Operator Theory

Abstract

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).