q-Frequently hypercyclic operators

TL;DR

This paper introduces the concept of q-frequently hypercyclic operators, providing criteria for their identification and demonstrating their existence across various topologies, including applications to specific operators on entire functions.

Contribution

It defines q-frequently hypercyclic operators, establishes a sufficient criterion for their identification, and applies this to various topologies and specific operators on entire functions.

Findings

Established a criterion for q-frequently hypercyclic operators.

Constructed examples of q-frequently hypercyclic operators in different topologies.

Proved the q-frequent hypercyclicity of a specific non-convolution operator on entire functions.

Abstract

We introduce q-frequently hypercyclic operators and derive a sufficient criterion for a continuous operator to be q-frequently hypercyclic on a locally convex space. Applications are given to obtain q-frequently hypercyclic operators with respect to the norm-, F-norm- and weak*- topologies. Finally, the frequent hypercyclicity of the non-convolution operator $T_\mu$ defined by $T_\mu(f)(z) = f'(\mu z)$, $\mu\ge1$ on the space $H(\mathbb{C})$ of entire functions equipped with the compact-open topology is shown.