Hellinger–Toeplitz theorem

Hellinger–Toeplitz theorem In functional analysis, ein Zweig der Mathematik, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product {displaystyle langle cdot |cdot-Bereich } ist begrenzt. Per Definition, an operator A is symmetric if {displaystyle langle Ax|yrangle =langle x|Ayrwinkel } für alle x, y in the domain of A. Note that symmetric everywhere-defined operators are necessarily self-adjoint, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded. The theorem is named after Ernst David Hellinger and Otto Toeplitz.

This theorem can be viewed as an immediate corollary of the closed graph theorem, as self-adjoint operators are closed. Alternative, it can be argued using the uniform boundedness principle. One relies on the symmetric assumption, therefore the inner product structure, in proving the theorem. Also crucial is the fact that the given operator A is defined everywhere (und, im Gegenzug, the completeness of Hilbert spaces).

The Hellinger–Toeplitz theorem reveals certain technical difficulties in the mathematical formulation of quantum mechanics. Observables in quantum mechanics correspond to self-adjoint operators on some Hilbert space, but some observables (like energy) sind unbegrenzt. By Hellinger–Toeplitz, such operators cannot be everywhere defined (but they may be defined on a dense subset). Take for instance the quantum harmonic oscillator. Here the Hilbert space is L2(R), the space of square integrable functions on R, and the energy operator H is defined by (assuming the units are chosen such that ℏ = m = ω = 1) {Anzeigestil [Hf](x)=-{frac {1}{2}}{frac {Mathrm {d} ^{2}}{Mathrm {d} x^{2}}}f(x)+{frac {1}{2}}x^{2}f(x).} This operator is self-adjoint and unbounded (its eigenvalues are 1/2, 3/2, 5/2, ...), so it cannot be defined on the whole of L2(R).

References Reed, Michael and Simon, Barry: Methoden der Mathematischen Physik, Volumen 1: Funktionsanalyse. Akademische Presse, 1980. See Section III.5. Teschl, Gerhard (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Vorsehung: Amerikanische Mathematische Gesellschaft. ISBN 978-0-8218-4660-5. verbergen vte Funktionsanalyse (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: Sätze in der Funktionalanalysis