Sums of Weakly Sequentially Recurrent Operators

An operator $T$ in a Banach space $X$ is said to be recurrent if the set
$$\begin{equation*} \{x\in X:\ x\in \overline{O(T,Tx)}\} \end{equation*}$$
is dense in $X$. The operator $T$ is said to be weakly sequentially recurrent if the set
$$\begin{equation*} \{x\in X:\ x\in \overline{O(T,Tx)}^w\} \end{equation*}$$
is weakly dense in $X$. Costakis et al. [Complex Anal. Oper. Theory 8 (8), 1601–1643] ask if $T\oplus T$ should be recurrent whenever so is $T$. This question has been answered negatively by Grivaux et al. [arXiv: 2212.03652]. In this paper, we prove the existence of an operator $T$ weakly sequentially recurrent such that $T\oplus T$ is not.