Von Neumann's theorem: Difference between revisions

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

Statement of the theorem

Let G and H be Hilbert spaces, and let T : dom(T) ⊆ G → H be a densely 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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 ==Statement of the theorem==
-Let ''G'' and ''H'' be Hilbert spaces, and let ''T''&nbsp;:&nbsp;dom(''T'')&nbsp;⊆&nbsp;''G''&nbsp;→&nbsp;''H'' be a partially defined operator from ''G'' into ''H''. Let ''T''<sup>∗</sup>&nbsp;:&nbsp;dom(''T''<sup>∗</sup>)&nbsp;⊆&nbsp;''H''&nbsp;→&nbsp;''G'' denote the [[adjoint operator|Hilbert adjoint]] of ''T''. Suppose that ''T'' is a [[closed operator]] and that ''T'' is densely defined, i.e. dom(''T'') is [[dense (topology)|dense]] in ''G''. Then ''T''<sup>∗</sup>''T'' is also densely defined and [[self-adjoint]]. That is,
+Let ''G'' and ''H'' be Hilbert spaces, and let ''T''&nbsp;:&nbsp;dom(''T'')&nbsp;⊆&nbsp;''G''&nbsp;→&nbsp;''H'' be a densely defined operator from ''G'' into ''H''. Let ''T''<sup>∗</sup>&nbsp;:&nbsp;dom(''T''<sup>∗</sup>)&nbsp;⊆&nbsp;''H''&nbsp;→&nbsp;''G'' denote the [[adjoint operator|Hilbert adjoint]] of ''T''. Suppose that ''T'' is a [[closed operator]] and that ''T'' is densely defined, i.e. dom(''T'') is [[dense (topology)|dense]] in ''G''. Then ''T''<sup>∗</sup>''T'' is also densely defined and [[self-adjoint]]. That is,
 :<math>(T^{*} T)^{*} = T^{*} T</math>