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 $G.$