The asymptotic behaviour of the Ces\`aro operator

We study the asymptotic behaviour of orbits (Tⁿx)n≥0 of the classical Ces\`aro 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∈ 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 Ces\`aro operator defined on spaces of continuous functions, establishing new and improved results in this setting, too.