Operator Theory Nonclassical Problems

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Overview

This monograph describes mathematical methods applicable to studying nonclassical problems of mathematical physics. The emphasis of the book is on applications of the interpolar theory of Banach spaces to the theory of linear operators to be expotentially dichotomous, to some continuity properties of linear operators in Hilbert scales, to the Riesz basis property of eigenelements and associated elements of linear pencils and the correspondending elliptic problems with indefinite weight functions, and to studying nonclassical boundary value problems for first order operator-differential equations.

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Product Details

Meet the Author

Sergei G. Pyatkov, Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia.

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Table of Contents

Ch. 1 Indefinite inner product spaces. Linear operators. Interpolation 1
1 Indefinite inner product spaces 2
2 The basic classes of operators in Krein spaces 14
3 Interpolation of Banach and Hilbert spaces and applications 23
4 The existence of maximal semidefinite invariant subspaces for J dissipative operators 52
5 First order equations. Decomposition of a solution 77
Ch. 2 Spectral theory for linear selfadjoint pencils 111
1 Examples 112
2 Basic assumptions. The structure of root subspaces 114
3 The Riesz basis property. Invariant subspaces 127
4 Sufficient conditions 147
Ch. 3 Elliptic eigenvalue problems with an indefinite weight function 151
1 Auxiliary function spaces. Interpolation 153
2 Preliminaries. Basic assumptions 209
3 Basisness theorems 220
4 Examples and counterexamples 236
Ch. 4 Operator-differential equations 253
1 Generalized solutions. Positive definite case 255
2 Degenerate case 268
3 The Fourier method 291
4 Some applications to partial differential equations 302
Bibliography 323
Index 347
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