Von Neumann's theorem explained

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 : \operatorname{dom}(T) \subseteq G \to H be an unbounded operator from G into H. Suppose that T is a closed operator and that T is densely defined, that is, \operatorname{dom}(T) is dense in G. Let T ^{*} : \operatorname{dom}\left(T ^{*\right)} \subseteq H \to G denote the adjoint of T. Then T ^{*} T is also densely defined, and it is self-adjoint. That is,$$\backslash left(T^*\; T\backslash right)^*\; =\; T^*\; T$$and the operators on the right- and left-hand sides have the same dense domain in G. ^[1]

Notes and References

1. Acuña. Pablo. 2021. von Neumann's Theorem Revisited. Foundations of Physics. en. 51. 3. 73. 10.1007/s10701-021-00474-5. 237887405 . 0015-9018. subscription.