Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.
is a linear transformation, then exactly one of the following holds: 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: In other words, A x = b has a solution
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
A sufficient condition for this statement to be true is for
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.
the Dirac delta function, considered as a distribution, or generalized function, in two variables.
induces a linear operator acting on a Banach space
In this language, the Fredholm alternative for integral equations is seen to be analogous to the Fredholm alternative for finite-dimensional linear algebra.
More generally, the Fredholm alternative is valid when
The Fredholm alternative may be restated in the following form: a nonzero
The Fredholm alternative can be applied to solving linear 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 L would be the Laplacian plus some lower order terms.
Combined with suitable boundary conditions and expressed on a suitable Banach space X (which encodes both the boundary conditions and the desired regularity of the solution), L becomes an unbounded operator from X to itself, and one attempts to solve where f ∈ X 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 Ω ⊆ Rn is a bounded open set with smooth boundary and h(x) is a fixed coefficient function (a potential, in the case of a Schrödinger operator).
The function f ∈ X is the variable data for which we wish to solve the equation.
Here one would take X to be the space L2(Ω) of all square-integrable functions on Ω, and dom(L) is then the Sobolev space W 2,2(Ω) ∩ W1,20(Ω), which amounts to the set of all square-integrable functions on Ω whose weak first and second derivatives exist and are square-integrable, and which satisfy a zero boundary condition on ∂Ω.
If X has been selected correctly (as it has in this example), then for μ0 >> 0 the operator L + μ0 is positive, and then employing elliptic estimates, one can prove that L + μ0 : dom(L) → X is a bijection, and its inverse is a compact, everywhere-defined operator K from X to X, with image equal to dom(L).
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: Let us explore the two alternatives as they play out for the boundary-value problem.
Replacing -μ0+λ−1 by λ, and treating the case λ = −μ0 separately, this yields the following Fredholm alternative for an elliptic boundary-value problem: The latter function u solves the boundary-value problem (*)–(**) introduced above.
By the spectral theorem for compact operators, one also obtains that the set of λ for which the solvability fails is a discrete subset of R (the eigenvalues of L).
The eigenvalues’ associated eigenfunctions can be thought of as "resonances" that block the solvability of the equation.