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]

References

1. Acuña, Pablo (2021). "von Neumann's Theorem Revisited". Foundations of Physics. 51 (3): 73. doi:10.1007/s10701-021-00474-5. ISSN 0015-9018.