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.