Analytic Fredholm theorem

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 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 the limit {displaystyle lim _{lambda to lambda _{0}}{frac {B(lambda )-B(lambda _{0})}{lambda -lambda _{0}}}} exists for all λ0 ∈ 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 {displaystyle B(lambda )psi =psi } has a finite-dimensional family of solutions. References Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 266. ISBN 0-387-00444-0. (Theorem 8.92) hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Fredholm theoryTheorems in functional analysisTheorems in complex analysis

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