  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.
