Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded \(C_{0}\)-semigroup of bi-contractions
type: Item
instance of: scholarly article
title: Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded \(C_{0}\)-semigroup of bi-contractions
zbMATH Open document ID: 1073.47043
author: A. I. Barsukov
author: Tomas Ya. Azizov
author: A. Dijksma
published in: Journal of Functional Analysis
publication date: 2004-10-01T00:00:00Z
review text: The authors extend to semigroups of bi-contractions some results contained in their previous paper [Oper. Theory, Adv. Appl. 124, 109--129 (2001; Zbl 1026.47026)] which concerned a single bi-contraction. Several intermediate properties involving a pair of commuting bi-contractions are contained in section 2. The main problem is to find similarities between a \(C_0\)-semigroup of bi-contractions on a Krein space and a semigroup of contractions in a Hilbert space, which preserves some spectral properties. Theorem 3.4 offers a twofold answer to this problem: One condition is the uniform boundedness of the semigroup, and the other is the possibility of decomposing the space as a direct sum of invariant maximal subspaces, one positive and the other negative. The class \(H\) of operators on a Krein space is defined in section 4 in terms of such \(T\)-invariant maximal subspaces. The general results from above are finally applied to study the stability of a \(C_0\)-semigroup of bounded operators (section 5), and the power boundedness of the co-generator of such a semigroup (section 6).
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 46C20
Mathematics Subject Classification ID: 47D60
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 2105588
zbMATH Keywords: bi-contractions
zbMATH Keywords: \(C\)-semigroups
zbMATH Keywords: power boundedness
zbMATH Keywords: similarity
reviewed by: Trandafir Balan
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.08.011
OpenAlex ID: W2086223535
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DOI: 10.1016/J.JFA.2003.08.011
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description: scientific article; zbMATH DE number 2105588