On the (C,α) uniform ergodic theorem
We improve a recent result of T. Yoshimoto about the uniform ergodic theorem with Cesàro means of order α. We give a necessary and sufficient condition for the (C,α) uniform ergodicity with α > 0. Introduction. In his classical paper [D], N. Dunford obtained several theorems about convergence of (fn(T ))n∈N, where T is a bounded linear operator on a Banach space and (fn)n∈N is a sequence of complex-valued functions, each of which is holomorphic on some open neighborhood of σ(T ). Different kinds of convergence (namely, convergence in B(X), strong and weak convergence) were treated. In connection with this, E. Hille [H] obtained, as an application of Abelian and Tauberian theorems, the uniform ergodic theorem as stated below with a view to relating the (C,α) ergodic theorem for an operator T and the properties of the resolvent R(·, T ). Theorem A (Hille [H, Theorem 6]). Let X be a Banach space and T ∈ B(X). A necessary condition for the existence of an operator E ∈ B(X) such that , for some fixed α > 0,