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.

Let G ⊆ C be a domain (an open and connected set). Let (H, 〈 , 〉) be a real or complex Hilbert space and let Lin(H) denote the space of bounded linear operators from H into itself; let I denote the identity operator. Let B : G → Lin(H) be a mapping such that

Then either