Norm convergence of some power series of operators in $L^p$ with applications in ergodic theory
Volume 200 / 2010
Studia Mathematica 200 (2010), 1-29
MSC: Primary 47B40, 37A30; Secondary 42B25, 42A45.
DOI: 10.4064/sm200-1-1
Abstract
Let $X$ be a closed subspace of $L^p(\mu)$, where $\mu$ is an arbitrary measure and $1< p< \infty$. Let $U$ be an invertible operator on $X$ such that $\sup_{n\in \mathbb{Z} }\|U^n\|< \infty$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $\sum_{n\ge 1} {(U^nf)}/{n^{1-\alpha}}$, $0\le \alpha < 1$, in terms of $\|f+\cdots +U^{n-1}f\|_p$, generalizing results for unitary (or normal) operators in $L^2(\mu)$. The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson–Bourgain–Gillespie.
