On the Speed of Convergence in the (C, α) Uniform Ergodic Theorem for Quasi-compact Operators

Let T be a quasi-compact operator on a complex Banach space X. We consider the Cesàro (C, α)-summability method (M(α)n, k) for a real α with 0 < α ≤ 1. Under the hypothesis that Tn/nα converges to zero in the weak operator topology as n approaches infinity, we prove that ∑∞k = 0M(α)n, kTk converges to a projection E(1; T) in the uniform operator topology as n approaches infinity and that [formula] where (B[X], || • ||B) is the Banach algebra of bounded linear transformations from X to X.