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 is a -dimensional vector space, with finite, and is a linear transformation, then exactly one of the following holds:

For each vector in there is a vector in so that. In other words: is surjective.

A more elementary formulation, in terms of matrices, is as follows. Given an matrix and a column vector, exactly one of the following must hold:

Either: has a solution
Or: has a solution with.

In other words, has a solution if and only if for any such that, it follows that .

Integral equations

Let be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation,
and the inhomogeneous equation
The Fredholm alternative is the statement that, for every 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 statement to be true is for to be square integrable on the rectangle . The integral operator defined by such a K is called a Hilbert–Schmidt integral operator.

Functional analysis

Results about Fredholm operators generalize these results to complete normed vector spaces of infinite dimensions; that is, Banach spaces.
The integral equation can be reformulated in terms of operator notation as follows. Write
to mean
with the Dirac delta function, considered as a distribution, or generalized function, in two variables.
Then by convolution, induces a linear operator acting on a Banach space of functions
given by
with given by
In this language, the Fredholm alternative for integral equations is seen to be analogous to the Fredholm alternative for finite-dimensional linear algebra.
The operator given by convolution with an kernel, as above, is known as a Hilbert–Schmidt integral operator.
Such operators are always compact. More generally, the Fredholm alternative is valid when is any compact operator. The Fredholm alternative may be restated in the following form: a nonzero either is an eigenvalue of or lies in the domain of the resolvent

Elliptic partial differential equations

The Fredholm alternative can be applied to solving linear elliptic [partial differential equations|elliptic boundary value problems]. The basic result is: if the equation and the appropriate Banach spaces have been set up correctly, then either
The argument goes as follows. A typical simple-to-understand elliptic operator would be the Laplacian plus some lower order terms. Combined with suitable boundary conditions and expressed on a suitable Banach space , becomes an unbounded operator from to itself, and one attempts to solve
where is some function serving as data for which we want a solution. The Fredholm alternative, together with the theory of elliptic equations, will enable us to organize the solutions of this equation.
A concrete example would be an elliptic boundary-value problem like
supplemented with the boundary condition
where is a bounded open set with smooth boundary and is a fixed coefficient function. The function is the variable data for which we wish to solve the equation. Here one would take to be the space of all square-integrable functions on, and is then the Sobolev space, which amounts to the set of all square-integrable functions on whose weak first and second derivatives exist and are square-integrable, and which satisfy a zero boundary condition on.
If has been selected correctly, then for the operator is positive, and then employing elliptic estimates, one can prove that is a bijection, and its inverse is a compact, everywhere-defined operator from to, with image equal to. We fix one such, but its value is not important as it is only a tool.
We may then transform the Fredholm alternative, stated above for compact operators, into a statement about the solvability of the boundary-value problem –. The Fredholm alternative, as stated above, asserts:

For each, either is an eigenvalue of, or the operator is bijective from to itself.

Let us explore the two alternatives as they play out for the boundary-value problem. Suppose. Then either
is an eigenvalue of there is a solution of is an eigenvalue of.
The operator is a bijection is a bijection is a bijection.
Replacing by, and treating the case separately, this yields the following Fredholm alternative for an elliptic boundary-value problem:

For each, either the homogeneous equation has a nontrivial solution, or the inhomogeneous equation possesses a unique solution for each given datum.

The latter function solves the boundary-value problem – introduced above. This is the dichotomy that was claimed in – above. By the spectral theorem for compact operators, one also obtains that the set of for which the solvability fails is a discrete subset of . The eigenvalues’ associated eigenfunctions can be thought of as "resonances" that block the solvability of the equation.