On Nörlund summation and Ergodic Theory, with applications to power series of Hilbert contractions

Abstract

We show that if $\mathbf{a}=(a_n{)}_{n\in \mathbb{N}}$ is a good weight for the dominated weighted ergodic theorem in $L^p$, $p>1$, then the Nörlund matrix $N_{\mathbf{a}}=\{a_{i-j}/A_i{\}}_{0\le j\le i}$, $A_i={\sum }_{k=0}^i|a_k|$ is bounded on ${\ell }^p(\mathbb{N})$. We study the regularity (convergence in norm, almost everywhere) of operators in ergodic theory: power series of Hilbert contractions, and power series ${\sum }_{n\in \mathbb{N}}{a}_n{P}_{n}f$ of $L^2$-contractions, and establish similar tight relations with the Nörlund operator associated to the modulus coefficient sequence $(|a_n|{)}_{n\in \mathbb{N}}$.

arXiv comment13 pages