Epsilon-hypercyclic operators

Abstract

For each fixed number $\varepsilon$ in $(0, 1)$ we construct a bounded linear operator on the Banach space $\ell_1$ having a certain orbit which intersects every cone of aperture $\varepsilon$, but with every orbit avoiding a certain ball of radius $d$, for every $d>0$. This answers a question from [8]. On the other hand, if $T$ is an operator on the Banach space $X$ such that for every $\varepsilon>0$ there is a point in $X$ whose orbit under the action of $T$ meets every cone of aperture $\varepsilon$, then $T$ has a dense orbit.