# Introduction to Linear Operator Theory

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## Book Description

This book is an introduction to the subject and is devoted to standard material on linear functional analysis, and presents some ergodic theorems for classes of operators containing the quasi-compact operators. It discusses various classes of operators connected with the numerical range.

## Table of Contents

Preface**1. Preliminaries: Set Theory and General Topology**1.1 The Algebra of Sets

1.2 Partially Ordered Sets

1.3 Topology and Topological Spaces

1.4 Baire’s Theorem

**2. Banach Spaces**

2.1 Linear Spaces

2.2 Linear Independence

2.3 Sets in Linear Spaces

2.4 Classes of Spaces: Isomorphic Spaces, Quotient Spaces, and Complementary Spaces

2.5 Seminorms and Norms on Linear Spaces

2.6 Linear Topological Spaces

2.7 Banach Spaces

2.8 Linear Operators on Banach Spaces

2.9 Uniformly Convex and Rotund Banach Spaces: Some Generalizations

2.10 The Hahn-Banach Extension Theorem

2.11 extension Theorems for complex Banach Spaces

2.12 Three Basic Theorems of Linear Analysis

2.13 Convergence in banach Spaces

2.14 The Adjoint of an Operator

2.15 The Spectrum of an Operator

2.16 The Local Spectrum of an Operator

2.17 Analytic Representation of the Dual of Some Banach Spaces

2.18 Measures of Noncompactness and Classes of Mappings on Banach Spaces

**3. Hilbert Spaces**

3.1 Inner Products on Linear Spaces

3.2 Orthonormal bases and the Bessel Inequality

3.3 Separable Hilbert Spaces: Gram-Schmidt Orthogonalization Method

3.4 Orthogonal Subspaces of a Hilbert Space

3.5 The Dual of a Hilbert Apace

3.6 Classes of a Bounded Linear Operators on Hilbert Spaces

**4. Banach Algebras**

4.1 Definitions and Some Examples

4.2 The Spectrum of an Element in a Banach Algebra with Unit

4.3 Representation Theorems for Commutative Banach Algebras

4.4 Structure Theorems for Commutative Banach Algebras

4.5 Representation Theorems for Noncommutative Banach Algebras

**5. Spectral Representation of Operators on Hilbert Spaces**

5.1 Semispectral and Spectral Families of Random Measures

5.2 Measurability and Integrability with Respect to Spectral Families

5.3 A Representation Theorem for L∞

5.4 Spectral Decomposition of Some Classes of Operators

5.5 Some Remarks on the Spectral Mapping Theorem for Hermitian and Normal Operators

**6. The Numerical Range**

6.1 The Numerical Range for Bounded Linear Operators on Hilbert Spaces

6.2 The Numerical Range and the Spectrum

6.3 The Numerical Range and It’s Closure

6.4 The Essential Numerical Range for Bounded Linear Operators on Hilbert Spaces

6.5 The Maximal Numerical Range of a Bounded Operator on a Hilbert Space

6.6 The Extreme Points of the Numerical Range for Hyponormal Operators and (WN) Operators

6.7 The Numerical Range and Some Classes of Operators

6.8 The Numerical Range and Tensor Products

6.9 The Numerical Range for Bounded Linear Operators on Banach Spaces

6.10 The Exponential Function on the Set of ALL Bounded Linear Operators on a Banach Space

6.11 The Numerical Radius, the Spectral Radius, and the Norm of a Bounded Linear Operator on a Banach Space

6.12 Hermitian and Normal Operators on Banach Spaces

6.13 Normal Operators on Banach Spaces

6.14 Classes of Elements in Banach Algebras with Unit: The Vidav-Palmer Theorem

6.15 Some Properties of Hermitian and Normal Elements of a Banach Algebra

6.16 The Numerical Radius and the Iterates of an Element

6.17 The Numerical Range of elements of Locally m-Convex Algebras

**7. Nonnormal Classes of Operators**

7.1 Classes of Nonnormal Operators

7.2 Spectral Sets and Dilations of Operators

7.3 Operators with G1 Property and Some Generalizations

7.4 Operators with Property Re σ (T) = σ (Re T)

7.5 The Class 1

7.6 Other Classes of Bounded Operators

**8. Conditions Implying Normality**

8.1 Conditions Implying Hermitianity

8.2 Conditions Implying Unitarity

8.3 Conditions Implying Normality

**9. Symmetrizable Operators: Generalizations and Applications**

9.1 Symmetrizable Operators on Hilbert Spaces

9.2 Symmetrizable Elements in Banach Algebras

9.3 Inner Products on Banach Spaces: Symmetrizable Operators and Some Generalizations

9.4 Some Applications of Symmetrizable Operators and Quasi-Normalizable Operators

9.5 Further Results on Symmetrizable Operators on Hilbert Spaces

**10 Invariant Subspaces and Some structure Theorems**

10.1 Invariant Subspaces: Some Existence Theorems

10.2 Reducing Invariant Subspaces

10.3 Some Structure Theorems

**11. The Weyl Spectrum of an Operator**

11.1 Preliminaries and Some General Results

11.2 Weyl’s Theorem

11.3 Weyl’s Theorem for Some Classes of Operators

11.4 The Weyl spectrum of an Element in a von Neumann Algebra

11.5 The von Neumann Theorem

**12. Analytic and Quasi-Analytic Vectors**

12.0 Introduction

12.1 Self-Adjoint Operators

12.2 Classes of Vectors for an Operator

12.3 Analytic and Quasi-Analytic Vectors and Essentially Self-Adjoint Operators

12.4 Quasi-Analytic Vectors and Semigroups of Operators

12.5 Analytic and Quasi-Analytic Elements in Commutative Banach Algebras

**13. Schwarz Norms**

13.1 Schwartz Norms

13.2 A New Class of Schwarz Norms

13.3 Schwarz Norms on Banach Spaces

**14. Maximum Theorems for Operator-Valued Holomorphic Functions**

14.1 Holomorphic Functions

14.2 Subharmonic Functions

14.3 Maximum Theorems for the Norm

14.4 Maximum Theorems for the Spectral Radius and for the Essential Spectral Radius

14.5 Maximum Theorems for Other Operator-Valued Holomorphic Functions

**15. Uniform Ergodic Theorems for Some Classes of Operators**

15.1 Classes of Operators

15.2 Applications to Markov Processes

**Contents**

Appendix. CP Classes

References

Symbol Index

Subject Index

Appendix. CP Classes

References

Symbol Index

Subject Index