Let $\mathbb{D}$ be an self-adjoint elliptic operator of a compact manifold and $G(x,y)$ the Green's function of $\mathbb{D}$. By definition $G(x,y)$ is the right inverse of $\mathbb{D}$ in the sense that $$ \mathbb{D} \int G(x,y)f(y)dy=f(x) $$ for any function $f$. What can we say about the left inverse of the Green's function? The naive guess is that $$ \int G(x,y)\mathbb{D}f(y)dy=f(x), $$ but I think this is not true in general as $f$ can be in the kernel of $\mathbb{D}$. My question is, what can we say about the left inverse of the Green's function?

I asked a simpler question in this post with no answer so far. In this case, it seems to me that the integration-by-part argument still works assuming that $c$ is not in the eigenvalue of the operator. Does this generalize to arbitrary elliptic operators?

Lastly, I would appreciate it if someone could suggest a good textbook to learn this kind of materials on the elliptic operators and Green's functions on compact manifolds.