If $P:E\rar E$ is a linear contraction, then
$$
\norm{e^{n(P-1)}x-P^nx}\leq\sqrt n\norm{(P-1)x}
$$
Let $N_t$ be a Poisson process with parameter $\l=1$. As $\E N_t=t$ and $\E N_t^2=t^2+t$ we infer that:
\begin{eqnarray*}
\tnorm{e^{n(P-1)}x-P^nx}
&\leq&\E\tnorm{P^{N_n}x-P^nx}
\leq\E|N_n-n|\norm{Px-x}\\
&\leq&(\E(N_n^2+n^2-2n N_n))^{\frac12}\norm{Px-x}
=\sqrt n\norm{Px-x}~.
\end{eqnarray*}