only a tool. We may then transform the Fredholm alternative, stated above for compact operators, into a statement about the solvability of the boundary-value problem (*)–(**). The Fredholm alternative, as stated above, asserts: For each λ ∈ R, either λ is an eigenvalue of K, or the operator K − λ is bijective from X to itself. Let us explore the two alternatives as they play out for the boundary-value problem. Suppose λ ≠ 0. Then either (A) λ is an eigenvalue of K ⇔ there is a solution h ∈ dom(L) of (L + μ0) h = λ−1h ⇔ –μ0+λ−1 is an eigenvalue of L. (B) The operator K − λ : X → X is a bijection ⇔ (K − λ) (L + μ0) = Id − λ (L + μ0) : dom(L) → X is a bijection ⇔ L + μ0 − λ−1 : dom(L) → X is a bijection. Replacing -μ0+λ−1 by λ, and treating the case λ = −μ0 separately, this yields the following Fredholm alternative for an elliptic boundary-value problem: For each λ ∈ R, either the homogeneous equation (L − λ) u = 0 has a nontrivial solution, or the inhomogeneous equation (L − λ) u = f possesses a unique solution u ∈ dom(L) for each given datum f ∈ X. The latter function u solves the boundary-value problem (*)–(**) introduced above. This is the dichotomy that was claimed in (1)–(2) above. By the spectral theorem for compact operators, one also obtains that the set of λ for which the solvability fails is a discrete subset of R (the eigenvalues of L). The eigenvalues’ associated eigenfunctions can be thought of as "resonances" that block the solvability of the equation. See also Spectral theory of compact operators Farkas' lemma References A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", American Mathematical Monthly, 108 (2001) p. 855. Fredholm theory Linear algebra