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

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

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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• http://mi.mathnet.ru/eng/faa2863
• https://doi.org/10.4213/faa2863
• http://mi.mathnet.ru/eng/faa/v41/i2/p93

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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
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
3. S. G. Pyatkov, “On the existence of maximal semidefinite invariant subspaces for $J$-dissipative operators”, Sb. Math., 203:2 (2012), 234–256
4. Markov V.G., “Nekotorye svoistva neznakoopredelennykh operatorov Shturma-Liuvillya”, Matematicheskie zametki YaGU, 19:1 (2012), 44–59
5. Wanjala G., “The Invariant Subspace Problem for Absolutely P-Summing Operators in Krein Spaces”, J. Inequal. Appl., 2012, 254
6. Kapitula T., Hibma E., Kim H.-P., Timkovich J., “Instability Indices for Matrix Polynomials”, Linear Alg. Appl., 439:11 (2013), 3412–3434
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
8. Makarov K.A., Schmitz S., Seelmann A., “On Invariant Graph Subspaces”, Integr. Equ. Oper. Theory, 85:3 (2016), 399–425
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