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A Uniform Ergodic Theorem for Some Norlund Means

Commun. Math. Anal.
Volume 21, Number 2 (2018), 1 - 34

A Uniform Ergodic Theorem for Some
Norlund Means

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Abstract

We obtain a uniform ergodic theorem for the sequence $\frac1{s(n)} \sum_{k=0}^n(\varDelta s)(n-k)\,T^k$, where $\varDelta$ 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 $\lim_{n\rightarrow+\infty} s(n+1)/s(n)=1$ and $\varDelta^qs\in\ell_1$ 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 $T$.