On invariant subspaces of dissipative operators in a space with indefinite metric

TL;DR

This paper proves a theorem on the existence of maximal nonnegative invariant subspaces for a class of dissipative operators in indefinite metric spaces, extending classical results by Pontrjagin, Langer, Krein, and Azizov.

Contribution

It generalizes existing theorems on invariant subspaces of dissipative operators to spaces with indefinite inner product, including spectral properties of these restrictions.

Findings

Existence of maximal nonnegative invariant subspaces is established.

Spectra of restricted operators lie in the closed upper half-plane.

The theorem extends classical results to indefinite metric spaces.

Abstract

The theorem on the existence of maximal nonnegative invariant subspaces for a special class of dissipative operators in Hilbert space with indefinite inner product is proved in the paper. It is shown in addition that the spectra of the restrictions of these operators on the corresponding invariant subspaces lie in the closed upper half-plane. The obtained theorem is a generalization of well-known results of L.S.Pontrjagin, H.K.Langer, M.G.Krein and T.Ja.Azizov devoted to this subject.