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 Fundam. Prikl. Mat., 1999, Volume 5, Issue 2, Pages 627–635 (Mi fpm397)

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

A. A. Shkalikov

M. V. Lomonosov Moscow State University

Abstract: 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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Bibliographic databases:
UDC: 517.43

Citation: A. A. Shkalikov, “On the existence of invariant subspaces of dissipative operators in space with indefinite metric”, Fundam. Prikl. Mat., 5:2 (1999), 627–635

Citation in format AMSBIB
\Bibitem{Shk99} \by A.~A.~Shkalikov \paper On the~existence of invariant subspaces of dissipative operators in space with indefinite metric \jour Fundam. Prikl. Mat. \yr 1999 \vol 5 \issue 2 \pages 627--635 \mathnet{http://mi.mathnet.ru/fpm397} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1803604} \zmath{https://zbmath.org/?q=an:0960.47020} 

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This publication is cited in the following articles:
1. Azizov, TY, “On the boundedness of Hamiltonian operators”, Proceedings of the American Mathematical Society, 131:2 (2003), 563
2. A. A. Shkalikov, “Invariant Subspaces of Dissipative Operators in a Space with Indefinite Metric”, Proc. Steklov Inst. Math., 248 (2005), 287–296
3. Alpay, D, “Basic classes of matrices with respect to quaternionic indefinite inner product spaces”, Linear Algebra and Its Applications, 416:2–3 (2006), 242
4. A. A. Shkalikov, “Dissipative Operators in the Krein Space. Invariant Subspaces and Properties of Restrictions”, Funct. Anal. Appl., 41:2 (2007), 154–167
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