On the existence of J-class operators

TL;DR

This paper demonstrates that not all non-separable Banach spaces admit J-class operators, providing examples where the J-set of every operator has empty interior, but such operators exist on reflexive non-separable spaces.

Contribution

It answers a longstanding question by showing that some non-separable Banach spaces lack J-class operators, contrasting with the existence of such operators on reflexive non-separable spaces.

Findings

Certain non-separable Banach spaces have no J-class operators.

On reflexive non-separable spaces, J-class operators always exist.

The J-set of every operator can have empty interior in specific non-separable spaces.

Abstract

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.