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

TL;DR

This paper extends the Hille-Yoshida-Phillips theorem to discrete semigroups on complete ultrametric locally convex spaces over p-adic fields, providing conditions for the equicontinuity of operator powers.

Contribution

It establishes a necessary and sufficient condition on the resolvent of an operator for the equicontinuity of its powers in ultrametric locally convex spaces, generalizing classical semigroup theory.

Findings

Characterization of resolvent conditions for equicontinuity

Extension of semigroup theorems to p-adic ultrametric spaces

New criteria for operator power behavior in non-Archimedean analysis

Abstract

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.