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

\left(T^{*}T\right)^{*}=T^{*}T

and the operators on the right- and left-hand sides have the same dense domain in $G.$ ^[1]

References

↑ Acuña, Pablo (2021). "von Neumann's Theorem Revisited". Foundations of Physics. 51 (3): 73. doi:10.1007/s10701-021-00474-5. ISSN 0015-9018. S2CID 237887405.

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[1] Acuña, Pablo (2021). "von Neumann's Theorem Revisited". Foundations of Physics. 51 (3): 73. doi:10.1007/s10701-021-00474-5. ISSN 0015-9018. S2CID 237887405.

[1]

Spectral theory and ^*-algebras
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Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
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Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F