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 [math]\displaystyle{ G }[/math] and [math]\displaystyle{ H }[/math] be Hilbert spaces, and let [math]\displaystyle{ T : \operatorname{dom}(T) \subseteq G \to H }[/math] be an unbounded operator from [math]\displaystyle{ G }[/math] into [math]\displaystyle{ H. }[/math] Suppose that [math]\displaystyle{ T }[/math] is a closed operator and that [math]\displaystyle{ T }[/math] is densely defined, that is, [math]\displaystyle{ \operatorname{dom}(T) }[/math] is dense in [math]\displaystyle{ G. }[/math] Let [math]\displaystyle{ T^* : \operatorname{dom}\left(T^*\right) \subseteq H \to G }[/math] denote the adjoint of [math]\displaystyle{ T. }[/math] Then [math]\displaystyle{ T^* T }[/math] is also densely defined, and it is self-adjoint. That is, [math]\displaystyle{ \left(T^* T\right)^* = T^* T }[/math] and the operators on the right- and left-hand sides have the same dense domain in [math]\displaystyle{ G. }[/math][1]
References
- ↑ Acuña, Pablo (2021). "von Neumann's Theorem Revisited" (in en). Foundations of Physics 51 (3): 73. doi:10.1007/s10701-021-00474-5. ISSN 0015-9018. https://link.springer.com/10.1007/s10701-021-00474-5.
![]() | Original source: https://en.wikipedia.org/wiki/Von Neumann's theorem.
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