Von Neumann's theorem

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

Statement of the theorem[edit]

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