A uniform ergodic theorem for some N\"orlund means

TL;DR

This paper establishes a uniform ergodic theorem for N"orlund means involving a sequence s and a bounded linear operator T, characterizing convergence conditions in terms of the resolvent set of T.

Contribution

It provides a new uniform ergodic theorem for N"orlund means with specific conditions on the sequence s and the operator T, extending classical ergodic results.

Findings

Convergence of N"orlund means depends on the spectral properties of T.

If T^n/s(n) converges to zero, the averages converge uniformly if 1 is in the resolvent set or a simple pole.

The theorem applies to sequences s with specific growth and summability conditions.

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 $T$.