Perturbation theory for linear operators : denseness and bases with applications / Aref Jeribi.

Format:

Book

Author/Creator:

Jeribi, Aref, author.

Language:

English

Subjects (All):

Spectral theory (Mathematics).

Linear operators.

Physical Description:

1 online resource (523 pages)

Place of Publication:

Singapore : Springer, [2021]

Summary:

This book discusses the important aspects of spectral theory, in particular, the completeness of generalised eigenvectors, Riesz bases, semigroup theory, families of analytic operators, and Gribov operator acting in the Bargmann space. Recent mathematical developments of perturbed non-self-adjoint operators are discussed with the completeness of the space of generalized eigenvectors, bases on Hilbert and Banach spaces and asymptotic behavior of the eigenvalues of these operators. Most results in the book are motivated by physical problems, such as the perturbation method for sound radiation by a vibrating plate in a light fluid, Gribov operator in Bargmann space and other applications in mathematical physics and mechanics. This book is intended for students, researchers in the field of spectral theory of linear non self-adjoint operators, pure analysts and mathematicians.

Contents:

Intro

Preface

Introduction

References

Contents

About the Author

Symbols Description

1 Basic Notations and Results

1.1 Spaces and Operators

1.1.1 Vector and Normed Spaces

1.1.2 Operators on Quasi-Banach Spaces

1.1.3 Closed and Closable Operators

1.1.4 Adjoint Operator

1.1.5 Fredholm Operators

1.2 Some Notions of Spectral Theory

1.2.1 Closed Graph Theorem

1.2.2 Resolvent Set and Spectrum

1.2.3 Bounded Operators

1.2.4 Numerical Range

1.3 Inequalities

1.4 Closed Operators

1.4.1 Closed Operator Perturbations

1.4.2 A-Bounded, A-Closed, and A-Closable

1.5 Lebesgue-Dominated Convergence Theorem

1.6 Compact, Weakly Compact, Strictly Singular ...

1.6.1 Compact Operator

1.6.2 Weakly Compact Operator

1.6.3 Strictly Singular Operator

1.6.4 Discrete Operator

1.6.5 Ascent and Descent Operators

1.6.6 Riesz Operator

1.7 A-Compact Operators

1.8 Dunford-Pettis Property

1.9 The Jeribi Essential Spectrum

1.9.1 Definition

1.9.2 A Characterization of the Jeribi Essential Spectrum

1.10 Jordan Chain for an Operator and Multiplicities

1.11 Laurent Series Expansion of the Resolvent

1.12 Bases

1.12.1 Algebraic Bases (Hamel Bases)

1.12.2 On a Schauder Basis

1.13 Normal Operator

1.14 Positive Operators

1.15 Spectrum of the Sum of Two Operators

1.16 Notes and Remarks

2 Analysis with Operators

2.1 Projections

2.1.1 Generalities

2.1.2 Orthogonal Projection

2.1.3 Spectral Projection

2.1.4 Sum of Spectral Projection

2.1.5 l2-Decomposition

2.2 Spectral Theory of Compact and Discrete Operators

2.2.1 Riesz-Schauder Theorem

2.2.2 Discrete Operators

2.3 Functions

2.3.1 Function of Finite Order

2.3.2 Function of Sine Type

2.3.3 Generating Function in L2(0, T)

2.4 Phragmén-Lindelöf Theorems.

2.5 Holomorphic Operator Functions

2.5.1 Spectrum and Multiplicities

2.5.2 Zeros of a Holomorphic Function

2.5.3 Determinant of Operator

2.6 Semigroup Theory

2.6.1 Definitions

2.6.2 Example

2.7 Concepts of Subordination and Fully Subordination

