On N\"orlund summation and Ergodic Theory, with applications to power series of Hilbert contractions

We show that if ${ \bf a}=(a_n)_{ n\in \N}$ is a good weight for the dominated weighted ergodic theorem in $L^p$, $p>1$, then the N\"orlund 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 $\ell^p(\N)$. 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 \N} a_nP_nf $ of $L^2$-contractions, and establish similar tight relations with the N\"orlund operator associated to the modulus coefficient sequence $(|a_n|)_{ n\in \N}$.

Categories: math.FA