On the Uniform Kreiss Resolvent Condition

Let $B$ be a Banach space with norm ${\|\cdot\|}$ and identity operator $I$. We prove that, for a bounded linear operator $T$ in $B$, the strong Kreiss resolvent condition
$$ \|(T-\lambda I)^{-k}\|\le\frac{M}{(|\lambda|-1)^k},\qquad|\lambda|>1,\ k=1,2,\dots, $$
implies the uniform Kreiss resolvent condition
$$ \bigg\|\sum_{k=0}^n \frac{T^k}{\lambda^{k+1}}\bigg\|\le\frac{L}{|\lambda|-1},\qquad|\lambda|>1,\ n=0,1,2,\dotsc. $$
We establish that an operator $T$ satisfies the uniform Kreiss resolvent condition if and only if so does the operator $T^m$ for each integer $m\ge 2$.