Fredholm alternative

If $T$ is compact and $\lambda \ne 0$, then $T-\lambda I$ is injective exactly when it is surjective. Equivalently, every nonzero point of the spectrum of $T$ is an eigenvalue. Compactness turns an approximate eigenvector sequence into a convergent eigenvector sequence.

For a symmetric uniformly elliptic divergence-form operator with zero conormal flux and zero reaction, Lax-Milgram theorem constructs the shifted inverse $K:L^2\to H^1$. The compact Rellich-Kondrachov theorem embedding makes it compact on $L^2$, and symmetry makes it self-adjoint. If the homogeneous kernel consists of constants, then $\ker (I-K)$ is the constant line and $K1=1$. The Fredholm alternative makes solvability equivalent to $Kf$ being orthogonal to constants, or $⟨Kf,1⟩=⟨f,K1⟩=\int f=0$. The weak solution is unique up to an additive constant. For a disconnected domain the identical argument gives one compatibility condition per connected component.