On some smoothening effects of the transition semigroup of a L\'evy process

Let $(P_t)$ be the transition semigroup of a L\'evy process $L$ taking values in a Hilbert space $H$. Let $\nu$ be the L\'evy measure of $L$. It is shown that for any bounded and measurable function $f$, $$ \int_H\left\vert P_tf(x+y)-P_tf(x)\right\vert ^2 \nu (\dif y)\le \frac 1 t P_tf^2(x) \qquad \text{for all $t>0$, $x\in H$.} $$ As $\nu$ can be infinite this formula establishes some smoothening effect of the semigroup $(P_t)$. In the paper some applications of the formula will be presented as well.