Alpha-admissibility for Ritt operators

Abstract

Let T : X --> X$ be a power bounded operator on Banach space. An operator C : X --> Y$ is called admissible for T if it satisfies an estimate $\sum_k\norm{CT^k(x)}^2\,\leq M^2\norm{x}^2$. Following Harper and Wynn, we study the validity of a certain Weiss conjecture in this discrete setting. We show that when X is reflexive and T is a Ritt operator satisfying a appropriate square function estimate, C is admissible for T if and only if it satisfies a uniform estimate $(1-| \omega|^2)^{1/2}\norm{C(I-\omega T)^{-1}}\,\leq K$ for $\omega\in \Cdb$, $|\omega|<1$. We extend this result to the more general setting of alpha-admissibility. Then we investigate a natural variant of admissibility involving R-boundedness and provide examples to which our general results apply.