On convergence of power series of $L_p$ contractions
Volume 112 / 2017
Abstract
Let $T$ be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the power series $\sum_{k=0}^\infty \beta_k T^k x$ when $\{\beta_k\}$ is a Kaluza sequence with divergent sum such that $\beta_k \to 0$ and $\sum_{k=0}^\infty \beta_k z^k$ converges in the open unit disk. We prove that weak and strong convergence are equivalent, and in a reflexive space also $\sup_n \|\sum_{k=0}^n \beta_k T^k x\| \lt \infty$ is equivalent to the convergence of the series. The last assertion is proved also when $T$ is a mean ergodic contraction of $L_1$.
For normal operators on a Hilbert space we obtain a spectral characterization of the convergence of $\sum_{n=0}^\infty \beta_n T^n x$, and a sufficient condition expressed in terms of norms of the ergodic averages, which in some cases is also necessary.
For $T$ Dunford–Schwartz of a $\sigma$-finite measure space or a positive contraction of $L_p$, ${1 \lt p \lt \infty}$, we prove that when $\{\beta_k\}$ is also completely monotone (i.e. a Hausdorff moment sequence) and $\beta_k=O(1/k)$, the norm convergence of $\sum_{k=0}^\infty \beta_k T^k f$ implies a.e. convergence.
For $T$ a positive contraction of $L_p$, $p \gt 1$, $f \in L_p$ and $\beta\in\mathbb R$, we show that if the series $\sum_{n=0}^\infty \frac{(\log(n+1))^\beta}{(n+1)^{1-1/r}}T^n f$ converges in $L_p$-norm for some $r\in (\frac{p}{p-1},\infty]$, then it converges a.e.
