Bourbaki

Remark

Tag 02G5, permanent. Cite it as [Bourbaki, Tag 02G5].

Théories spectrales, chapter I, § 7. Spectre des endomorphismes des espaces de Banach

Let H be a subset open and closed in Sp(uu), whose associated spectral decomposition is E=EH(u)⊕E~H(u)E = E_H(u)\oplus \widetilde{E}_H(u). It follows from assertion c) that if one identifies E′E' with EH(u)′⊕E~H(u)′E_H(u)'\oplus \widetilde{E}_H(u)', then we have EH′(tu)=EH(u)′E'_H(^tu) = E_H(u)' and E~H′(tu)=E~H(u)′\widetilde{E}'_H(^tu) =\widetilde{E}_H(u)'.

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-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.