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

This article is cited in 4 scientific papers (total in 4 papers)

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.

Full text: PDF file (413 kB)

Bibliographic databases:
UDC: 517.43
Received: 01.03.1999

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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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  crossref  mathscinet  zmath  isi
    2. A. A. Shkalikov, “Invariant Subspaces of Dissipative Operators in a Space with Indefinite Metric”, Proc. Steklov Inst. Math., 248 (2005), 287–296  mathnet  mathscinet  zmath
    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  crossref  mathscinet  zmath  isi
    4. A. A. Shkalikov, “Dissipative Operators in the Krein Space. Invariant Subspaces and Properties of Restrictions”, Funct. Anal. Appl., 41:2 (2007), 154–167  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Фундаментальная и прикладная математика
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