Frequently hypercyclic $C_0$-semigroups indexed with complex sectors

Abstract

In this paper, we study frequent hypercyclicity for strongly continuous semigroups of operators ${\left\{T_t\right\}}_{t\in \hat{I}”}$ indexed with complex sectors. We propose a revised and more natural definition of frequent hypercyclicity compared to the one in [Chaouchi et al.,2020]. Additionally, we establish a sufficient condition and a necessary condition for a $C_0$-semigroup $\{T_t{\}}_{t\in \hat{I}”}$ to be frequently hypercyclic. Moreover, we derive a practical and applicable criterion for translation semigroups $\{T_t{\}}_{t\in \hat{I}”}$ on $L_{\ddot{I}}^p(\hat{I}”,\mathbb{K})$ spaces, expressed in terms of the integral of the weight function. As a result, we provide explicit examples of frequently hypercyclic translation semigroups on $L_{\ddot{I}}^p(\hat{I}”,\mathbb{K})$. Lastly, we present a necessary condition on the weight function for the translation semigroups, under which it is demonstrated that Example I (i) [Chaouchi,2020] is not frequently hypercyclic under the revised definition.

13 pages