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.


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


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 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 noncompact state space. I would also appreciate partial answers which work for some concrete example of $G$ (say elliptic partial differential operators, or discrete operators). 



Symmetric Feller processes and Dirichlet forms
I've decided to post an incomplete preliminary answer. 


I ran into your problem when I was writing [1]. On page
258 you will see my resolution. 


I should point out that in my case, the underlying space
$X$ was compact, and that $m$ was a finite measure with full
support. Thus, $C(X)$ embeds into $L^2(X;m)$ with a continuous, 
 linear injection in the obvious way. This may not hold in the 
locally compact case, and I'm not sure how serious a problem that is.


Translated into your notation,
 and letting $\tilde G$ be the Friedrichs extension we note 
that $\bar G$ and $\tilde G$ agree on $\cal D$
and so the resolvent operators $\bar R\_\lambda$ and $\tilde R\_\lambda$
 agree on $(\lambda-G)({\cal D})$. We deduce that $\bar R\_\lambda= \tilde R\_\lambda$
on $C(X)$ and using the Yosida approximation conclude the same 
about the semigroup operators $\bar T\_t$ and $\tilde T\_t$.


I hope this is of some help. If anything is unclear, let me know.


[1] A result on the infinitely many neutral alleles diffusion model. 
*Journal of Applied Probability* 28, 253-267 (1991). 


