Fredholm's theorem

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.

Contents 1 Linear 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: {displaystyle (operatorname {row} M)^{bot }=ker M.} Similarly, the orthogonal complement of the column space of M is the null space of the adjoint: {displaystyle (operatorname {col} M)^{bot }=ker M^{*}.} Integral equations Fredholm's theorem for integral equations is expressed as follows. Let {displaystyle K(x,y)} be an integral kernel, and consider the homogeneous equations {displaystyle int _{a}^{b}K(x,y)phi (y),dy=lambda phi (x)} and its complex adjoint {displaystyle int _{a}^{b}psi (x){overline {K(x,y)}},dx={overline {lambda }}psi (y).} Here, {displaystyle {overline {lambda }}} denotes the complex conjugate of the complex number {displaystyle lambda } , and similarly for {displaystyle {overline {K(x,y)}}} . Then, Fredholm's theorem is that, for any fixed value of {displaystyle lambda } , these equations have either the trivial solution {displaystyle psi (x)=phi (x)=0} or have the same number of linearly independent solutions {displaystyle phi _{1}(x),cdots ,phi _{n}(x)} , {displaystyle psi _{1}(y),cdots ,psi _{n}(y)} .

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

Here, 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, including, for example, 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,y)phi (y),dy=f(x).} Solutions to this equation exist if and only if the function {displaystyle f(x)} is orthogonal to the complete set of solutions {displaystyle {psi _{n}(x)}} of the corresponding homogeneous adjoint equation: {displaystyle int _{a}^{b}{overline {psi _{n}(x)}}f(x),dx=0} where {displaystyle {overline {psi _{n}(x)}}} is the complex conjugate of {displaystyle psi _{n}(x)} and the former is one of the complete set of solutions to {displaystyle lambda {overline {psi (y)}}-int _{a}^{b}{overline {psi (x)}}K(x,y),dx=0.} A sufficient condition for this theorem to hold is for {displaystyle K(x,y)} to be square integrable on the rectangle {displaystyle [a,b]times [a,b]} .

References E.I. Fredholm, "Sur une classe d'equations fonctionnelles", Acta Math., 27 (1903) pp. 365–390. Weisstein, Eric W. "Fredholm's Theorem". MathWorld. B.V. Khvedelidze (2001) [1994], "Fredholm theorems", Encyclopedia of Mathematics, EMS Press 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 theoryLinear algebraTheorems in functional analysis