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Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions

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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.

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Authors and Affiliations

  1. Department of Mathematic and Mechanics, M. V. Lomonosov Moscow State University, Moscow

    A. A. Shkalikov

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__________

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 41, No. 2, pp. 93–110, 2007

Original Russian Text Copyright © by A. A. Shkalikov

This work was supported by the program “Leading Scientific Schools,” grant no. NSh-5247.2006.1, by the Russian Foundation for Basic Research, grant no. 07-01-00283, and by the INTAS, grant no. 05-1000008-7883.

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Shkalikov, A.A. Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions. Funct Anal Its Appl 41, 154–167 (2007). https://doi.org/10.1007/s10688-007-0014-y

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  • DOI: https://doi.org/10.1007/s10688-007-0014-y

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