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 $$T : \operatorname{dom}(T) \subseteq G \to H$$ 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, $$\left(T^* T\right)^* = T^* T$$ and the operators on the right- and left-hand sides have the same dense domain in $$G.$$