Alpha-admissibility for Ritt operators

Let T : X --> X$ be a power bounded operator on Banach space. An operator C : X --> Y$iscalledadmissibleforTifitsatisfiesanestimate$\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 $(1-\mathrm{\mid }\omega {\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\text{norm}C(I-\omega T{)}^{-1}\le K$ for $\omega \in \text{Cdb}$, $\mathrm{\mid }\omega \mathrm{\mid }<1$. 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.