Norm convergence of some power series of operators in $L^p$ with applications in ergodic theory

Christophe Cuny¹

¹ Équipe ERIM University of New Caledonia B.P. R4 98800 Nouméa, New Caledonia

Studia Mathematica, Tome 200 (2010) no. 1, pp. 1-29

Résumé

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.

Détail

DOI : 10.4064/sm200-1-1

Keywords: closed subspace where arbitrary measure infty invertible operator sup mathbb infty motivated applications ergodic theory obtain optimal conditions convergence series sum alpha alpha terms cdots n generalizing results unitary normal operators proofs make spectral integration initiated berkson gillespie particularly results paper berkson bourgain gillespie