I have been studying Fredholm Alternative for PDE's in the book Evans - Partial Differential Equations. The result is:
Theorem 4 (page 321) Precisely one of the following statements holds:
either
$(\alpha)$ For each $f \in L^2(U)$ there exists a unique weak solution $u$ of $$ (P_1) \quad \begin{cases} Lu = f, U \\ u = 0, \partial U \end{cases} $$ or else
$(\beta)$ there exists a weak solution $u \neq 0$ of $$ (P_2) \quad \begin{cases} Lu = 0, U \\ u = 0, \partial U \end{cases} $$
I want to understand this Evans's proof. Let me write some steps until I get into my doubt.
It is known that there exists a $\gamma \geq 0$ such that the problem $$ (P_\gamma) \quad \begin{cases} L_\gamma := Lu + \gamma u = f , U\\ u = 0, \partial U \end{cases} $$ has a unique solution for each $f \in L^2(U)$. So, we can define the solution operator $$S_\gamma : L^{2}(U) \rightarrow H^1_0(U)$$ such that $$ B_\gamma(S_\gamma(f), \varphi) = \int_U f \varphi, \quad \forall \varphi \in H^1_0(U), $$ where $B_\gamma$ is the bilinear form associated to the problem $(P_\gamma)$ which Dirichlet condition. Also consider $i : H^1_0(U) \rightarrow L^2(U)$ the usual compact immersion and define $\overline{S_\gamma} = i \circ S_\gamma$. This operator is continous and compact. I know that, given $f \in L^2(U)$, $u \in H^1_0(U)$ is a weak solution of $(P_1)$ if and only if $$ u - \gamma \overline{S_\gamma} u = S_\gamma(f). $$ Also, $u \in H^1_0(U)$ is a weak solution of $(P_2)$ if and only if $$ u - \gamma \overline{S_\gamma} u = 0. $$ We can apply Fredholm Alternative to $\gamma \overline{S_\gamma}$ in order to obtain:
Proposition: One of the following statements holds:
either
$(1) \quad$ For all $g \in L^2(U)$ the equation $u - \gamma \overline{S_\gamma} u = g$ has a unique solution
or else
$(2) \quad$ The equation $u - \gamma \overline{S_\gamma} u = 0$ has non trivial solution.
My doubt: I could not realize if Evans used the above Proposition to obtain Theorem 4. If he used, how to obtain the result?
With your notations, as you indeed remarked $K := \gamma \overline{S_\gamma}$ is a compact operator on $H := L^2(U)$. Hence, you are trying to solve an equation of the form $$ (\mathrm{Id} - K) u = g $$ with $u, g \in H$ and a compact operator $K : H \to H$. Such equations are studied in Section D.5: Compact operators, Fredholm theory of Appendix D: Linear functional analysis of Evans' book.
The fundamental Fredholm alternative at the abstract level (of which you are trying to prove a consequence for the study of PDEs) precisely corresponds to what you stated as your proposition (see item (iv) of Theorem 5 in Appendix D.5 in Evans's book): $$ \operatorname{kernel} (\mathrm{Id} - K) = \{ 0 \} \quad \text{ if and only if } \quad \operatorname{range} (\mathrm{Id} - K) = H. $$ So either the kernel is not reduced to $\{0\}$ and you have a non-zero solution to $u = Ku$, or the range is equal to $H$ and for every $g \in H$ you have a solution to $u - K u = g$ (which is unique since the kernel is $\{ 0\}$).