## Communications in Mathematical Analysis

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

### A Uniform Ergodic Theorem for Some Nörlund Means

#### 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 $T$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 5 October 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.cma/1538704836

**Mathematical Reviews number (MathSciNet)**

MR3866091

**Zentralblatt MATH identifier**

07002173

**Subjects**

Primary: 47A35: Ergodic theory [See also 28Dxx, 37Axx] 47A10: Spectrum, resolvent

**Keywords**

bounded linear operators uniform ergodic theorem Nörlund means of operator iterates spectrum poles of the resolvent concave real sequences least concave majorant of a real sequence

#### Citation

Burlando, Laura. A Uniform Ergodic Theorem for Some Nörlund Means. Commun. Math. Anal. 21 (2018), no. 2, 1--34. https://projecteuclid.org/euclid.cma/1538704836