Hellinger–Toeplitz theorem

Hellinger–Toeplitz theorem In functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product {displaystyle langle cdot |cdot rangle } is bounded. By definition, an operator A is symmetric if {displaystyle langle Ax|yrangle =langle x|Ayrangle } for all 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. Alternatively, 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 (and, in turn, 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) are unbounded. 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) {displaystyle [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: Methods of Mathematical Physics, Volume 1: Functional Analysis. Academic Press, 1980. See Section III.5. Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Providence: American Mathematical Society. ISBN 978-0-8218-4660-5. 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: Theorems in functional analysis

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