On the existence of J-class operators

In this note we answer in the negative the question raised by G.Costakis and A.Manoussos, whether there exists a J-class operator on every non-separable Banach space. In par- ticular we show that there exists a non-separable Banach space constructed by A.Arvanitakis, S.Argyros and A.Tolias such that the J-set of every operator on this space has empty interior for each non-zero vector. On the other hand, on non-separable spaces which are reflexive there always exist a J-class operator.