Fredholm alternative

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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[edit]

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[edit]

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[edit]

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

Correspondence[edit]

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[edit]

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}.

See also[edit]

References[edit]