2.7.1 Concepts of Subordination

2.7.2 Concepts of Fully Subordination

2.8 Notes and Remarks

3 Series of Complex Terms

3.1 Identity Results

3.1.1 Technical Results

3.1.2 Proof of Eq. (3.0.1) When (ak)k equiv1

3.1.3 General Case

3.2 Duality Bracket

3.2.1 Proof of Eq. (3.2.1) When (ak)k equiv1

3.2.2 Proof of Eq. (3.2.1) When (ak)k1 is Any Sequence in mathbbC

3.3 Notes and Remarks

4 Carleman-Class

4.1 Singular Values

4.1.1 Singular Values of a Compact Operator

4.1.2 Polar Representation of a Bounded Operator

4.1.3 The Dimension of an Operator

4.1.4 The Schmidt Expansion of a Compact Operator

4.1.5 Some Properties of Singular Values

4.1.6 Intermediate Ideals Between F(X) and mathcalK(X)

4.2 Spectral Theory of Compact Operators

4.2.1 Quasi-Nilpotent Operator

4.2.2 Entire Function

4.3 Generalized Eigenvectors Associated with the Non-zero Eigenvalues

4.3.1 Holomorphic Function

4.3.2 Norm of the Resolvent

4.4 calCp Carleman-Class

4.4.1 Definition

4.4.2 The Resolvent Representation

4.4.3 Some Properties of calCp Carleman-Class

4.5 Fredholm Determinant

4.6 Notes and Remarks

5 The Evolutionary Problem

5.1 Semigroups

5.1.1 Basic Elementary Properties of Semigroups

5.1.2 The Infinitesimal Generator of a Continuous Semigroup

5.1.3 Hille-Yosida Theorem

5.1.4 The Differentiability of the Semigroup

5.2 Fractional Operators

5.2.1 Dunford Integral

5.2.2 Fractional of Carleman-Class Operators.

5.3 Expansions on Generalized Eigenvectors of Operators in Hilbert Space

5.3.1 Hypotheses

5.3.2 Basic Properties

5.3.3 Representation of the Solutions

5.3.4 The Simple Case of an Operator with Nuclear Resolvent

5.3.5 The Limit Case of an Operator with an Almost Nuclear Resolvent

5.4 Notes and Remarks

6 Completeness Criteria of the Space of Generalized Eigenvectors of Non-Self-Adjoint Operators

6.1 Keldysh Results

6.1.1 In Hilbert Space

6.1.2 In Banach Space

6.2 Denseness of the Generalized Eigenvectors of a Compact Operator or an Operator with Compact Resolvent

6.2.1 Subspace Attached to an Operator with Compact Resolvent

6.2.2 Completeness Criteria of the System of Generalized Eigenvectors of an Operator with Compact Resolvent

6.2.3 A Density Result of the Space Generated by the Generalized Eigenvectors of a Compact Operator

6.3 Completeness of the System of Root Subspaces

6.3.1 Riesz Projection

6.3.2 Root Subspaces

6.3.3 System of Subspaces

6.4 Notes and Remarks

7 Bases on Hilbert and Banach Spaces

7.1 Some Notions on the Bases of a Vector Space

7.1.1 On a Schauder Basis

7.1.2 The Coefficient Functionals

7.2 Orthonormal Bases in Hilbert Space

7.3 Examples of Compact Operators

7.3.1 Finite-Rank Operator

7.3.2 Hilbert-Schmidt Operator

7.4 Equivalent Bases

7.4.1 Image of a Basis Under a Topological Isomorphism

7.4.2 Definitions

7.4.3 Characterization of Equivalent Bases

7.4.4 Near Bases

7.5 Hilbert Bases

7.6 Riesz Bases

7.6.1 On Riesz Bases in a Separable Hilbert Space

7.6.2 Riesz Basis of Jordan Chains

7.6.3 Basis Property of the Exponential Family

7.6.4 Hilbert-Schmidt Operators

7.6.5 Perturbation of Riesz Bases in a Separable Hilbert Space

7.6.6 Riesz Basis of Operator-Valued Functions.

7.6.7 Riesz Basis of Subspaces

7.7 mathcalL-Basis in L2(0,T)

