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.

[edit] Statement of the theorem

Let G and H be Hilbert spaces, and let T : dom(T) ⊆ G → H be a partially 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 T^∗T is also densely defined and self-adjoint. That is,

(T ^* T) ^* = T ^* T

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

[edit] References

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