Sums of Weakly Sequentially Recurrent Operators

Abstract

An operator $$T$$ in a Banach space $$X$$ is said to be recurrent if the set $$\{x\in X:\ x\in \overline{O(T,Tx)}\}$$ is dense in $$X$$ . The operator $$T$$ is said to be weakly sequentially recurrent if the set $$\{x\in X:\ x\in \overline{O(T,Tx)}^w\}$$ 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.