Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded C0-semigroup of bi-contractions

Abstract

We give necessary and sufficient conditions under which a C 0 -semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded. r 2004 Elsevier Inc. All rights reserved. MSC: primary 47D60; 47B50; secondary 47A20; 47A15

Growth Estimates for exp(A-1t) on a Hilbert Space

“…In Azizov, Barsukov, and Dijksma [1], Gomilko [4], and in Guo and Zwart [6], it is shown that the solutions of (4) are bounded if both A and A −1 generate a uniformly bounded semigroup. The question whether the uniform boundedness of the semigroup generated by A is sufficient is still open.…”

Об Обратном Операторе Генератора $C_0$-Полугруппы

“…[7], [8]). Так, в работах [8], [9], [10] было независимо показано, что в случае гильбертова пространства X справедливо утверждение…”

Stability analysis in continuous and discrete time

“…This result was first proved in [2]. We present a new proof with a sharper estimate in the sup-norm of A n d .…”

“…However, for some groups of systems stable solutions of the differential equation lead to stable solutions of the difference equation. From [21], [20], and [2], we know the following result Lemma 1.6 Let A ∈ L(X) be the generator of a bounded semigroup, that is e…”

“…As a corollary we state, that if a bounded operator in the Hilbert space is the generator of a bounded semigroup, then the powers of its cogenerator are power bounded, see also [21], [20], [2]. …”