Von Neumann's theorem

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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 {\displaystyle G} {\displaystyle G} and H {\displaystyle H} {\displaystyle H} be Hilbert spaces, and let T : dom ⁡ ( T ) ⊆ G → H {\displaystyle T:\operatorname {dom} (T)\subseteq G\to H} {\displaystyle T:\operatorname {dom} (T)\subseteq G\to H} be an unbounded operator from G {\displaystyle G} {\displaystyle G} into H . {\displaystyle H.} {\displaystyle H.} Suppose that T {\displaystyle T} {\displaystyle T} is a closed operator and that T {\displaystyle T} {\displaystyle T} is densely defined, that is, dom ⁡ ( T ) {\displaystyle \operatorname {dom} (T)} {\displaystyle \operatorname {dom} (T)} is dense in G . {\displaystyle G.} {\displaystyle G.} Let T ∗ : dom ⁡ ( T ∗ ) ⊆ H → G {\displaystyle T^{*}:\operatorname {dom} \left(T^{*}\right)\subseteq H\to G} {\displaystyle T^{*}:\operatorname {dom} \left(T^{*}\right)\subseteq H\to G} denote the adjoint of T . {\displaystyle T.} {\displaystyle T.} Then T ∗ T {\displaystyle T^{*}T} {\displaystyle T^{*}T} is also densely defined, and it is self-adjoint. That is, ( T ∗ T ) ∗ = T ∗ T {\displaystyle \left(T^{*}T\right)^{*}=T^{*}T} {\displaystyle \left(T^{*}T\right)^{*}=T^{*}T} and the operators on the right- and left-hand sides have the same dense domain in G . {\displaystyle G.} {\displaystyle G.}[1]

References