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 Template:Math be a domain (an open and connected set). Let Template:Math 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 Template:Math be a mapping such that

• B is analytic on G in the sense that the limit $$\underset{\lambda \to {\lambda }_0}{\lim }\frac{B(\lambda )-B({\lambda }_0)}{\lambda -{\lambda }_0}$$ exists for all Template:Math; and
• the operator B(λ) is a compact operator for each Template:Math.

Then either

• Template:Math does not exist for any Template:Math; or
• Template:Math exists for every Template:Math, where S is a discrete subset of G (i.e., S has no limit points in G). In this case, the function taking λ to Template:Math is analytic on Template:Math and, if Template:Math, then the equation $$B(\lambda )\psi =\psi$$ has a finite-dimensional family of solutions.

References

• Template:Cite book (Theorem 8.92)