Coarse topological transitivity on open cones and coarsely J-class and D-class operators

We generalize the concept of coarse hypercyclicity, introduced by Feldman in [13], to that of coarse topological transitivity on open cones. We show that a bounded linear operator acting on an infinite dimensional Banach space with a coarsely dense orbit on an open cone is hypercyclic and a coarsely topologically transitive (mixing) operator on an open cone is topologically transitive (mixing resp.). We also “localize” these concepts by introducing two new classes of operators called coarsely J-class and coarsely D-class operators and we establish some results that may make these classes of operators potentially interesting for further studying. Namely, we show that if a backward unilateral weighted shift on l2(N) is coarsely J-class (or D-class) on an open cone then it is hypercyclic. Then we give an example of a bilateral weighted shift on l∞(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∞(Z) does not support J-class bilateral weighted shifts, see [10]. Finally, we show that there exists a non-separable Banach space which supports no coarsely D-class operators on open cones. Some open problems are added.