7.8 Notes and Remarks

8 On a Riesz Basis of Finite-Dimensional Invariant Subspaces

8.1 Location of the Spectrum

8.2 Riesz Basis

8.2.1 On a Riesz Basis of Finite-Dimensional Invariant Subspaces

8.2.2 A Large Gap in σ(G) Yields a Gap in σ(T)

8.2.3 Riesz Basis

8.2.4 Sum of Multiplicities

8.2.5 Spectral Riesz Basis of Subspaces

8.2.6 A Riesz Basis Associated to a Block Operator Matrix

8.2.7 Gap in the Spectrum Around the Imaginary Axis

8.3 The Evolutionary Equation

8.3.1 C0-Semigroup

8.3.2 Riesz Basis of Subspaces

8.3.3 Riesz Basis

8.4 Notes and Remarks

9 Analytic Operators in Feki-Jeribi-Sfaxi's Sense

9.1 Family of Operators Dependent of Several Parameters

9.2 Invariance of the Closure

9.3 Eigenvalues

9.4 Eigenvectors

9.5 Notes and Remarks

10 On a Schauder and Riesz Bases of Eigenvectors of an Analytic Operator

10.1 Completeness of the System of Root Vectors of T(ε)

10.1.1 In Banach Space

10.1.2 In Hilbert Space

10.2 On Riesz Bases in a Separable Hilbert Space

10.3 On a Finitely Spectral Riesz Basis of a Family of Non-normal Operators

10.3.1 Spectrum of T(ε)

10.3.2 Riesz Basis of Subspaces

10.4 Riesz Basis in L2(0, T)

10.5 Notes and Remarks

11 On the Asymptotic Behavior of the Eigenvalues of an Analytic Operator in the Sense of Kato

11.1 Perturbation of T0

11.2 Behavior of the Spectrum of Perturbed Operator T(ε) Under a Finite Rank Perturbation

11.2.1 Discrete Spectrum

11.2.2 Estimate Norm

11.2.3 Sum of Multiplicities of All Eigenvalues of T(ε)-Kr

11.3 Behavior of the Spectrum of Perturbed Operator T(ε)

11.3.1 Argument of the Function Dε(λ)

11.3.2 Sum of Multiplicities of All Eigenvalues of T(ε).

11.4 Notes and Remarks

12 On the Basis Property of Root Vectors Related to a Non-self-adjoint Analytic Operator

12.1 Completeness of the System of Root Vectors of T(ε)

12.2 Basis with Parentheses of Root Vectors of T(ε)

12.2.1 Localization of the Spectrum of T(ε)

12.2.2 Basis with Parentheses

12.3 Notes and Remarks

13 Perturbation Method for Sound Radiation by a Vibrating Plate in a Light Fluid

13.1 Perturbation Method for Sound Radiation by a Vibrating Plate in a Light Fluid

13.1.1 Position of the Problem

13.1.2 Open Questions Introduced in ch1313Filippi

13.1.3 Spectral Properties of the Operator T0

13.1.4 Spectral Properties of the Resolvent of the Operator T0

13.1.5 Compactness Results

13.1.6 Completeness of the System of Root Vectors

13.1.7 On a Riesz Basis in L2(-L,L)

13.2 Vibrating Plate in a Light Fluid

13.2.1 Elementary Results

13.2.2 Completeness Results

13.2.3 Basis with Parentheses

13.3 Notes and Remarks

14 Gribov Operator in Bargmann Space

14.1 Finite Sum of Gribov Operators on Null Transverse Dimension (n=1)

14.2 Infinite Sum of Gribov Operators on Null Transverse Dimension (n=1)

14.2.1 Riesz Basis of Subspaces in the Case Where γ=1

14.2.2 On the Asymptotic Behavior of the Eigenvalue of Gribov Operator in the Case Where γ=0

14.2.3 Basis with Parentheses of Gribov Operator in the Bargmann Space in the Case Where γ=0

14.3 Notes and Remarks

15 Applications in Mathematical Physics and Mechanics

15.1 Time-Dependent Rectilinear Transport Equation

15.1.1 Resolvent and Spectrum of A

15.1.2 Distribution of the Eigenvalues of the Operator A

15.1.3 Differentiability of the Semigroup Generated by A

15.2 Behavior of Resolvent in the Case of the Lamé System.

15.2.1 Explicit Expression for the Operator A.

Notes:

Description based on print version record.

ISBN:

981-16-2528-X