Fredholm alternative

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In mathematics, the Fredholm alternative, name 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.

1 Linear algebra
2 Integral equations
3 Functional analysis
- 3.1 Correspondence
- 3.2 Alternative
4 See also
5 References

[edit] 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.
$\dim(\ker(T)) > 0$ .

[edit] 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).

[edit] Functional analysis

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

[edit] Correspondence

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

T = λ - K

or, in index notation,

T (x, y) = λδ(x - y) - K (x, y)

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

$T:V\to V$

is given by

$\phi \mapsto \psi$

with $ψ$ 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.

[edit] 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 $λ$ is either an eigenvalue of K, or it lies in the domain of the resolvent

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