Stability of \(C\)-regularized semigroups
Let \(X\) be a Banach space, let \(T=\{T(t)\}_{t \geq 0}\) be a bounded \(C\)-regularized semigroup generated by \(A\), where \(C\) is a bounded injective linear operator on \(X\) such that \(R(C)\) is dense in \(X\). Denoting by \(\sigma_u(A, Cx)\) the set of all points \(\lambda \in i {\mathbb R}\) such that \((\lambda - A)^{-1}Cx\) cannot be extended holomorphically, the authors prove that if there exists a dense subspace \(Y \subset X\) such that for every \(x \in Y\), the set \(\sigma_u(A, Cx)\) is at most countable and \(\sigma_r(A) \cap i {\mathbb R} = \emptyset\), then the semigroup \(T\) is stable. The also study the case when \(R(C)\) is non-dense.
scientific article; zbMATH DE number 2174195