Английская Википедия:Analytic Fredholm theorem

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

• B is analytic on G in the sense that the limit <math display="block">\lim_{\lambda \to \lambda_{0}} \frac{B(\lambda) - B(\lambda_{0})}{\lambda - \lambda_{0}}</math> exists for all Шаблон:Math; and
• the operator B(λ) is a compact operator for each Шаблон:Math.

Then either

• Шаблон:Math does not exist for any Шаблон:Math; or
• Шаблон:Math exists for every Шаблон: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 Шаблон:Math is analytic on Шаблон:Math and, if Шаблон:Math, then the equation <math display="block">B(\lambda) \psi = \psi</math> has a finite-dimensional family of solutions.

References

• Шаблон:Cite book (Theorem 8.92)