For $t \geq 0$ let $P_{t}(x,\cdot)=N(x,t)$ be the transition kernels of a one dimensional Brownian Motion. Then $(\bar{P}_{t})_{t \geq 0}$ with
$ \bar{P}_{t}:C_{0}(\mathbb{R},\mathbb{R}) \rightarrow C_{0}(\mathbb{R},\mathbb{R}), f \mapsto (x \mapsto \int f(y) P_{t}(x,dy)) $
is a $C_{0}$-semigroup. We define the generator L of $(\bar{P}_{t})_{t \geq 0}$ by
$ Lf=lim_{t \downarrow 0} \frac{\bar{P}_{t}f-f}{t} $
for $f \in D(L)=\{f \in C_{0}(\mathbb{R}):lim_{t \downarrow 0} \frac{\bar{P}_{t}f-f}{t} exists\}$. I already know that $C_{K}^{\infty}(\mathbb{R})$ is a core of L and that $Lf=\frac{1}{2}f''$ for $f \in C_{K}^{\infty}(\mathbb{R}).$
Question: How can one proof that $(\bar{P}_{t})_{t \geq 0}$ has the generator $Lf=\frac{1}{2}f''$ with $D(L)=\{f \in C_{0}(\mathbb{R}):f'' \in C_{0}(\mathbb{R})\}$.