Almost everywhere convergence and boundedness of cesàro-$\alpha$ ergodic averages

Abstract

We study the almost everywhere convergence of the ergodic Cesàro-$\alpha$ averages $R_{n,\alpha}f = \frac{1}{A^{\alpha}_{n}}\Sigma^{n}_{i=0}{A^{\alpha-1}_{n-i}T^{i}f}$ and the boundedness of the ergodic maximal operator $M_{\alpha}f = \mathrm{sup}_{n \in \mathbb{N}}|R_{n, \alpha}f|$, associated with a positive linear operator $T$ with positive inverse on some $L^{p}(\mu)$, $1 \lt p \lt \infty$, $0 \lt \alpha \leq 1$.