Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: He, Shengnan, Yin, Zongbin
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415082315
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