In Villani's Hypocoercity, I am faced with proving a statement which is indicated by the title.
Consider a $C^\infty$ function $V: \mathbb{R}^n \to \mathbb{R}$, convergeing to $+\infty$ fast enough at infinity. For $x, v \in \mathbb{R}^n \times \mathbb{R}^n$, set
$$ f_\infty(x,v) := \frac{e^{[V(x) + \frac{\vert v \vert^2}{2}]}}{Z}, \quad \mu(dxdv) = f_\infty(x,v)dxdv, $$ where $Z$ is chosen in such a way that $\mu$ is a probability measure. Define
$$ \mathcal{H}:= L^2(\mu),\quad L := -\Delta_v + v\cdot \nabla_v + v\cdot \nabla_x - \nabla V(x)\cdot \nabla_v $$
where $L$ is an unbounded operator on $\mathcal{H}$ with its domain $D(L)\subset \mathcal{H}$.
The associated equation is the kinetic Fokker-Planck equation with confinement potential $V$, in the form $$ \partial_t h + Lh = 0 \cdots (*) $$
It is known that the above equation $(*)$ generates a $C^\infty$ regularizing contraction semigroup in $L^2(\mu)$.
Later in the book, it says since $Lh \in \mathcal{K}'$ for $h \in \mathcal{K}' \cap \mathcal{S}$, $L$ leaves $\mathcal{K}'$ invariant and therefore so does $e^{-tL}$. Here, $\mathcal{K}'$ is some specific subspace of $\mathcal{H}$ and $\mathcal{S}$ is a dense topological vector space of $\mathcal{H}$, for example, the Schwartz space.
(1) I really don't understand why the invariance of $\mathcal{K}'$ under $L$ leads to that of $\mathcal{K}'$ under $e^{-tL}$.
(2) Also, I am wondering if $e^{-tL}$ leaves the Schwartz space $\mathcal{S}$ invariant, too.
I'd really appreciate it if you'd help me. Thanks!