Remark

Tag 03BB, permanent. Cite it as [Bourbaki, Tag 03BB].

Théories spectrales, chapter V, § 2. Représentations des groupes localement compacts

Suppose that $u$ is an endomorphism of E. The unitary representation of $\mathbf{R}$ in E defined by $\varrho (t) =e^{itu}$ then satisfies the inequality $\|\varrho (t)-1_E\|\leqslant |t| \|u\|$ for every $t\in \mathbf{R}$ , and the mapping $\varrho$ of $\mathbf{R}$ into the Banach space $\mathscr{L}(E)$ is the unique solution of the linear differential equation

1d\varrho

=u\circ \varrho

idt

(cf. FVR, IV, p. 26, §6).

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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