Let $E$ be a locally compact Hausdorff space.
I want to show that a linear operator $(\mathcal D(A),A)$ on $C_0(E)$$^1$ is closable and the closure $(\mathcal D(\overline A),\overline A)$ is the generator of a Feller$^2$ semigroup on $C_0(E)$ if and only if
- $\mathcal D(A)$ is dense;
- $(\mathcal D(A),A)$ satisfies the nonnegative maximum principle, i.e. $$\forall f\in\mathcal D(A):\forall x_0\in E:f(x_0)=\sup_{x\in E}f(x)\ge0\Rightarrow (Af)(x_0)\le0;\tag1$$
- $A_\lambda:=\lambda\operatorname{id}_{\mathcal D(A)}-A$ has dense range for some $\lambda>0$; and
Now, I know that if $F$ is a $\mathbb R$-Banach space, a linear operator $(\mathcal D(B),B)$ on $F$ is closable and $(\mathcal D(\overline B),\overline B)$ is the generator of a strongly continuous contraction semigroup on $F$ if and only if
- $\mathcal D(B)$ is dense;
- $\mathcal D(B)$ is dissipative; and
- $B_\lambda\mathcal D(B)$ is dense for some (and hence all) $\lambda>0$.
This is the Lumer-Phillips theorem. So, the desired claim seems to be a simple reformulation of this equivalence. I know that a linear operator on $C_0(E)$ satisfying the nonnegative maximum principle $(1)$ is dissipative. On the other hand, the generator of a contractive nonnegativity preserving semigroup on $C_0(E)$ satisfies the nonnegative maximum principle.
So, it seems like (please tell me if anything is wrong) the only missing piece is the sub-Markovity. How can we embed this into the equivalence?
$^1$ $C_0(E)$ denotes the space of continuous functions $E\to\mathbb R$ vanishing at infinity equipped with the supremum norm.
$^2$ A semigroup $(T(t))_{t\ge0}$ is called Feller if it is contractive (i.e. $\left\|T(t)\right\|_{C_0(E)}\le1$ for all $t\ge0$), sub-Markov (i.e. $0\le T(t)f\le 1$ for all $f\in C_0(E)$ with $0\le f\le1$) and $$(T(t)f)(x)\xrightarrow{t\to0}f(x)\;\;\;\text{for all }x\in\mathbb R\text{ and }f\in C_0(\mathbb R)\tag4.$$
$^3$ $1$ stands for the function $E\to\mathbb R$ which is constantly $1$ here.