A generalization of the uniform ergodic theorem to poles of arbitrary order

We obtain a generalization of the uniform ergodic theorem to the sequence ${\displaystyle (1/n^p){⅀}_{k=0}^{n-1}{T}^k}$, where T is a bounded linear operator on a Banach space and p is a positive integer. Indeed, we show that uniform convergence of the sequence above, together with an additional condition which is automatically satisfied for p = 1, is equivalent to 1 being a pole of the resolvent of T plus convergence to zero of ${\displaystyle \parallel T^n\parallel /n^p}$. Furthermore, we show that the two conditions above, together, are also equivalent to 1 being a pole of order less than or equal to p of the resolvent of T, plus a certain condition ℇ(k,p), which is less restrictive than convergence to zero of ${\displaystyle \parallel T^n\parallel /n^p}$ and generalizes the condition (called condition (ℇ-k)) introduced by K. B. Laursen and M. Mbekhta in their paper [LM2] (dealing with the case p=1).