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Table of Contents
Intro
Preface
Contents
1 Introduction
2 Some Aspects of the Spectral Theory of Unbounded Operators
2.1 Preliminary Information About Unbounded Operators
2.2 Extensions of Hermitian Operators to Self-AdJoint Operators
2.3 The Spectral Decomposition of Bounded Operators
2.4 The Spectral Decomposition of Unbounded Operators
2.5 Representations of One-Parameter Unitary Groups
2.6 Self-Adjointness Criteria
2.7 Rigged Spaces
2.8 Tensor Products
2.9 Representations of Continuous Multi-Linear Forms
2.10 Semi-Bounded Bilinear Forms
2.11 The Generalized Eigenvectors Expansion
2.12 A General Case with a Quasi-Scalar Product
Bibliographical Notes
3 Jacobi Matrices and the Classical Moment Problem
3.1 Difference Operators, Jacobi Matrices, and Self-Adjointness Conditions
3.2 The Generalized Eigenvectors Expansion and the Fourier Transform Corresponding to Jacobi Matrices
3.3 The Inverse Problem
3.4 Further Spectral Analysis of the Difference Operator
3.5 The Classical Moment Problem
3.6 Some Other Generalizations of the Moment Problem
3.7 Connections with the Theory of Jacobi Matrices
Bibliographical Notes
4 The Strong Moment Problem
4.1 Preliminaries to the Strong Moment Problem
4.2 The Solution of the Strong Moment Problem
4.3 The Orthogonalization Procedure and the Construction of a Tri-Diagonal Block Matrix
4.4 Direct and Inverse Spectral Problems Corresponding to Tri-Diagonal Block Jacobi-Laurent Matrices Generating Self-Adjoint Operators
4.5 Considerations of Hermitian Block Jacobi-Laurent Type Matrices
4.6 The Connection Between the Strong Moment Problem and the Spectral Theory of Jacobi-Laurent Matrices
4.7 Two Addition Facts
4.8 The Inner Structure of the Jacobi-Laurent Matrix
Bibliographical Notes
5 Block Jacobi Type Matrices in the Complex Moment Problem
5.1 Construction of the Tri-Diagonal Block Jacobi Type Matrix of a Bounded Normal Operator
5.2 Direct and Inverse Spectral Problems for the Tri-Diagonal Block Jacobi Type Matrices of Bounded Normal Operators
5.3 Normality Conditions of Block Jacobi Type Matrices
5.4 The Solution of the Complex Moment Problem
5.5 The Weyl Function and Polynomials of the Second Kind in the Complex Moment Problem
Bibliographical Notes
6 Unitary Block Jacobi Type Matrices and the Trigonometric Moment Problem
6.1 Introduction to the Unitary Block Jacobi Type Matrices
6.2 Construction of the Tri-Diagonal Block Jacobi Type Matrices of a Unitary Operator
6.3 Direct and Inverse Spectral Problems for the Tri-Diagonal Block Jacobi Type Matrices of Unitary Operators
6.4 The Detailed Internal Structure of the Block Matrix of the Unitary Operator
6.5 The Solution of the Trigonometric Moment Problem
Bibliographical Notes
7 Block Jacobi Type Matrices and the Complex Moment Problem in the Exponential Form
Preface
Contents
1 Introduction
2 Some Aspects of the Spectral Theory of Unbounded Operators
2.1 Preliminary Information About Unbounded Operators
2.2 Extensions of Hermitian Operators to Self-AdJoint Operators
2.3 The Spectral Decomposition of Bounded Operators
2.4 The Spectral Decomposition of Unbounded Operators
2.5 Representations of One-Parameter Unitary Groups
2.6 Self-Adjointness Criteria
2.7 Rigged Spaces
2.8 Tensor Products
2.9 Representations of Continuous Multi-Linear Forms
2.10 Semi-Bounded Bilinear Forms
2.11 The Generalized Eigenvectors Expansion
2.12 A General Case with a Quasi-Scalar Product
Bibliographical Notes
3 Jacobi Matrices and the Classical Moment Problem
3.1 Difference Operators, Jacobi Matrices, and Self-Adjointness Conditions
3.2 The Generalized Eigenvectors Expansion and the Fourier Transform Corresponding to Jacobi Matrices
3.3 The Inverse Problem
3.4 Further Spectral Analysis of the Difference Operator
3.5 The Classical Moment Problem
3.6 Some Other Generalizations of the Moment Problem
3.7 Connections with the Theory of Jacobi Matrices
Bibliographical Notes
4 The Strong Moment Problem
4.1 Preliminaries to the Strong Moment Problem
4.2 The Solution of the Strong Moment Problem
4.3 The Orthogonalization Procedure and the Construction of a Tri-Diagonal Block Matrix
4.4 Direct and Inverse Spectral Problems Corresponding to Tri-Diagonal Block Jacobi-Laurent Matrices Generating Self-Adjoint Operators
4.5 Considerations of Hermitian Block Jacobi-Laurent Type Matrices
4.6 The Connection Between the Strong Moment Problem and the Spectral Theory of Jacobi-Laurent Matrices
4.7 Two Addition Facts
4.8 The Inner Structure of the Jacobi-Laurent Matrix
Bibliographical Notes
5 Block Jacobi Type Matrices in the Complex Moment Problem
5.1 Construction of the Tri-Diagonal Block Jacobi Type Matrix of a Bounded Normal Operator
5.2 Direct and Inverse Spectral Problems for the Tri-Diagonal Block Jacobi Type Matrices of Bounded Normal Operators
5.3 Normality Conditions of Block Jacobi Type Matrices
5.4 The Solution of the Complex Moment Problem
5.5 The Weyl Function and Polynomials of the Second Kind in the Complex Moment Problem
Bibliographical Notes
6 Unitary Block Jacobi Type Matrices and the Trigonometric Moment Problem
6.1 Introduction to the Unitary Block Jacobi Type Matrices
6.2 Construction of the Tri-Diagonal Block Jacobi Type Matrices of a Unitary Operator
6.3 Direct and Inverse Spectral Problems for the Tri-Diagonal Block Jacobi Type Matrices of Unitary Operators
6.4 The Detailed Internal Structure of the Block Matrix of the Unitary Operator
6.5 The Solution of the Trigonometric Moment Problem
Bibliographical Notes
7 Block Jacobi Type Matrices and the Complex Moment Problem in the Exponential Form