Jacobi Matrices and the Moment Problem

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
	7.1 Construction of the Tri-Diagonal Block Matrix Corresponding to the Complex Moment Problem in the Exponential Form
	7.2 Direct and Inverse Spectral Problems for Tri-Diagonal Block Jacobi Type Matrices Corresponding to the Complex Moment Problem in the Exponential Form
	7.3 Conditions of Unitarity and Commutativity of Matrices Corresponding to the Complex Moment Problem in the Exponential Form
	7.4 The Solution of the Complex Moment Problem in the Exponential Form
	Bibliographical Notes
8 Block Jacobi Type Matrices and the Two Dimensional Real Moment Problem
	8.1 Construction of the Tri-Diagonal Block Jacobi Type Matrices Corresponding to the Two Dimensional Real Moment Problem
	8.2 Direct and Inverse Spectral Problems for the Tri-Diagonal Block Jacobi Type Matrices Corresponding to the Two-Dimensional Real Moment Problem
	8.3 An Analogue of the Weyl Function Corresponding to the Two-Dimensional Real Moment Problem
	8.4 The Solution of the Two-Dimensional Real Moment Problem
	8.5 Examples of Matrices Corresponding to the Two-Dimensional Real Moment Problem
	Bibliographical Notes
9 Applications of the Spectral Theory of Jacobi Matrices and Their Generalizations to the Integration of Nonlinear Equations
	9.1 The Integration of the Toda Chain on the Semi-axis Using the Spectral Theory of Jacobi Matrices
	9.2 The Doubly Infinite Toda Chain and Its Equivalent Lax Equation for Block Jacobi Type Matrices
	9.3 The Spectral Theory of Corresponding to Toda Chains Block Jacobi Type Matrices
	9.4 Equations for the Weyl Function and Spectral Matrices
	9.5 The Basic Statement and Applications to Hamiltonian Systems
	Bibliographical Notes
References
Subject Index
Notation Index