  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.
