On the~existence of invariant subspaces of dissipative operators in space with indefinite metric

A. A. Shkalikov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 2, pp. 627-635

Résumé

Let $\mathcal H$ be Hilbert space with fundamental symmetry $J=P_+-P_-$, where $P_\pm$ are mutualy orthogonal projectors such that $J^2$ is identity operator. The main result of the paper is the following: if $A$ is a maximal dissipative operator in the Krein space $\mathcal K=\{\mathcal H,J\}$, the domain of $A$ contains $P_+(\mathcal H)$, and the operator $P_+AP_-$ is compact, then there exists an $A$-invariant maximal non-negative subspace $\mathcal L$ such that the spectrum of the restriction $A|_{\mathcal L}$ lies in the closed upper-half complex plain. This theorem is a modification of well-known results of L. S. Pontrjagin, H. Langer, M. G. Krein and T. Ja. Azizov. A new proof is proposed in this paper.