Regular Extensions of Hermitian Operators

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A. V. Kuzhel, S. a. Kuzhel
VSP, Jan 1, 1998 - Science - 272 pages
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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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Contents

Hermitian Operators
9
Symmetric Operators
18
Regular Extensions of Hermitian Operators
26
Accretive Operators
53
REGULAR EXTENSIONS WITH RESTRICTIONS
61
SelfAdjoint NormPreserving Extensions of Hermitian Contractions
67
Normal NormPreserving Extensions of Hermitian Contractions
81
SelfAdjoint BoundPreserving Extensions of Semibounded Hermitian
96
Description of Regular Extensions of Hermitian Operators
146
Characteristic Functions of Hermitian Operators
159
Spectral Properties of Regular Extensions
166
EXAMPLES AND APPLICATIONS
177
BoundaryValue Spaces for Model of ZeroRange Potentials
189
Some Properties of Nonperturbed Operators
201
Abstract Wave Equation
215
Elements of the LaxPhillips Scattering Theory for pPerturbed
234

Regular UInvariant Extensions of Hermitian Operators
106
Canonical Dissipative Extensions of Hermitian Operators
118
BOUNDARYVALUE SPACES OF HERMITIAN OPERATORS
129

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