I am trying to prove that $P_t := e^{\lambda t (P-I)}$ (where $Pf:= \int f(y) P(\cdot , dy)\in \mathcal{C}_0(\mathbb{R}^d)$, for $f\in \mathcal{C}_0(\mathbb{R}^d)$, $P$ being a probability kernel), is a strongly-continuous contraction semigroup (on $(\mathcal{C}_0(\mathbb{R}^d),||\, .\, ||_{\infty})$).
It remains for me to prove that it is a contraction : $||P_t(f)||_{\infty}\leq || f||_{\infty}$.
At best I could find that $$||P_t(f)||_{\infty}\leq e^{|\lambda t | ||P-I ||}||f||_{\infty} $$ and $||P-I||\leq 2$, wich is not sufficient... (since, of course, in all case $e^{|\lambda t | ||P-I ||}\geq 1$)
