Journal of Operator Theory

Volume 67, Issue 1, Winter 2012 pp. 101-119.

$J$-class operators and hypercyclicity

Authors: George Costakis (1) and Antonios Manoussos (2)
Author institution: (1) Department of Mathematics, University of Crete, Knossos Avenue, GR-714 09 Heraklion, Crete, Greece
(2) Fakultaet fuer Mathematik, SFB 701, Universitaet Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Summary: 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 some non-separable Banach spaces which do not support topologically transitive operators, as for example $l^{\infty}(\mathbb{N})$, do admit $J$-class operators.

Keywords: Hypercyclic operators, $J$-class operators, cyclic vectors, unilateral and bilateral weighted shifts

Contents Full-Text PDF