Inverse of $\mathcal{U}$-frequently hypercyclic operators

TL;DR

This paper demonstrates that within the class of invertible -frequently hypercyclic operators on ^p spaces, the inverse operator may not retain the -frequently hypercyclic property, highlighting a nuanced behavior of such operators.

Contribution

It provides the first example of an invertible -frequently hypercyclic operator whose inverse is not -frequently hypercyclic, revealing new insights into the stability of hypercyclicity under inversion.

Findings

Existence of invertible -frequently hypercyclic operators on ^p

Inverse operators may not be -frequently hypercyclic

Highlights complexity of hypercyclicity properties under inversion

Abstract

We show that there exists an invertible $\mathcal{U}$-frequently hypercyclic operator on $\ell^p(\mathbb{N})$ ($1\le p <\infty$) whose inverse is not $\mathcal{U}$-frequently hypercyclic.