In the proof of Theorem 1.6.1 of Tromba's Teichmüller Theory in Riemannian geometry, it says:
For $(M,g)$ a closed Riemann Surface of genus $g\ge 2$, the Fredholm alternative holds for the operator $$\Delta_g-id:H^s(M,\mathbb{R})\rightarrow H^{s-2}(M,\mathbb{R}).$$
And then the author goes on in showing its injectivity.
Similarly, in another paper but in the same setting, I have the operator $$\delta_g\delta_g^\ast:\mathfrak{X}_s(M)\rightarrow \mathfrak{X}_{s-2}(M),$$ $\mathfrak{X}_s(M)$ being $H^s$ vector fields, and again the author states that it has trivial kernel (which is easy to show) - and thus I assume that its surjectivity follows as above.
I am quite new to the subject, so I was wondering if you could explain why the Fredholm alternative applies to these particular operators, or provide a reference for this kind of problems.
Many thanks in advance
It's not $\Delta$ which is compact but the map $I_u:H^1\rightarrow (H^1)^*$ given by $$I_u(v): v\mapsto \int_M uv d\mu$$
Tromba refers to Gilbarg and Trudinger, Elliptic Partial Differential Equations (2nd edition) - the relevant Theorem is Theorem 8.3 in that book. There you will find a similar result for elliptic boundary value problems of second order (for the case $s=0$). The basic reasoning in your situation is the same, however. The compactness of the aforementioned map is Lemma 8.5 in that book, the Fredholm alternative for bilinear forms applied to elliptic partial differential equations is Theorem 8.6.
Since all arguments involve only estimates for integrals they carry over more or less directly to the case of compact surfaces.