Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded \(C_{0}\)-semigroup of bi-contractions
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).
scientific article; zbMATH DE number 2105588
scientific article