J-class operators and hypercyclicity

TL;DR

This paper introduces and studies J-class operators, a localized variant of hypercyclic operators, exploring their properties, examples, and analogues to classical hypercyclicity results, even in non-separable spaces.

Contribution

It defines J-class operators as a new concept in linear dynamics, extending hypercyclicity notions and providing characterizations and examples, including in non-separable spaces.

Findings

J-class operators generalize hypercyclic operators.

Many hypercyclic results have analogues for J-class operators.

Non-separable spaces like l^∞ support J-class operators.

Abstract

The purpose of the present work is to treat a new notion related to linear dynamics, which can be viewed as a "localization" of the notion of hypercyclicity. In particular, let $T$ be a bounded linear operator acting on a Banach space $X$ and let $x$ be a non-zero vector in $X$ such that for every open neighborhood $U\subset X$ of $x$ and every non-empty open set $V\subset X$ there exists a positive integer $n$ such that $T^{n}U\cap V\neq\emptyset$. In this case $T$ will be called a $J$-class operator. We investigate the class of operators satisfying the above property and provide various examples. It is worthwhile to mention that many results from the theory of hypercyclic operators have their analogues in this setting. For example we establish results related to the Bourdon-Feldman theorem and we characterize the $J$-class weighted shifts. We would also like to stress that even non-separable Banach spaces which do not support topologically transitive operators, as for example $l^{\infty}(\mathbb{N})$, do admit $J$-class operators.