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.