Further results on $p$-adic semigroup of contractions

TL;DR

This paper establishes new necessary and sufficient conditions on the resolvent of an operator in a $p$-adic ultrametric Banach space to ensure its powers are uniformly bounded by 1, advancing understanding of $p$-adic contraction semigroups.

Contribution

It provides novel criteria linking the resolvent of an operator to the boundedness of its powers in $p$-adic ultrametric Banach spaces.

Findings

Derived new resolvent conditions for contractions

Characterized when powers of operators are bounded by 1

Enhanced understanding of $p$-adic semigroup behavior

Abstract

Let $E$ be an ultrametric Banach space over $\mathbb{C}_{p}.$ In this note, we obtain a new necessary and sufficient conditions on the resolvent of $A$ such that for each $m\in\mathbb{N},\;\|A^{m}\|\leq1.$