The asymptotic behaviour of the Cesàro operator

We study the asymptotic behaviour of orbits $(T^{n}x{)}_{n\ge 0}$ of the classical Cesàro operator $T$ for sequences $x$ in the Banach space $c$ of convergent sequences. We give new non-probabilistic proofs, based on the Katznelson-Tzafriri theorem and one of its quantified variants, of results which characterise the set of sequences $x\in c$ that lead to convergent orbits and, for sequences satisfying a simple additional condition, we provide a rate of convergence. These results are then shown, again by operator-theoretic techniques, to be optimal in different ways. Finally, we study the asymptotic behaviour of the Cesàro operator defined on spaces of continuous functions, establishing new and improved results in this setting, too.