Effective one-particle hamiltonian from many-body Green function

Consider a many-body Green function in terms of localized orbitals, $G_{mu nu}(omega).$ The underlying hamiltonian is something like (I will write it with a great generality and omitting spin, this could be many particular things, of course):
$$H = sum t_{mu nu} c^dagger_{mu} c_{nu} + text{h.c} + sum_{mu nu alpha beta} U_{mu nu alpha beta}c^dagger_{mu} c^dagger_{alpha}c_{nu}c_{beta}. $$
I was toying with the object
$$h(omega)= omega-G^{-1}(omega),$$
where $G(omega)$ is the matrix with coefficients $G_{mu nu}(omega).$ In a one-electron problem, this would be exactly the hamiltonian. My question then is, is this object, $h(omega)$, something in the many-body case?
In particular, I was interested in the following problem: Assume that we now connect our system to a different system, described by a one particle hamiltonian, $h_m$, and that the interaction between the old system and the new one is decribed also by a one-particle hamiltonian, $h_{text{int}}.$
Then, an effective one-particle hamiltonian of the whole system should be the block matrix:

$$begin{pmatrix}
h(omega) & h_text{int}
h_text{int}^prime & h_m
end{pmatrix}$$

Call this matrix $h_text{new}.$ One of my questions is if now the many-body Green function of the whole new system could be calculated as:
$$G_text{new}(omega) = (omega-h_text{new}(omega)^{-1},$$
but I haven’t been able to prove it (is it probably false in general?)

It would be interesting if that true, for, in that case, elementary algebra for block diagonalization would lead to the fact that the new Green function in the diagonal sites of the old system is:
$$left(omega-h(omega)- h_text{int}^prime g_m(omega)h_text{int} right)^{-1},$$
a result that would be interesting to formulate perturbation theory when the unperturbed system for which we know a exact solution is (rather unusually) a many-body hamiltonian, and the perturbation has the form of a one-particle hamiltonian.

Edit: This is closely related to this question I asked here some time ago, and which didn’t receive answers.