$$D_{F}(x,y) = <0|T\{\phi_{0}(x) \phi_{0}(y)\}|0> $$

is the Green's function of the operator (except maybe for a constant):

$$ (\Box + m^2)$$

In other words:

$$ (\Box + m^2) D_{F}(x,y) = - i \hbar \delta^{4}(x-y)$$

My question is:

Which is the operator that corresponds to:

$$<\Omega |T\{\phi(x) \phi(y)\}|\Omega> $$

being the Green's function (understood as "operator times Green's function equals to delta") in QFT?

I assume that the answer to my question has something to do with the Schwinger-Dyson equations, but I cannot find it out.