Weakly Sequentially Recurrent Shifts Operators

Abstract

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.