FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

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

A. A. Shkalikov

Abstract

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Let $\$\; \backslash mathcal\; H\; \$$ be Hilbert space with fundamental symmetry $J=P$₊-P_-, where $P$_± are mutualy orthogonal projectors such that $J2$ is identity operator. The main result of the paper is the following: if $A$ is a maximal dissipative operator in the Krein space $\$\; \backslash mathcal\; K=\backslash \{\backslash mathcal\; H,J\backslash \}\; \$$, the domain of $A$ contains $\$\; P\_+(\backslash mathcal\; H)\; \$$, and the operator $P$₊AP_- is compact, then there exists an $A$-invariant maximal non-negative subspace $\$\; \backslash mathcal\; L\; \$$ such that the spectrum of the restriction $\$\; A|\_\{\backslash mathcal\; L\}\; \$$ 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.

All articles are published in Russian.

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Last modified: July 6, 1999