Advanced Search

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.

For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by $M{ₙ}^{\alpha }={\left(A{ₙ}^{\alpha }\right)}^{-1}{\sum }_{k=0}^n{A}_{n-k}^{\alpha -1}{T}^k$. For α = 1, we find $Mₙ¹={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$, the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.