Operator Theory

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).