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.

Statement of the theorem Let (H, ⟨ , ⟩) be a real or complex Hilbert space and let A : H → H be a bounded, compact, 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, such that |λi| is monotonically non-increasing and, if N = +∞, {displaystyle lim _{ito +infty }lambda _{i}=0.} Furthermore, 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, i.e., {displaystyle Avarphi _{i}=lambda _{i}varphi _{i}{mbox{ for }}i=1,dots ,N.} Moreover, 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 _{i}langle varphi _{i},urangle varphi _{i}{mbox{ for all }}uin H.} References Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 356. ISBN 0-387-00444-0. (Theorem 8.94) Royden, Halsey; Fitzpatrick, Patrick (2017). Real Analysis (Fourth ed.). New York: MacMillan. ISBN 0134689496. (Section 16.6) 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: Operator theoryTheorems in functional analysis