Weakly Sequentially Recurrent Shifts Operators

This paper studies the weakly sequentially recurrence property of shifts operators. In the case of $\ell^p(\mathbb{N})$, $1\leq p<\infty$, we show that the weak recurrence, recurrence, hypercyclicity, and weak hypercyclicity are equivalent. In the case of $\ell^\infty(\mathbb{N})$ (resp. $\ell^\infty(\mathbb{Z})$), we prove that the unilateral backward (resp. bilateral backward) can never be weakly sequentially recurrent.