Hilbert–Schmidt theorem

Hilbert–Schmidt theorem In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems.

Enunciado do teorema Seja (H, ⟨ , ⟩) be a real or complex Hilbert space and let A : H → H be a bounded, compactar, self-adjoint operator. Then there is a sequence of non-zero real eigenvalues λi, i = 1, …, N, with N equal to the rank of A, de tal modo que |λi| is monotonically non-increasing and, if N = +∞, {displaystyle lim _{ito +infty }lambda _{eu}=0.} Além disso, if each eigenvalue of A is repeated in the sequence according to its multiplicity, then there exists an orthonormal set φi, i = 1, …, N, of corresponding eigenfunctions, ou seja, {displaystyle Avarphi _{eu}= lambda _{eu}varphi_{eu}{mbox{ por }}i=1,dots ,N.} Além disso, the functions φi form an orthonormal basis for the range of A and A can be written as {displaystyle Au=sum _{i=1}^{N}lambda _{eu}langle varphi _{eu},urangle varphi _{eu}{mbox{ para todos }}uin H.} References Renardy, Michael; Roger, Roberto C. (2004). Uma introdução às equações diferenciais parciais. Textos em Matemática Aplicada 13 (Second ed.). Nova york: Springer-Verlag. pp. 356. ISBN 0-387-00444-0. (Teorema 8.94) Royden, Halsey; Fitzpatrick, Patrick (2017). Real Analysis (Fourth ed.). Nova york: MacMillan. ISBN 0134689496. (Seção 16.6) hide vte Functional analysis (tópicos – glossário) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algébrico/topológico)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 (ou funções generalizadas) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Teoria dos operadoresTeoremas em análise funcional