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:


 * 1) 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).
 * 2) $$\dim(\ker(T)) > 0.$$

A more elementary formulation, in terms of matrices, is as follows. Given an m&times;n matrix A and a m&times;1 column vector b, exactly one of the following must hold:


 * 1) Either: A x = b has a solution x
 * 2) Or: AT y = 0 has a solution y with yTb ≠ 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 AT y = 0, it follows that yTb = 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


 * $$L u = f,\qquad u\in \operatorname{dom}(L) \subseteq X, $$

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 ⊆ Rn 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 L2(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 μ0 >> 0 the operator L + μ0 is positive, and then employing elliptic estimates, one can prove that L + μ0 : 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 μ0, 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 + 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.