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The concept of regular extensions of an Hermitian (non-densely defined) operator was introduced by A. Kuzhel in 1980. This concept is a natural generalization of proper extensions of symmetric (densely defined) operators. The use of regular extensions enables one to study various classes of extensions of Hermitian operators without using the method of linear relations. The central question in this monograph is to what extent the Hermitian part of a linear operator determines its properties. Various properties are investigated and some applications of the theory are given.
Chapter 1 deals with some results from operator theory and the theory of extensions. Chapter 2 is devoted to the investigation of regular extensions of Hermitian (symmetric) operators with certain restrictions. In chapter 3 regular extensions of Hermitian operators with the use of boundary-value spaces are investigated. In the final chapter, the results from chapters 1-3 are applied to the investigation of quasi-differential operators and models of zero-range potential with internal structure.
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978-90-6764-294-1 (9789067642941)
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Preface
CHAPTER 1: REGULAR EXTENSIONS
Linear Operators
Spectrum of a Linear Operator
Hermitian Operators
Symmetric Operators
Regular Extensions of Hermitian Operators
Dissipative Extensions of Hermitian Operators
Accretive Operators
CHAPTER 2: REGULAR EXTENSIONS WITH RESTRICTIONS
Self-Adjoint Bound-Preserving Extensions of Semibounded Symmetric Operators. Theorem and Hypothesis of Von Neumann
Proof of the Von Neumann Hypothesis (Friedrichs Method)
Self-Adjoint Norm-Preserving Extensions of Hermitian Contractions
Normal Norm-Preserving Extensions of Hermitian Contractions
Krein's Proof of the Von Neumann Hypothesis
Squared Symmetric Operators
Self-Adjoint Bound-Preserving Extensions of Semibounded Hermitian Operators
Regular U-Invariant Extensions of Hermitian Operators
Canonical Dissipative Extensions of Hermitian Operators
CHAPTER 3: BOUNDARY-VALUE SPACES OF HERMITIAN OPERATORS
Definition and General Properties of Boundary-Value Spaces
Description of Regular-Extensions of Hermitian Operators in Terms of Boundary-Value Spaces
Characteristic Functions of Hermitian Operators
Spectral Properties of Regular Extensions
CHAPTER 4: EXAMPLES AND APPLICATIONS
Quasidifferential Operators
Boundary-Value Spaces for Model of Zero-Range Potentials with Internal Structure
Some Properties of Nonperturbed Operators
Abstract Wave Equation
Elements of the Lax-Phillips Scattering Theory for -Perturbed Abstract Wave Equation
References
Subject Index
Notation
