Remarks on uniform ergodic theorems

Let T be a bounded linear operator on a Banach space X. In this paper we study uniform Cesaro ergodicity when T is not necessarily powerbounded, and relate it to the uniform convergence of the Abel averages. When X is over the complex field, we show that uniform Abel ergodicity is equivalent to the uniform convergence of the powers of all (one of) the Abel averages A_α, α ∈ (0, 1). This is equivalent to uniform Cesàro ergodicity of T when ∥Tⁿ∥/n → 0. For positive operators on real or complex Banach lattices, uniform Abel ergodicity is equivalent to uniform Cesaro ergodicity. An example shows that this is not true in general. For a C₀-semi-group {Tt}_t≥0 on X complex satisfying limt_t→∞/t = 0, we show that uniform ergodicity is equivalent to uniform convergence of (R)_n for every (one) > 0, where R is the resolvent family of the generator of the semi-group.