Frequently hypercyclic C₀-semigroups indexed with complex sectors

In this paper, we study frequent hypercyclicity for strongly continuous semigroups of operators ₜ∈Δ 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 C₀-semigroup ₜ ∈ Δ to be frequently hypercyclic. Moreover, we derive a practical and applicable criterion for translation semigroups ₜ ∈ Δ on Lᵖ_ρ(Δ, 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 Lᵖρ(Δ, 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.