von Neumann's theorem

Let $G$ and $H$ be Hilbert spaces, and let $T\; :\; \backslash operatorname\{dom\}(T)\; \backslash subseteq\; G\; \backslash to\; H$ be an unbounded operator from $G$ into $H.$ Suppose that $T$ is a closed operator and that $T$ is densely defined, that is, $\backslash operatorname\{dom\}(T)$ is dense in $G.$ Let $T^*\; :\; \backslash operatorname\{dom\}\backslash left(T^*\backslash right)\; \backslash subseteq\; H\; \backslash to\; G$ denote the adjoint of $T.$ Then $T^*\; T$ is also densely defined, and it is self-adjoint. That is,

$$\backslash left(T^*\; T\backslash right)^*\; =\; T^*\; T$$

and the operators on the right- and left-hand sides have the same dense domain in $G.${{Cite journal|last=Acuña|first=Pablo|date=2021|title=von Neumann's Theorem Revisited|url=https://link.springer.com/10.1007/s10701-021-00474-5|journal=Foundations of Physics|language=en|volume=51|issue=3|pages=73|doi=10.1007/s10701-021-00474-5|bibcode=2021FoPh...51...73A |s2cid=237887405 |issn=0015-9018|url-access=subscription}}

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Category:Operator theory

Category:Theorems in functional analysis