Fredholm's Theorem - Integral Equations

Integral Equations

Fredholm's theorem for integral equations is expressed as follows. Let be an integral kernel, and consider the homogeneous equations

and its complex adjoint

Here, denotes the complex conjugate of the complex number, and similarly for . Then, Fredholm's theorem is that, for any fixed value of, these equations have either the trivial solution or have the same number of linearly independent solutions, .

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).

Here, the integral is expressed as a one-dimensional integral on the real number line. In Fredholm theory, this result generalizes to integral operators on multi-dimensional spaces, including, for example, Riemannian manifolds.