On Nörlund summation and ergodic theory, with applications to power series of Hilbert contractions

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