ON THE EXISTENCE OF J-CLASS OPERATORS ON BANACH SPACES

In this paper we answer in the negative the question raised by C. Costakis and A. Manoussos whether there exists a J-class operator on every non-separable Banach space. In particular we show that there exists a non-separable Banach space constructed by S. Argyros, A. Arvanitakis and A. To has 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 exists a J-class operator.