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Let $(G, \mathcal D)$ be a densely defined operator on $C_0$ (continuous functions vanishing at infinity on some nice topological space) whose closure $\bar G$ generates a Feller semigroup and let $X$ be a Markov process corresponding to it.

Assume that $\mathcal D\subset C_K$ (continuous functions with compact support), that $G(\mathcal D)\subset C_K$ and that $G$ is symmetric with respect to a Radon measure $m$(Edit: with full support, but not necessarily finite), i.e.

$\int Gf\ g \ dm = \int f \ Gg\ dm$ for every $f,g\in \mathcal D$.

I guess that the Dirichlet form $\mathcal E$ of $X$ (defined as in the book of Fukushima/Oshima/Takeda by using the transition kernel, see (1.4.13) on p.30 in the last edition) is given by the closure of

$\mathcal D\ni f,g \mapsto \int Gf\ g dm$.

In other terms the Friedrichs extension of $G$ in $L^2(dm)$ should be the generator of the $L^2$ semigroup induced by $X$. (Edit: by $L^2$ semigroup induced by $X$ I mean the semigroup corresponding to the Dirichlet form $\mathcal E$ )

Is this true? I didn't find a reference nor a simple argument for showing this.

Or is it possible that a selfadjoint extension other than the Friedrichs one generates the $L^2$ semigroup induced by $X$?

Edit: From the answer of Byron Schmuland it is clear to me that the guess is true if $m$ is finite (this happens for example when the state space is compact)compact. Observe that in this case $G$ is essentially selfadjoint in $L^2$, so the Friedrichs extension is just the closure of $G$ and there are no other selfadjoint extensions. I'm still confused about the case of $m$ being not finitenoncompact state space. I would also appreciate partial answers which work for some concrete example of $G$ (say elliptic partial differential operators, or discrete operators).

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Let $(G, \mathcal D)$ be a densely defined operator on $C_0$ (continuous functions vanishing at infinity on some nice topological space) whose closure $\bar G$ generates a Feller semigroup and let $X$ be a Markov process corresponding to it.

Assume that $\mathcal D\subset C_K$ (continuous functions with compact support), that $G(\mathcal D)\subset C_K$ and that $G$ is symmetric with respect to a Radon measure $m$(not m$(Edit: with full support, but not necessarily finite), i.e.$\int Gf\ g \ dm = \int f \ Gg\ dm$for every$f,g\in \mathcal D$. I guess that the Dirichlet form$\mathcal E$of$X$(defined as in the book of Fukushima/Oshima/Takeda by using the transition kernel, see (1.4.13) on p.30 in the last edition) is given by the closure of$\mathcal D\ni f,g \mapsto \int Gf\ g dm$. In other terms the Friedrichs extension of$G$in$L^2(dm)$should be the generator of the$L^2$semigroup induced by$X$. (Edit: by$L^2$semigroup induced by$X$I mean the semigroup corresponding to the Dirichlet form$\mathcal E$) Is this true? I didn't find a reference nor a simple argument for showing this. Or is it possible that a selfadjoint extension other than the Friedrichs one generates the$L^2$semigroup induced by$X$? Edit: From the answer of Byron Schmuland it is clear to me that the guess is true if$m$is finite (this happens for example when the state space is compact). Observe that in this case$G$is essentially selfadjoint in$L^2$, so the Friedrichs extension is just the closure of$G$and there are no other selfadjoint extensions. I'm still confused about the case of$m$being not finite. I would also appreciate partial answers which work for some concrete example of$G$(say elliptic partial differential operators, or discrete operators). 4 small spelling correction: it's Friedrichs not Friedrich. Let$(G, \mathcal D)$be a densely defined operator on$C_0$(continuous functions vanishing at infinity on some nice topological space) whose closure$\bar G$generates a Feller semigroup and let$X$be a Markov process corresponding to it. Assume that$\mathcal D\subset C_K$(continuous functions with compact support), that$G(\mathcal D)\subset C_K$and that$G$is symmetric with respect to a Radon measure$m$(not necessarily finite), i.e.$\int Gf\ g \ dm = \int f \ Gg\ dm$for every$f,g\in \mathcal D$. I guess that the Dirichlet form of$X$(defined as in the book of Fukushima/Oshima/Takeda by using the transition kernel, see (1.4.13) on p.30 in the last edition) is given by the closure of$\mathcal D\ni f,g \mapsto \int Gf\ g dm$. In other terms the Friedrich Friedrichs extension of$G$in$L^2(dm)$should be the generator of the$L^2$semigroup induced by$X$. Is this true? I didn't find a reference nor a simple argument for showing this. Or is it possible that a selfadjoint extension other than the Friedrich Friedrichs one generates the$L^2$semigroup induced by$X$? 3 Added another question at the end. 2 I added the assumption that the range of G is ocntained in$\mathcal D$so that$Gf\$ is in L^2(dm) and everything is well defined.
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