alpha-Admissibility for Ritt Operators

Let be a power bounded operator on Banach space. An operator is called admissible for if it satisfies an estimate . Following Harper and Wynn, we study the validity of a certain Weiss conjecture in this discrete setting. We show that when is reflexive and is a Ritt operator satisfying a appropriate square function estimate, is admissible for if and only if it satisfies a uniform estimate for , . We extend this result to the more general setting of -admissibility. Then we investigate a natural variant of admissibility involving -boundedness and provide examples to which our general results apply.