Functional near-infrared spectroscopy

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: A^T y = 0 has a solution y with y^Tb ≠ 0.

In other words, A x = b has a solution $(\mathbf{b}\in \mathrm{\Im }(A))$ if and only if for any y s.t. A^T y = 0, y^Tb = 0 $(\mathbf{b}\in \ker (A^T{)}^{\mathrm{\perp }})$.

Integral equations

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

$\lambda \varphi (x)-\underset{a}{\overset{b}{\int }}K(x,y)\varphi (y)dy=0$

and the inhomogeneous equation

$\lambda \varphi (x)-\underset{a}{\overset{b}{\int }}K(x,y)\varphi (y)dy=f(x).$

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 $\varphi (x)$, so that

$T:V\to V$

is given by

$\varphi \mapsto \psi$

with $\psi$ given by

$\psi (x)=\underset{a}{\overset{b}{\int }}T(x,y)\varphi (y)dy$

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

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

See also

• Spectral theory of compact operators

References

• E.I. Fredholm, "Sur une classe d'equations fonctionnelles", Acta Math., 27 (1903) pp. 365–390.
• A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", American Mathematical Monthly, 108 (2001) p. 855.
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