Hypercyclicity and chaoticity spaces of \(C_0\) semigroups
The authors improve a criterion for hypercyclicity of strongly continuous semigroups \((T_t)_{t\geq 0}\) on a separable Banach space \(X\) due to [\textit{S.\,El Mourchid}, Semigroup Forum 73, No.\,2, 313--316 (2006; Zbl 1115.47009)]. Using ideas of [\textit{F.\,Bayart} and \textit{S.\,Grivaux}, J.~Funct.\ Anal.\ 226, No.\,2, 281--300 (2005; Zbl 1089.47008)], El Mourchid proved that the semigroup \((T_t)_{t\geq 0}\) is hypercyclic in \(X\) provided that the generator \(A\) is of the form \(Af(s)=isf(s)\) for all \(s \in \Omega:=]\omega_1,\omega_2[ \subset \mathbb{R}\), for a strongly measurable function \(f: \Omega \rightarrow X\) such that the span of \(f(\Omega \setminus \Omega')\) is dense in \(X\) for each \(\Omega' \subset \mathbb{R}\) of measure zero. The authors prove in the present paper that if the condition on \(f(\Omega \setminus \Omega')\) is replaced by nontriviality, it follows that the semigroup is hypercyclic on an invariant closed subspace of \(X\), and if the selection of eigenvalues \(f\) is additionally weakly continuous, then it is chaotic on an invariant closed subspace of \(X\).\ Examples show that the assumptions on \(f\) cannot be relaxed.\ A characterization of the subspace of hypercyclicity or chaoticity in terms of the essential range of the selection \(f\) is obtained.\ These results are applied to conclude the chaoticity of translation semigroups on weighted \(L_p\)-spaces.\ It might be informative for the reader in connection with the present research to now that \textit{J.\,A.\thinspace Conejero, V.\,Müller} and \textit{A.\,Peris} [J.\ Funct.\ Anal.\ 244, No.\,1, 342--348 (2007; Zbl 1123.47010)], proved that a strongly continuous semigroup \((T_t)_{t\geq 0}\) on a separable Banach space is hypercyclic if and only if each operator \(T_t\), \(t>0\), is hypercyclic.
scientific article; zbMATH DE number 5290022