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 Tr. Mat. Inst. Steklova, 2005, Volume 248, Pages 294–303 (Mi tm139)

Invariant Subspaces of Dissipative Operators in a Space with Indefinite Metric

A. A. Shkalikov

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

Abstract: A theorem on the existence of maximal nonnegative invariant subspaces is proved for a special class of dissipative operators in a Hilbert space with indefinite inner product. It is shown that the spectra of the restrictions of these operators on the corresponding invariant subspaces lie in the closed upper half-plane. The theorem obtained is a generalization of the well-known results of L. S. Pontryagin, H. K. Langer, M. G. Krein, and T. Ya. Azizov devoted to this subject.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2005, 248, 287–296

Bibliographic databases:
UDC: 517.9+517.43

Citation: A. A. Shkalikov, “Invariant Subspaces of Dissipative Operators in a Space with Indefinite Metric”, Studies on function theory and differential equations, Collected papers. Dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol'skii, Tr. Mat. Inst. Steklova, 248, Nauka, MAIK «Nauka/Inteperiodika», M., 2005, 294–303; Proc. Steklov Inst. Math., 248 (2005), 287–296

Citation in format AMSBIB
\Bibitem{Shk05} \by A.~A.~Shkalikov \paper Invariant Subspaces of Dissipative Operators in a~Space with Indefinite Metric \inbook Studies on function theory and differential equations \bookinfo Collected papers. Dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol'skii \serial Tr. Mat. Inst. Steklova \yr 2005 \vol 248 \pages 294--303 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm139} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2165936} \zmath{https://zbmath.org/?q=an:1146.47005} \transl \jour Proc. Steklov Inst. Math. \yr 2005 \vol 248 \pages 287--296 

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This publication is cited in the following articles:
1. A. A. Shkalikov, “Dissipative Operators in the Krein Space. Invariant Subspaces and Properties of Restrictions”, Funct. Anal. Appl., 41:2 (2007), 154–167
2. S. G. Pyatkov, “On the existence of maximal semidefinite invariant subspaces for $J$-dissipative operators”, Sb. Math., 203:2 (2012), 234–256
3. Markov V.G., “Nekotorye svoistva neznakoopredelennykh operatorov Shturma-Liuvillya”, Matematicheskie zametki YaGU, 19:1 (2012), 44–59
4. 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
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