Von Neumann's theorem

Von Neumann's theorem

In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces.

tatement of the theorem

Let "G" and "H" be Hilbert spaces, and let "T" : dom("T") ⊆ "G" → "H" be a partially defined operator from "G" into "H". Let "T" : dom("T") ⊆ "H" → "G" denote the Hilbert adjoint of "T". Suppose that "T" is a closed operator and that "T" is densely defined, i.e. dom("T") is dense in "G". Then "T""T" is also densely defined and self-adjoint. That is,

:(T^{*} T)^{*} = T^{*} T

and the operators on the right- and let-hand sides have the same dense domain in "G".

References


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