Bourbaki

Remark 3

Tag 02WW, permanent. Cite it as [Bourbaki, Tag 02WW].

Théories spectrales, chapter IV, § 1. Opérateurs compacts sur un espace hilbertien

If uu is positive, then αn(u)=λn(u)\alpha_n(u) =\lambda_n(u) for every n∈IEn\in I_E (indeed, one then has αn(u)2=λn(u∗u)=λn(u2)=λn(u)2\alpha_n(u)^2=\lambda_n(u^*u) =\lambda_n(u^2) =\lambda_n(u)^2 by Prop. 4 of IV, p. 153).

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-5-6-mini, gpt-5-mini, gpt-5.4, 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.