An epsilon-hypercyclicity criterion and its application on classical Banach spaces

TL;DR

This paper introduces a new criterion for epsilon-hypercyclicity and extends existing methods to construct such operators in classical Banach spaces, broadening the scope of hypercyclicity theory.

Contribution

It develops an epsilon-hypercyclicity criterion and applies it to construct operators in classical Banach spaces that are epsilon-hypercyclic but not hypercyclic.

Findings

Established a new epsilon-hypercyclicity criterion.

Constructed epsilon-hypercyclic operators in classical Banach spaces.

Extended hypercyclicity concepts to broader classes of Banach spaces.

Abstract

We provide a criterion for $\varepsilon$-hypercyclicity. Also, we extend the ideas of Badea, Grivaux, M\"uller and Bayart to construct $\varepsilon$-hypercyclic operators which are not hypercyclic in a wider class of separable Banach spaces, including several classical Banach spaces. For instance, our result can be applied to separable infinite dimensional $L^p$ spaces and $C(K)$ spaces.