01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. This monograph deals with mathematical methods applicable to studying nonclassical problems of mathematical physics. Many problems of this type are reduced to equations, where the operators involved are noninvertible. In this case, the author uses special decompositions of an operator, generalized resolvents, semigroups with kernels, and some other approaches. The simplest model of this type is the first order operator-differential equation with a noninvertible operator in front of the derivative. The corresponding spectral problems arising here are the well-known problems for linear pencils. These and other problems are studied with the use of methods, which are based on the interpolation theory for Banach spaces. The emphasis is on applications of this theory to the theory of linear operators in indefinite inner product spaces, to studying the property of a linear operator to be exponentially dichotomous, to some continuity properties of linear operators in Hilbert scales, to the Riesz basis properties of eigenelements and associated elements of linear pencils and the corresponding elliptic problems with an indefinite weight functions, and to studying nonclassical boundary value problems for first order operator-differential equations.

Table of Contents:
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-self-adjoint 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 self-adjoint pencils; examples; self-adjoint 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. (Part contents).