bounded Linear Operators By Means of Fredholm and Semi-Fredholm Perturbations By Means of Fredholm Inverse By Means of Demicompact Operators Unbounded Linear Operators Essential Spectra for the Sum of Closed and Bounded Linear Operators Essential Spectra for the Product of Closed and Bounded Linear Operators Invariance of the Essential Spectra Characterization of the Rako cević and Schmoeger Essential Spectra Invariance of the Kato Spectrum by Commuting Nilpotent Perturbation Invariance of Schechter s Essential Spectrum Characterization of Schechter s Essential Spectrum Invariance by Means of Demicompact Operators Invariance by Means of Noncompactness Measure Invariance of the Schechter s Essential Spectrum in Dunford Pettis Space Invariance Under Perturbation of Polynomially Compact Operators Invariance by Means of Weak Noncompactness Measure Stability of the Essential Spectra By Means of Measure of Weak Noncompactness By Means of the Graph Measure of Weak Noncompactness By Measure of Non-upper Semi-Fredholm Perturbations Generalized Convergence Convergence to Zero Compactly Borel Mappings Spectral Mapping Theorem A Characterization of Polynomially Riesz Strongly Continuous Semigroups Polynomially Fredholm Perturbations Polynomially Riesz Operator A Spectral Characterization of the Uniform Continuity of Strongly Continuous Groups

13 Contents xiii 7.13 Some Results on Strongly Continuous Semigroups of Operators Arbitrary Banach Spaces Hereditarily Indecomposable Banach Spaces Pseudo-Spectra Pseudo-Spectrum of Linear Operator Approximation of "-Pseudo-Spectrum Approximation of the Spectrum Pseudo-Browder s Essential Spectrum Extending the Resolvent to a Browder Resolvent b.:/ Definition of the Pseudo-Browder Essential Spectrum Pseudo-Browder s Essential Spectrum of Linear Operators Characterization of the Pseudo-Browder Essential Spectrum Stability of the Pseudo-Browder s Essential Spectrum Pseudo-Jeribi and Pseudo-Schechter Essential Spectra By Means of a Fredholm and Semi-Fredholm Perturbations By Means of Noncompactness Measure S -Essential Spectra Definitions and Preliminary Results Characterization of S-Essential Spectra The S-Browder s Essential Spectrum S-Essential Spectra of the Sum of Bounded Linear Operators S-Essential Spectra by Means of Demicompact Operators Characterization of the Relative Schechter s and Approximate Essential Spectra Essential Spectra of 2 2 Block Operator Matrices Case Where the Resolvent of the Operator A Is a Fredholm Perturbation Frobenius Schur s Decomposition Closability and Closure of the Block Operator Matrix Essential Spectra of L Sufficient Conditions Case Where the Operator A Is Closed Closability and Closure of the Block Operator Matrix

14 xiv Contents Essential Spectra Particular Cases Case Where the Operator A Is Closable Closability and Closure of the Block Operator Matrix Essential Spectra of A The Operator A as an Infinitesimal Generator of a Holomorphic Semigroup Relative Boundedness for Block Operator Matrices Stability of the Wolf Essential Spectrum of Some Matrix Operators Acting in Friedrichs Module Wolf Essential Spectrum of Block Operator Matrix Acting in Friedrichs Module The M -Essential Spectra of Block Operator Matrices Closability and Closure of the Block Operator Matrix Essential Spectra of L Essential Spectra of 3 3 Block Operator Matrices Case Where the Operator A Is Closed The Operator L 0 and Its Closure Essential Spectra of the Operator L Case Where the Operator A Is Closable Closability and Closure of the Operator L Gustafson, Weidman, Kato, Wolf, Schechter, Browder, Rakočević, and Schmoeger s Essential Spectra of L Block Operator Matrices Using Browder Resolvent The Operator A 0 and Its Closure Rakočević and Schmoeger Essential Spectra of A Perturbations of Unbounded Fredholm Linear Operators The Operator A and Its Closure Index of A Spectral Graph Theory Line graph Operators on Graphs Lower Local Complexity On the Persson s Lemma Spectral Measure Persson s Lemma Essential Self-Adjointness Unbounded Properties A Counter Example

15 Contents xv Nelson Commutator Theorem Application on Schur Test The Adjacency Matrix on Line Graph Oriented Graph Case of Bipartite Graph Applications in Mathematical Physics and Biology Time-Asymptotic Description of the Solution for a Transport Equation Preliminaries and Notations Compactness and Generation Results Auxiliary Lemmas Solution for the Cauchy Problem (13.1.1) Generation Results Time-Asymptotic Behavior Time-Asymptotic Behavior of the Solution for a Cauchy Problem Given by a One-Velocity Transport Operator with Maxwell Boundary Condition Auxiliary Results The Resolvent of the Operator B Generation Result Asymptotic Behavior of the Solution The Time-Asymptotic Behavior of a Transport Operator with a Diffuse Reflection Boundary Condition The Resolvent of the Operator B Compactness and Generation Results Asymptotic Behavior of the Solution Essential Spectra of Transport Operator Arising in Growing Cell Populations The Resolvent of the Operator S K Spectral Properties of S K Generation Results Compactness Results The Irreducibility of the Semigroup e ta k Existence of the Leading Eigenvalues of A K The Strict Monotonicity of the Leading Eigenvalue of A K Essential Spectra of A K Some Applications of the Regularity and Irreducibility on Transport Theory Weak Compactness Results Essential Spectra Existence Results of Eigenvalues Monotonicity of the Spectral Bound Singular Neutron Transport Operator

16 xvi Contents 13.7 Systems of Ordinary Differential Operators Essential Spectra of Two-Group Transport Operators The Expression of the Resolvent of T H Compactness Results Essential Spectra Elliptic Problems with -Dependent Boundary Conditions The Problem Verification of the Assumptions.J1/.J 8/ of Chap. 10, Sect The Closure of the Operator A Spectrum of the Operator A Semigroup Generation Delay Differential Equations A -Rational Sturm Liouville s Problem Two-Group Radiative Transfer Equations in a Channel Functional Setting The Expression of the Resolvent of the Operator T2 H Essential Spectra Three-Group Transport Equation Bibliography Index