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Funktsional. Anal. i Prilozhen., 2007, Volume 41, Issue 2, Pages 93–110 (Mi faa2863)  

This article is cited in 8 scientific papers (total in 8 papers)

Dissipative Operators in the Krein Space. Invariant Subspaces and Properties of Restrictions

A. A. Shkalikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a $C_0$-semigroup.

Keywords: dissipative operator, Pontryagin space, Krein space, invariant subspace, $C_0$-semigroup, holomorphic semigroup

DOI: https://doi.org/10.4213/faa2863

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English version:
Functional Analysis and Its Applications, 2007, 41:2, 154–167

Bibliographic databases:

UDC: 517.9+517.43
Received: 12.01.2007

Citation: A. A. Shkalikov, “Dissipative Operators in the Krein Space. Invariant Subspaces and Properties of Restrictions”, Funktsional. Anal. i Prilozhen., 41:2 (2007), 93–110; Funct. Anal. Appl., 41:2 (2007), 154–167

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Strauss M., “Spectral estimates and basis properties for self-adjoint block operator matrices”, Integral Equations Operator Theory, 67:2 (2010), 257–277  crossref  mathscinet  zmath  isi  elib  scopus
    2. Azizov T.Ya., Behrndt J., Jonas P., Trunk C., “Spectral points of definite type and type $\pi$ for linear operators and relations in Krein spaces”, J. Lond. Math. Soc. (2), 83:3 (2011), 768–788  crossref  mathscinet  zmath  isi  scopus
    3. S. G. Pyatkov, “On the existence of maximal semidefinite invariant subspaces for $J$-dissipative operators”, Sb. Math., 203:2 (2012), 234–256  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Markov V.G., “Nekotorye svoistva neznakoopredelennykh operatorov Shturma-Liuvillya”, Matematicheskie zametki YaGU, 19:1 (2012), 44–59  zmath  elib
    5. Wanjala G., “The Invariant Subspace Problem for Absolutely P-Summing Operators in Krein Spaces”, J. Inequal. Appl., 2012, 254  crossref  mathscinet  zmath  isi  elib  scopus
    6. Kapitula T., Hibma E., Kim H.-P., Timkovich J., “Instability Indices for Matrix Polynomials”, Linear Alg. Appl., 439:11 (2013), 3412–3434  crossref  mathscinet  zmath  isi  scopus
    7. Pyatkov S.G., “Existence of Maximal Semidefinite Invariant Subspaces and Semigroup Properties of Some Classes of Ordinary Differential Operators”, Oper. Matrices, 8:1 (2014), 237–254  crossref  mathscinet  zmath  isi  elib  scopus
    8. Makarov K.A., Schmitz S., Seelmann A., “On Invariant Graph Subspaces”, Integr. Equ. Oper. Theory, 85:3 (2016), 399–425  crossref  mathscinet  zmath  isi  elib  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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