Editing Normal operator (section)

==Unbounded normal operators==

The definition of normal operators naturally generalizes to some class of unbounded operators. Explicitly, a closed operator ''N'' is said to be normal if 
:<math>N^*N = NN^*.</math>
Here, the existence of the adjoint ''N*'' requires that the domain of ''N'' be dense, and the equality includes the assertion that the domain of ''N*N'' equals that of ''NN*'', which is not necessarily the case in general.

Equivalently normal operators are precisely those for which<ref>Weidmann, Lineare Operatoren in Hilberträumen, Chapter 4, Section 3</ref>
:<math>\|Nx\|=\|N^*x\|\qquad</math> 
with
:<math>\mathcal{D}(N)=\mathcal{D}(N^*).</math>
The spectral theorem still holds for unbounded (normal) operators. The proofs work by reduction to bounded (normal) operators.<ref name=Frei>Alexander Frei, Spectral Measures, Mathematics Stack Exchange, [https://math.stackexchange.com/q/1332154 Existence] {{Webarchive|url=https://web.archive.org/web/20210626022512/https://math.stackexchange.com/questions/1332154/spectral-measures-existence |date=2021-06-26 }}, [https://math.stackexchange.com/q/1112508 Uniqueness] {{Webarchive|url=https://web.archive.org/web/20210626022607/https://math.stackexchange.com/questions/1112508/spectral-measures-uniqueness |date=2021-06-26 }}</ref><ref name="Conway">[[John B. Conway]], A Course in Functional Analysis, Second Edition, Chapter X, Section §4</ref>