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