The imaginary point spectrum and hypercyclicity
Let \(T(\cdot)\) be a hypercyclic \(C_0\)-semigroup on a separable Banach space \(X\); in other words, for any open sets \(U\) and \(V\) there exists \(t>0\) such that \(T(t)U\cap V\neq\varnothing\). This kind of semigroups has been studied in several papers [see, e.g., \textit{W.\,Desch, W.\,Schappacher} and \textit{G.\,F.\,Webb}, J.~Ergodic Theory Dyn.\ Syst.\ 17, No.\,4, 793--819 (1997; Zbl 0910.47033); \textit{T.\,Bermúdez, A.\,Bonilla} and \textit{A.\,Martinón}, Proc.\ Am.\ Math.\ Soc.\ 131, No.\,8, 2435--2441 (2003; Zbl 1044.47006)]. The author proves the following Theorem. Let \(A\) be the infinitesimal generator of a \(C_0\)-semigroup \(T(\cdot)\) on \(X\). Assume that for \(-\infty\leq w_1 <w_2 \leq \infty\) there is an integrable function \(f:(w_1,w_2)\longrightarrow X\) such that (i) \(f(s) \in\text{Ker}(isI-A)\) almost everywhere for \(s\) in \((w_1,w_2)\); (ii) \(\text{span}\{f(s):~s\in (w_1,w_2)\setminus \Omega\}\) is dense in \(X\) for every subset \(\Omega\) of measure zero. Then the semigroup is hypercyclic. A key ingredient in the proof is a classical spectral mapping theorem, which in particular implies that if \(\lambda \in \sigma_0(A)\) and \(A(x_{\lambda})=\lambda x_{\lambda}\), then \(T(t)(x_{\lambda})=e^{t\lambda}x_{\lambda}\). As an application, the author shows a differential equation whose associated semigroup is hypercyclic.
scientific article; zbMATH DE number 5118899