Satz von Fredholm

Fredholm's theorem In mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear algebra, or in terms of the Fredholm operator on Banach spaces.

The Fredholm alternative is one of the Fredholm theorems.

Inhalt 1 Lineare Algebra 2 Integral equations 3 Existence of solutions 4 References Linear algebra Fredholm's theorem in linear algebra is as follows: if M is a matrix, then the orthogonal complement of the row space of M is the null space of M: {Anzeigestil (Name des Bedieners {row} M)^{bot }=ker M.} Ähnlich, the orthogonal complement of the column space of M is the null space of the adjoint: {Anzeigestil (Name des Bedieners {col} M)^{bot }=ker M^{*}.} Integral equations Fredholm's theorem for integral equations is expressed as follows. Lassen {Anzeigestil K(x,j)} be an integral kernel, and consider the homogeneous equations {Anzeigestil int _{a}^{b}K(x,j)Phi (j),dy=lambda phi (x)} and its complex adjoint {Anzeigestil int _{a}^{b}psi (x){überstreichen {K(x,j)}},dx={überstreichen {Lambda }}psi (j).} Hier, {Anzeigestil {überstreichen {Lambda }}} denotes the complex conjugate of the complex number {Display-Lambda } , and similarly for {Anzeigestil {überstreichen {K(x,j)}}} . Dann, Fredholm's theorem is that, for any fixed value of {Display-Lambda } , these equations have either the trivial solution {Anzeigestil psi (x)=phi (x)=0} or have the same number of linearly independent solutions {Anzeigestil Phi _{1}(x),cdots ,Phi _{n}(x)} , {Anzeigestil psi _{1}(j),cdots ,psi _{n}(j)} .

A sufficient condition for this theorem to hold is for {Anzeigestil K(x,j)} to be square integrable on the rectangle {Anzeigestil [a,b]mal [a,b]} (where a and/or b may be minus or plus infinity).

Hier, the integral is expressed as a one-dimensional integral on the real number line. In Fredholm theory, this result generalizes to integral operators on multi-dimensional spaces, einschließlich, zum Beispiel, Riemannian manifolds.

Existence of solutions One of Fredholm's theorems, closely related to the Fredholm alternative, concerns the existence of solutions to the inhomogeneous Fredholm equation {displaystyle lambda phi (x)-int _{a}^{b}K(x,j)Phi (j),dy=f(x).} Solutions to this equation exist if and only if the function {Anzeigestil f(x)} is orthogonal to the complete set of solutions {Anzeigestil {psi _{n}(x)}} of the corresponding homogeneous adjoint equation: {Anzeigestil int _{a}^{b}{überstreichen {psi _{n}(x)}}f(x),dx=0} wo {Anzeigestil {überstreichen {psi _{n}(x)}}} is the complex conjugate of {Anzeigestil psi _{n}(x)} and the former is one of the complete set of solutions to {Display-Lambda {überstreichen {psi (j)}}-int _{a}^{b}{überstreichen {psi (x)}}K(x,j),dx=0.} A sufficient condition for this theorem to hold is for {Anzeigestil K(x,j)} to be square integrable on the rectangle {Anzeigestil [a,b]mal [a,b]} .

References E.I. Fredholm, "Sur une classe d'equations fonctionnelles", Acta Math., 27 (1903) pp. 365–390. Weißstein, Erich W. "Fredholm's Theorem". MathWorld. B.V. Khvedelidze (2001) [1994], "Fredholm theorems", Enzyklopädie der Mathematik, EMS Press hide vte Functional analysis (Themen – Glossar) Leerzeichen BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraisch/topologisch)lokal konvexreflexivseparable Theoreme Hahn-BanachRiesz-Darstellunggeschlossener Graphgleichmäßiges BeschränktheitsprinzipKakutani-FixpunktKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatoren adjointboundcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebren Banach-AlgebraC*-AlgebraSpektrum einer C*-AlgebraOperator-Algebravon Gruppenalgebra einer lokalvariant-kompakten Gruppe SubraumproblemMahlersche Vermutung Anwendungen Hardy-RaumSpektraltheorie gewöhnlicher DifferentialgleichungenWärmekernindexsatzVariationsrechnungFunktionsrechnungIntegraloperatorJones-PolynomTopologische QuantenfeldtheorieNichtkommutative GeometrieRiemann-HypotheseVerteilung (oder verallgemeinerte Funktionen) Fortgeschrittene Themen Approximation PropertyBalanced SetChoquet-TheorieSchwache TopologieBanach-Mazur-AbstandTomita-Takesaki-Theorie Kategorien: Fredholm theoryLinear algebraTheorems in functional analysis