FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1999, VOLUME 5, NUMBER 2, PAGES 627-635

A. A. Shkalikov

Abstract

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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.

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Last modified: July 6, 1999