This is an example from Davies' One-Parameter Semigroups.
Define the bounded operators $T_t$ on $L^p(\mathbb{R}^n)$ by $$(T_tf)(x)=(2\pi t)^{-n/2} \int_{\mathbb{R}^n} e^{-(x-y)^2/2t}f(y)dy$$ then one may easily show that $T_t$ is a one-parameter semigroup on $L^p(\mathbb{R}^n)$. By using the invariance of $\mathcal{S}$ under Fourier transforms one sees that $T_t(\mathcal{S})\subset \mathcal{S}$ for all $t\ge 0$, where $\mathcal{S}$ is the Schwartz Space. Then it says that $$(Zf)(x)=\frac{1}{2}\Delta f(x)$$ for all $f\in \mathcal{S}$.
I don't understand how to get this last statement. How can we show that the infinitesimal generator of $f$ becomes half the Lapalcian? I would greatly appreciate any help.