Bourbaki

Corollary

Tag 035E, permanent. Cite it as [Bourbaki, Tag 035E].

Théories spectrales, chapter IV, § 5. Opérateurs partiels normaux et théorème spectral

The endomorphism b(u)b(u) is hermitian if and only if uu is self-adjoint.

This results from the injectivity of the mapping u↦b(u)u\mapsto b(u) (prop. 2) and from the formula b(u∗)=b(u)∗b(u^*) =b(u)^* (cor. of prop. 1).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

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Machine translation of the English, by gpt-5.4, gpt-5-6-mini, against glossary version 34. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.