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

A. A. Shkalikov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 41 (2007) no. 2, pp. 93-110.

Résumé

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.

Détail

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

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