The concept of multivalued linear operators―or linear relations―is 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 focuses on the theory of linear relations, 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 expert readers will also find sources of inspiration.
Author(s): Aymen Ammar, Aref Jeribi
Publisher: Apple Academic Press
Year: 2021
Language: English
Pages: 314
City: Palm Bay
Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
About the Authors
Preface
Symbol Description
1 Introduction
2 Fundamentals
2.1 Banach Space
2.1.1 Direct Sum
2.1.2 Distance Function
2.1.3 Normed Vector Space
2.1.4 Banach Space
2.2 Relations on Sets
2.3 Linear Relations
2.3.1 The Algebra of Linear Relations
2.3.2 Inverse of Linear Relation
2.3.3 Sum and Product of Linear Relations
2.3.4 Restrictions and Extensions of Linear Relations
2.4 Index and Co-Index of Linear Relations
2.4.1 Properties of the Quotient Map QT
2.5 Generalized Kernel and Range of Linear Relations
2.5.1 Ascent and Descent and Singular Chain of Linear Relations
2.5.2 Norm of a Linear Relation
2.5.3 Continuity and Openness of a Linear Relation
2.5.4 Selections of Linear Relation
2.6 Relatively Boundedness of Linear Relations
2.7 Closed and Closable Linear Relations
2.7.1 Closed Linear Relations
2.7.2 Closable Linear Relations
2.8 Adjoint of Linear Relations
2.9 Minimum Modulus of Linear Relations
2.10 Quantities for Linear Relations
2.10.1 A Formula for Gap Between Multivalued Linear Operators
2.10.2 Measures of Noncompactness
2.11 Precompact and Compact Linear Relations
2.12 Strictly Singular Linear Relations
2.13 Polynomial Multivalued Linear Operators
2.14 Some Classes of Multivalued Linear Operators
2.14.1 Multivalued Fredholm and Semi-Fredholm Linear Operators
2.14.2 Multivalued Fredholm and Semi-Fredholm of Closed Linear Operators in Banach Space
2.14.3 Atkinson Linear Relations
2.15 Quasi-Fredholm and Semi Regular Linear Relations
2.15.1 Quasi-Fredholm Linear Relations
2.15.2 Semi Regular Linear Relations
2.15.3 Essentially Semi Regular Linear Relations
2.16 Perturbation Results for Multivalued Linear Operators
2.16.1 Small Perturbation Theorems of Multivalued Linear Operators
2.17 Fredholm Perturbation Classes of Linear Relations
2.18 Spectrum and Pseudospectra of Linear Relations
2.18.1 Resolvent Set and Spectrum of Linear Relations
2.18.2 Subdivision of the Spectrum of Linear Relations
2.18.3 Essential Spectra of Linear Relations
2.19 S-Spectra of Linear Relations in Normed Space
2.19.1 Some Properties of S-Resolvent of Linear Relations in Normed Space
2.19.2 The Augmented S-Spectrum
2.19.3 S-Essential Resolvent Sets of Multivalued Linear Operators
2.20 Pseudospectra and Essential Pseudospectra of Linear Relations
2.20.1 Pseudospectra of Linear Relations
2.20.2 Essential Pseudospectra of Linear Relations
2.20.3 The Essential "-Pseudospectra of Linear Relations
2.20.4 S-Pseudospectra and S-Essential Pseudospectra of Linear Relations
3 The Stability Theorems of Multivalued Linear Operators
3.1 Stability of Closeness of Multivalued Linear Operators
3.1.1 Sum of Two Closed Linear Relations
3.1.2 Sum of Three Closed Linear Relations
3.1.3 Sum of Two Closable Linear Relations
3.1.4 Product of Closable Linear Relations
3.2 Sequence of Multivalued Linear Operators Converging in the Generalized Sense
3.2.1 The Gap Between Two Linear Relations
3.2.2 Relationship Between the Gap of Linear Relation and their Selection
3.2.3 Generalized Convergence of Closed Linear Relations
3.3 Perturbation Classes of Multivalued Linear Operators
3.3.1 Fredholm and Semi-Fredholm Perturbation Classes of Multivalued Linear Operators
3.4 Atkinson Linear Relations
3.5 The α-and β-Atkinson Perturbation Classes
3.6 Index of a Linear Relations
3.6.1 Index of Upper Semi-Fredholm Relation Under Strictly Singular Perturbation
3.6.2 Index of an Lower Semi-Fredholm Relation Under Strictly Singular Perturbations
3.7 Demicompact Linear Relations
3.7.1 Auxiliary Results on Demicompact Linear Relations
3.7.2 Demicompactness and Fredholm Linear Relations
3.8 Relatively Demicompact Linear Relations
3.8.1 Auxiliary Results on Relatively Demicompact Linear Relations
3.8.2 Fredholm Theory by Means Relatively Demicompact Linear Relations
3.9 Essentially Semi Regular Linear Relations
3.9.1 Some Proprieties of Essentially Semi Regular Linear Relations
3.9.2 Perturbation Results of Essentially Semi Regular Linear Relations
4 Essential Spectra and Essential Pseudospectra of a Linear Relation
4.1 Characterization of the Essential Spectrum of a Linear Relations
4.1.1 σ- and β-Essential Spectra of Linear Relations
4.1.2 Invariance of Essential Spectra of Linear Relations
4.2 The Essential Spectrum of a Sequence of Linear Relations
4.3 Spectral Mapping Theorem of Essential Spectra
4.3.1 Essential Spectra of the Sum of Two Linear Relations
4.4 S-Essential Spectra of Linear Relations
4.4.1 Characterization of S-Essential Spectra of Linear Relations
4.4.2 Characterization of σe5,S(·)
4.4.3 Relationship Between σe4,S(·) and σe5,S(·)
4.5 Racočević and Schmoeger S-Essential Spectra of a Linear Relations
4.5.1 Stability of Racočević and Schmoeger S-Essential Spectra of a Linear Relations
4.6 S-Essential Spectra of the Sum of Two Linear Relations
4.7 Pseudospectra and ε-Pseudospectra of Linear Relations
4.7.1 Some Properties of Pseudospectra and ε-Pseudospectra of Linear Relations
4.7.2 Characterization of Pseudospectra of Linear Relations
4.7.3 Stability of Pseudospectra of Linear Relations
4.8 Localization of Pseudospectra of Linear Relations
4.9 Characterization of ε-Pseudospectra of Linear Relations
4.9.1 The Pseudospectra and the ε-Pseudospectra of a Sequence of Closed Linear Relations
4.10 Essential Pseudospectra of Linear Relations
4.10.1 Stability of Essential Pseudospectra of Linear Relations
4.11 The Essential ε-Pseudospectra of Linear Relations
4.11.1 The Essential ε-Pseudospectra of a Sequence of Linear Relations
4.12 S-Pseudospectra of Linear Relations
4.12.1 S-Essential Pseudospectra of Linear Relations
Bibliography
Index
