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

If V is an n-dimensional vector space and 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 . In other words: T is surjective (and so also bijective, since V is finite-dimensional).
  2. .

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 if and only if for any y s.t. AT y = 0, yTb = 0 .

Integral equations

Let be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation,

and the inhomogeneous equation

The Fredholm alternative states that, for any non-zero fixed complex number , either the first equation has a non-trivial solution, or the second equation has a solution for all .

A sufficient condition for this theorem to hold is for to be square integrable on the rectangle (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

or, in index notation,

with the Dirac delta function. Here, T can be seen to be an linear operator acting on a Banach space V of functions , so that

is given by

with given by

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 is either an eigenvalue of K, or it lies in the domain of the resolvent

See also

References

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