Frequently hypercyclic semigroups

TL;DR

This paper establishes a criterion for frequent hypercyclicity in strongly continuous semigroups of operators, linking it to the Pettis integral and the generator, with applications to Ornstein-Uhlenbeck and translation semigroups.

Contribution

It introduces a new sufficient condition for frequent hypercyclicity based on the Pettis integral, applicable via the generator of the semigroup.

Findings

Criterion verified for Ornstein-Uhlenbeck semigroups.

Applied to translation semigroups on weighted function spaces.

Demonstrated frequent hypercyclicity in specific operator classes.

Abstract

We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be verified in certain cases in terms of the infinitesimal generator of semigroup. Applications are given for semigroups generated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of $p$-integrable functions, or continuous functions that, multiplied by the weight, vanish at infinity.