Von Neumann's theorem
Statement of the theorem
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 T∗T is also densely defined and self-adjoint. That is,
and the operators on the right- and left-hand sides have the same dense domain in G.
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