Analytic Fredholm theorem

In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and important theorems, the Fredholm alternative and the Hilbert–Schmidt theorem. The result is named after the Swedish mathematician Erik [Ivar Fredholm].

Statement of the theorem

Let be a domain. Let be a real or complex Hilbert space and let Lin denote the space of bounded linear operators from H into itself; let I denote the identity operator. Let be a mapping such that

B is analytic on G in the sense that the limit exists for all ; and
the operator B is a compact operator for each.

Then either

does not exist for any ; or
exists for every, where S is a discrete subset of G. In this case, the function taking λ to is analytic on and, if, then the equation has a finite-dimensional family of solutions.