On convergence of power series of $L_p$ contractions

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