PUB - Publikationen an der Universität Bielefeld

We generalize the concept of coarse hypercyclicity, introduced by Feldman in\cite{Fe1}, to that of coarse topological transitivity on open cones. We showthat a bounded linear operator acting on an infinite dimensional Banach spacewith a coarsely dense orbit on an open cone is hypercyclic and a coarselytopologically transitive (mixing) operator on an open cone is topologicallytransitive (mixing resp.). We also "localize" these concepts by introducing twonew classes of operators called coarsely $J$-class and coarsely $D$-classoperators and we establish some results that may make these classes ofoperators potentially interesting for further studying. Namely, we show that ifa backward unilateral weighted shift on $l^2(\mathbb{N})$ is coarsely $J$-class(or $D$-class) on an open cone then it is hypercyclic. Then we give an exampleof a bilateral weighted shift on $l^{\infty}(\mathbb{Z})$ which is coarsely$J$-class, hence it is coarsely $D$-class, and not $J$-class. Note that,concerning the previous result, it is well known that the space$l^{\infty}(\mathbb{Z})$ does not support $J$-class operators, see\cite{CosMa2}. Finally, we show that there exists a non-separable Banach spacewhich supports no coarsely $D$-class operators on open cones. Some openproblems are added.