Spectral conditions for stability of one-parameter semigroups
The author proves the following theorem: Let \(\{S(t)\), \(t\geq 0\}\) be a \(C_0\)-semigroup of linear operators on a Banach space \(Y\) with generator \(B\) and let \(\{T(t)\), \(t\geq 0\}\) be a bounded \(C_0\)-semigroup of linear operators on a Banach space \(X\) with generator \(A\). Let \(\sigma(B)\cap i\mathbb{R}\) be countable, let \(P\sigma(A^*)\cap i\mathbb{R}\) be empty, and let there exist a bounded linear operator \(C:Y\to X\) with dense range which interwines the two semigroups (that is, \(CS(t)=T(t)C\) for all \(t\geq O\)). Then \(|T(t)x|\to 0\) as \(t\to\infty\) for each \(x\) in \(X\).
\(C_ 0\)-semigroup of linear operators on a Banach space
scientific article; zbMATH DE number 881702