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