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Spectral Theory of Multivalued Linear Operators



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ISBN 9781771889667
September 15, 2021 Forthcoming by Apple Academic Press
345 Pages

 
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Book Description

The concept of multivalued linear operators—or linear relations, the one of the most exciting and influential fields of research in modern mathematics. Applications of this theory can be found in economic theory, noncooperative games, artificial intelligence, medicine, and more. This new book, Spectral Theory of Multivalued Linear Operators, focuses on the theory of multivalued linear operators, responding to the lack of resources exclusively dealing with the spectral theory of multivalued linear operators.

The subject of this book is the study of linear relations over real or complex Banach spaces. The main purposes are the definitions and characterization of different kinds of spectra and extending the notions of spectra that are considered for the usual one single-valued operator bounded or not bounded. The volume introduces the theory of pseudospectra of multivalued linear operators. The main topics include demicompact linear relations, essential spectra of linear relation, pseudospectra, and essential pseudospectra of linear relations.

The volume 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. Beginning graduate students who wish to enter the field of spectral theory of multivalued linear operators will benefit from the material covered, and an expert reader will also find some of the results interesting enough to be sources of inspiration. Prerequisites for the book are the basic courses in classical real and complex analysis and some knowledge of basic functional analysis. In fact, this theory constitutes a harmonious mixture of analysis (pure and applied), topology, and geometry.

Table of Contents

1. Introduction

 

2. Fundamentals

Banach Space

Relations on Sets

Linear Relations

Index and Co-Index of Linear Relation

Generalized Kernel and Range of Linear Relations

Closed and Closable Linear Relation

Adjoint of Linear Relation

Minimum Modulus of a Linear Relation

Quantities for Linear Relation

Precompact and Compact Linear Relations

Strictly Singular Linear Relations

Demicompact and Relatively Demicompact Linear Relation

Polynomial Multivalued Linear Operators

Some Classes of Multivalued Linear Operator

Semi Regular and Essentially Semi Regular Linear Relations

Perturbation Theorems of Linear Relation

Fredholm Perturbation Classes of Linear Relation

Spectrum and Pseudospectra of Linear Relation

S-Spectra of Linear Relations in Normed Space

Pseudospectra and Essential Pseudospectra of Linear Relation

 

3. The Stability Theorems of Multivalued Linear Operators

Relatively Boundedness of Linear Relation

Generalized Convergence of Multivalued Linear Operators

Generalized Convergence of Closed Linear Relations

Fredholm Perturbation Classes

Atkinson Linear Relation

The α and α Atkinson Perturbation Classes

Index of a Linear Relations

Demicompact Linear Relation

Relatively Demicompact Linear Relations

Essentially Semi Regular Linear Relation

Relationship between Quasi-Fredholm and Semi Regular Linear Relations

 

4. Essential Spectra of a Linear Relation

Characterization of the Essential Spectrum of a Linear Relation

The Essential Spectrum of a Sequence of Linear Relations

Spectral Mapping Theorem of Essential Spectra

S-Essential Spectra of Linear Relation

Racoevic and Schmoeger S-Essential Spectra of a Linear Relation

S-Essential Spectra of the Sum of Two Linear Relations

Pseudospectra and ε Pseudospectra of Linear Relations

Localization of Pseudospectra of Linear Relations

Characterization of ε Pseudospectra of Linear Relations

Essential Pseudospectra of Linear Relations

The Essential ε-Pseudospectra of Linear Relations

S-Pseudospectra of Linear Relations

...
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Author(s)

Biography

Aymen Ammar, PhD, is currently with the Department of Mathematics, Faculty of Sciences of Sfax at the University of Sfax, Tunisia, where he is an Assistant Professor. He has published many articles in international journals. His areas of interest include spectral theory, matrice operators, and linear relations.

Aref Jeribi, PhD, is a 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), 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 (2015), the author of the book Denseness, Bases and Frames in Banach Spaces and Applications (2018), the author of the book Linear Operators and Their Essential Pseudospectra (2018), co-author of the book Analyse numérique matricielle méthodes et algorithmes, exercices et problèmes corrigés (2020). He has published many journal articles in international journals. His areas of interest include spectral theory, matrice operators, transport theory, Gribov operator, Bargmann space, fixed point theory, Riesz basis, and linear relations.