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 Hilbert-Schmidt theorems. The result is named after the Swedish mathematician Erik Ivar Fredholm.

tatement of the theorem

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

* "B" is analytic on "G" in the sense that that the limit

::$lim\_\{lambda\; o\; lambda\_\{0\; frac\{B(lambda)\; -\; B(lambda\_\{0\})\}\{lambda\; -\; lambda\_\{0$

: exists for all "λ"₀ ∈ "G"; and

* the operator "B"("λ") is a compact operator for each "λ" ∈ "G".

Then either

* (I − "B"("λ"))⁻¹ does not exist for any "λ" ∈ "G"; or

* (I − "B"("λ"))⁻¹ exists for every "λ" ∈ "G" "S", 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 (I − "B"("λ"))⁻¹ is analytic on "G" "S" and, if "λ" ∈ "S", then the equation

::$B(lambda)\; psi\; =\; psi$

: has a finite-dimensional family of solutions.

References

* cite book
author = Renardy, Michael and Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = 356
id = ISBN 0-387-00444-0 (Theorem 7.92)