On the uniform ergodic theorem for some Norlund means

We obtain a uniform ergodic theorem for the sequence $ rac1{s(n)} sum_{k=0}^n(arDelta s)(n-k),T^k$, where $arDelta$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence of its partial sums, $T$ is a bounded linear operator on a Banach space and $s$ is a divergent nondecreasing sequence of strictly positive real numbers, such that $lim_{n ightarrow+infty} s(n+1)/s(n)=1$ and $arDelta^qsinell_1$ for some positive integer $q$. Indeed, we prove that if $T^n/s(n)$ converges to zero in the uniform operator topology, then the sequence of averages above converges in the same topology if and only if 1 is either in the resolvent set of $T$, or a simple pole of the resolvent function of $T$.