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
References
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
References
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
References
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)