Von Neumann's theorem

In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces.

Statement of the theorem

Let G and H be Hilbert spaces, and let T : dom(T) ⊆ G → H be an unbounded operator from G into H. Suppose that T is a closed operator and that T is densely defined, i.e. dom(T) is dense in G. Let T^∗ : dom(T^∗) ⊆ H → G denote the adjoint of T. Then T^∗T is also densely defined, and it is self-adjoint. That is,

(T^{*}T)^{*}=T^{*}T

and the operators on the right- and left-hand sides have the same dense domain in G.

References

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Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory