PUB - Publikationen an der Universität Bielefeld

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