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