Further results on $p$-adic semigroup of contractions

Let $E$ be an ultrametric Banach space over ${\mathbb{C}}_p\mathrm{.}$ In this note, we obtain a new necessary and sufficient conditions on the resolvent of $A$ such that for each $m\in \mathbb{N},\mathrm{\parallel }{A}^m\mathrm{\parallel }\le 1.$