Operator Theory. Nonclassical Problems

Inverse and Ill-Posed Problems Series

Contents:

Introduction

INDEFINITE INNER PRODUCT SPACES. LINEAR OPERATORS. INTERPOLATION

Indefinite inner product spaces

Definitions; Krein spaces; The Gram operator. W-spaces; J-orthogonal complements. Projective completeness; J-orthonormalized systems

The basic classes of operators in Krein spaces

J-dissipative operators; J-selfadjoint operators

Interpolation of Banach and Hilbert spaces and applications

Preliminaries; Continuity of some functionals in a Hilbert scale; Separation of the spectrum of an unbounded operator; Interpolation properties of bases

The existence of maximal semidefinite invariant subspaces for J-dissipative operators

First order equations. Decomposition of a solution

Function spaces; The Cauchy problem; Auxiliary definitions. Some properties of imaginary powers of operators; Solvability of the Cauchy problem in the original Banach space; Adjoint problems; Arbitrary operators. Phase spaces; Remarks and examples

SPECTRAL THEORY FOR LINEAR SELFADJOINT PENCILS

Examples

Selfadjoint pencils; Elliptic eigenvalue problems with an indefinite weight function

Basic assumptions. The structure of the root subspaces

The Riesz basis property. Invariant subspaces

Basis property; Invariant subspaces. Some applications

Sufficient conditions

ELLIPTIC EIGENVALUE PROBLEMS WITH AN INDEFINITE WEIGHT FUNCTION

Auxiliary function spaces. Interpolation

Definitions; Interpolation of weighted Sobolev spaces; Inequalities of the Hardy type

Preliminaries. Basic assumptions

Variational statement; Elliptic problems

Basiness theorems

The general case; The one-dimensional case

Examples and counterexamples

OPERATOR-DIFFERENTIAL EQUATIONS

Generalized solutions. Positive definite case

Preliminaries; Uniqueness and existence theorems

Degenerate case

Preliminaries; Solvability theorems. The case of a bounded interval; Solvability problems. The case of the interval (0, ); Smoothness of solutions. Orthogonality conditions; The periodic problem. Linear inverse problems

The Fourier method

Representation of solutions. First order equations; Some problems for the second order equation

Some applications to partial differential equations

Higher order parabolic equations with changing time direction; Second order parabolic equations with changing time direction; Orthogonality conditions. Parabolic equations; Second order mixed type equations

Orthogonality conditions

Bibliography

Index

2002; xiv pages
ISBN 90-6764-363-7
Price: EUR 196/US$ 228