Frequently hypercyclic $C_0$-semigroups indexed with complex sectors
Shengnan He, Zongbin Yin
TL;DR
The paper refines the study of frequent hypercyclicity for $C_0$-semigroups indexed by complex sectors by introducing a natural lower-density notion on sectors $\Delta(\alpha)$ and proving a general sufficient condition (Frequent Hypercyclicity Criterion) and a necessary condition for $C_0$-semigroups on Banach spaces to be frequently hypercyclic. It shows that frequent hypercyclicity implies $\mathcal{F}_{pld}$-transitivity and develops a practical criterion for translation semigroups on weighted $L^p$ spaces $L^{p}_{\rho}(\Delta)$, governed by the finiteness of $\int_{\Delta} \rho(s)\,ds$. The authors provide explicit examples of frequently hypercyclic translation semigroups under the revised framework and give a weight-condition-based necessary condition demonstrating that a prior Chaouchi example is not FH in this setting. Overall, the work advances understanding of linear chaos in sector-indexed semigroups and furnishes testable criteria for constructing FH semigroups in applied contexts.
Abstract
In this paper, we study frequent hypercyclicity for strongly continuous semigroups of operators $\left\{T_{t}\right\}_{t\inΔ}$ 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 $\{T_{t}\}_{t \in Δ}$ to be frequently hypercyclic. Moreover, we derive a practical and applicable criterion for translation semigroups $\{T_{t}\}_{t \in Δ}$ on $L^p_ρ(Δ, \mathbb{K})$ spaces, expressed in terms of the integral of the weight function. As a result, we provide explicit examples of frequently hypercyclic translation semigroups on $L^{p}_ρ(Δ, \mathbb{K})$. 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.