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2018 A Uniform Ergodic Theorem for Some Nörlund Means
Laura Burlando
Commun. Math. Anal. 21(2): 1-34 (2018).

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 $T$.

Citation

Download Citation

Laura Burlando. "A Uniform Ergodic Theorem for Some Nörlund Means." Commun. Math. Anal. 21 (2) 1 - 34, 2018.

Information

Published: 2018
First available in Project Euclid: 5 October 2018

zbMATH: 07002173
MathSciNet: MR3866091

Subjects:
Primary: 47A10, 47A35

Rights: Copyright © 2018 Mathematical Research Publishers

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Vol.21 • No. 2 • 2018
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