For $-\Delta$ considered as a self-adjoint operator on $L^2(\mathbb{R}^d)$, one may write it's resolvent as the Fourier multiplier $g(\xi)=(|\xi|^2-z)^{-1}$. Now let $\Omega\subset \mathbb{R}^d$ be a bounded domain and let $-\Delta^\Omega$ be the Dirichlet Laplacian on $\Omega$, defined, for instance, via the Friedrichs Extension Theorem. Is there an expression for $(-\Delta^\Omega-z)^{-1}$ in terms of the operator $g(\xi)$ and some multiplication operator $f(x)$? If not, can we approximate $(-\Delta^\Omega-z)^{-1}$ by operators of the form $f(x)g(\xi)$? It seems that something like $f(x)g(\xi)f(x)$ might work for $f$ some bump function of $\Omega$, but the fact that $-\Delta^\Omega$ has to be defined abstractly leaves me unsure of the details.
My motivation is that I would like to prove that $-\Delta^\Omega$ has compact resolvent from the fact that an operator of the form $f(x)g(\xi)$ is compact if $f$ and $g$ vanish at $\infty$. Since one usually proves that the resolvent is compact from the Rellich embedding theorem and one can prove this theorem from the above compactness criterion, I have reason to believe this should work.
Try writing the Fourier multiplier as an expression involving the heat kernel: $$ \int_0^{\infty}e^{-t(|\xi|^2-z)}dt=\frac{1}{|\xi|^2-z}. $$