# 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×n matrix A and a m×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 s.t. AT y = 0, yTb = 0 $(\mathbf{b} \in \ker(A^T)^{\bot})$.

## Integral equations

Let $K(x,y)$ be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation,

$\lambda \phi(x)- \int_a^b K(x,y) \phi(y) \,dy = 0$

and the inhomogeneous equation

$\lambda \phi(x) - \int_a^b K(x,y) \phi(y) \,dy = f(x).$

The Fredholm alternative states that, for any 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 theorem to hold 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).

## Functional analysis

Results on the Fredholm operator generalize these results to vector spaces of infinite dimensions, Banach spaces.

### Correspondence

Loosely speaking, the correspondence between the linear algebra version, and the integral equation version, is as follows: Let

$T=\lambda - K$

or, in index notation,

$T(x,y)=\lambda \delta(x-y) - K(x,y)$

with $\delta(x-y)$ the Dirac delta function. Here, T can be seen to be an linear operator acting on a Banach space V of functions $\phi(x)$, so that

$T:V\to V$

is given by

$\phi \mapsto \psi$

with $\psi$ given by

$\psi(x)=\int_a^b T(x,y) \phi(y) \,dy$

In this language, the integral equation alternatives are seen to correspond to the linear algebra alternatives.

### Alternative

In more precise terms, the Fredholm alternative only applies when K is a compact operator. From Fredholm theory, smooth integral kernels are compact operators. The Fredholm alternative may be restated in the following form: a nonzero $\lambda$ is either an eigenvalue of K, or it lies in the domain of the resolvent

$R(\lambda; K)= (K-\lambda \operatorname{Id})^{-1}.$