# Fredholm alternative

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

## Linear algebra

If V is an n-dimensional vector space and $T:V\to V$ is a linear transformation, then exactly one of the following holds:
* For each vector v in V there is a vector u in V so that $T(u)=v$. In other words: T is surjective (and so also bijective, since V is finite-dimensional).
* $\dim(\ker(T))>0.$
A more elementary formulation, in terms of matrices, is as follows. Given an m×n matrix A and a m×1 column vector b, exactly one of the following must hold:
* Either: A x = b has a solution x
* Or: A$^T$ y = 0 has a solution y with y$^T$b ≠ 0.
In other words, A x = b has a solution $(\mathbf {b} \in \operatorname {Im} (A))$ if and only if for any y such that A$^T$ y = 0, it follows that y$^T$b = 0 $(i.e.,\mathbf {b} \in \ker(A^{T})^{\bot })$.

## Integral equations

Let $K(x,y)$ be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation,
    $\lambda \varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy=0$
and the inhomogeneous equation
    $\lambda \varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy=f(x).$
The Fredholm alternative is the statement that, for every non-zero fixed complex number $\lambda \in \mathbb {C} ,$ either the first equation has a non-trivial solution, or the second equation has a solution for all $f(x)$.
A sufficient condition for this statement to be true is for $K(x,y)$ to be square integrable on the rectangle $[a,b]\times [a,b]$ (where a and/or b may be minus or plus infinity). The integral operator defined by such a K is called a Hilbert–Schmidt integral operator.

## Functional analysis

Results about Fredholm operators generalize these results to complete normed vector spaces of infinite dimensions; that is, Banach spaces.
The integral equation can be reformulated in terms of operator notation as follows. Write (somewhat informally) $T=\lambda -K$ to mean $T(x,y)=\lambda \;\delta (x-y)-K(x,y)$ with $\delta (x-y)$ the Dirac delta function, considered as a distribution, or generalized function, in two variables. Then by convolution, $T$ induces a linear operator acting on a Banach space $V$ of functions $\varphi (x)$ $V\to V$ given by $\varphi \mapsto \psi$ with $\psi$ given by $\psi (x)=\int _{a}^{b}T(x,y)\varphi (y)\,dy=\lambda \;\varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy.$
In this language, the Fredholm alternative for integral equations is seen to be analogous to the Fredholm alternative for finite-dimensional linear algebra.
The operator $K$ given by convolution with an $L^{2}$ kernel, as above, is known as a Hilbert–Schmidt integral operator. Such operators are always compact. More generally, the Fredholm alternative is valid when $K$ is any compact operator. The Fredholm alternative may be restated in the following form: a nonzero $\lambda$ either is an eigenvalue of $K,$ or lies in the domain of the resolvent $R(\lambda ;K)=(K-\lambda \operatorname {Id} )^{-1}.$

## Elliptic partial differential equations

The Fredholm alternative can be applied to solving linear elliptic boundary value problems. The basic result is: if the equation and the appropriate Banach spaces have been set up correctly, then either
    (1) The homogeneous equation has a nontrivial solution, or
    (2) The inhomogeneous equation can be solved uniquely for each choice of data.
The argument goes as follows. A typical simple-to-understand elliptic operator L would be the Laplacian plus some lower order terms. Combined with suitable boundary conditions and expressed on a suitable Banach space X (which encodes both the boundary conditions and the desired regularity of the solution), L becomes an unbounded operator from X to itself, and one attempts to solve
    $Lu=f,\qquad u\in \operatorname {dom} (L)\subseteq X,$
where f ∈ X is some function serving as data for which we want a solution. The Fredholm alternative, together with the theory of elliptic equations, will enable us to organize the solutions of this equation.
A concrete example would be an elliptic boundary-value problem like
    $(*)\qquad Lu:=-\Delta u+h(x)u=f\qquad {\text{in }}\Omega ,$
supplemented with the boundary condition
    $(**)\qquad u=0\qquad {\text{on }}\partial \Omega ,$
where Ω ⊆ R$^n$ is a bounded open set with smooth boundary and h(x) is a fixed coefficient function (a potential, in the case of a Schrödinger operator). The function f ∈ X is the variable data for which we wish to solve the equation. Here one would take X to be the space L$^2$(Ω) of all square-integrable functions on Ω, and dom(L) is then the Sobolev space W $^{2,2}$(Ω) ∩ W$^{1,2}$ $_0$(Ω), which amounts to the set of all square-integrable functions on Ω whose weak first and second derivatives exist and are square-integrable, and which satisfy a zero boundary condition on ∂Ω.
If X has been selected correctly (as it has in this example), then for μ$_0$ >> 0 the operator L + μ$_0$ is positive, and then employing elliptic estimates, one can prove that L + μ$_0$ : dom(L) → X is a bijection, and its inverse is a compact, everywhere-defined operator K from X to X, with image equal to dom(L). We fix one such μ$_0$, but its value is not important as it is 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 + μ$_0$) h = λ$^{−1}$h ⇔ –μ$_0$+λ$^{−1}$ is an eigenvalue of L.
(B) The operator K − λ : X → X is a bijection ⇔ (K − λ) (L + μ$_0$) = Id − λ (L + μ$_0$) : dom(L) → X is a bijection ⇔ L + μ$_0$ − λ$^{−1}$ : dom(L) → X is a bijection.
Replacing -μ$_0$+λ$^{−1}$ by λ, and treating the case λ = −μ$_0$ 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.