Mathematics

Spectral Theory of Multivalued Linear Operators
Aymen Ammar, PhD
Aref Jeribi, PhD

Spectral Theory of Multivalued Linear Operators

Published. Available now.
Pub Date: September 2021
Hardback Price: see ordering info
Hard ISBN: 9781771889667
E-Book ISBN: 9781003131120
Pages: 314 w/index
Binding Type: Hardback

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.

CONTENTS:
Preface

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 ? and ? 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 ? Pseudospectra of Linear Relations
Localization of Pseudospectra of Linear Relations
Characterization of ? Pseudospectra of Linear Relations
Essential Pseudospectra of Linear Relations
The Essential ?-Pseudospectra of Linear Relations
S-Pseudospectra of Linear Relations

Index


About the Authors / Editors:
Aymen Ammar, PhD
Associate Professor, Department of Mathematics, University of Sfax, Tunisia

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
Professor, Department of Mathematics, University of Sfax, Tunisia

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 (SpringerVerlag, New-York, 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 (Taylor and Francis, 2015), the author of the book Denseness, Bases and Frames in Banach Spaces and Applications (De Gruyter, Berlin, 2018), the author of the book Linear Operators and Their Essential Pseudospectra (Apple Academic Press, 2018), co-author of the book Analyse numérique matricielle méthodes et algorithmes, exercices et problèmes corrigés (Paris: Ellipses, 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.




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