A Hille-Yoshida-Phillips theorem for discrete semigroups on complete ultrametric locally convex spaces

Let $E$ be a complete Hausdorff locally convex space over $\mathbb{C}_{p},$ let $A\in\mathcal{L}(E)$ such that $(I-\lambda A)^{-1}$ is analytic on its domain. In this paper, we give a necessary and sufficient condition on the resolvent of $A$ such that $(A^{n})_{n\in\mathbb{N}}$ is equi-continuous.