On the existence of \(J\)-class operators on Banach spaces (Q2845871)

Let \(X\) be a Banach space and \(L(X)\) the space of bounded linear operators acting on \(X\). We define the \(J\)-set of \(x\) under \(T\), \(J_T(x)\), as the set of elements \(y\in X\) such that there exists a strictly increasing sequence of natural numbers \((k_n)_n\) and a sequence \((x_n)_n\) in \(X\) such that \(\lim_{n\rightarrow \infty}x_n=x\) and \(\lim_{n\rightarrow \infty}T^{k_n}x_n=y\). If \(J_T(x)=X\) for some \(x\in X\setminus\{0\}\), then \(T\) is said to be a \(\mathcal{J}\)-class operator. On separable spaces, every hypercyclic operator is \(\mathcal{J}\)-class, but the converse does not hold. Nevertheless, the authors show that on non-separable spaces which are reflexive, there always exists a \(\mathcal{J}\)-class operator.NEWLINENEWLINENEWLINE\(\mathcal{J}\)-class operators were introduced by \textit{G. Costakis} and \textit{A. Manoussos} in [J. Oper. Theory 67, No. 1, 101--119 (1929; Zbl 1243.47021)]. In that paper, they raised the question whether there exists a \(\mathcal{J}\)-class operator on every non-separable Banach space. The authors give a negative question using a non-separable Banach space constructed by \textit{S. A. Argyros} et al. in [Lond. Math. Soc. Lect. Note Ser. 337, 1--90 (2006; Zbl 1131.46006)]. Furthermore, they even show the \(J\)-set of every operator on this space has empty interior for each non-zero vector.