Corollary 2

Tag 02B7, permanent. Cite it as [Bourbaki, Tag 02B7].

Théories spectrales, chapter I, § 4. Calcul fonctionnel holomorphe

Let $\mu$ be an isolated point of Sp $_A(x)$ . Let Γ be the oriented boundary of an open disk Δ with center $\mu$ such that

Sp $_A(x)\cap (\Gamma \cup \Delta ) =\{\mu\}$ .

Then the idempotent $j_{\{\mu\}}$ associated with $x$ and $\{\mu\}$ is given by

j_{\{\mu\}}=\frac{1}{2i\pi}\int_{\Gamma}(z-x)^{-1}dz

*In other words, the idempotent $j_{\{\mu\}}$ is the residue at $\mu$ of the resolvent of $x.*$

For $z\in \mathbf{C}-$ Sp $_A(x)$ , one has

(z-u)^{-1}= R(x, z) = R_{\{\mu\}}(x, z) + R_H(x, z)

where H = Sp $_A(x)-\{\mu\}$ (formula (19)). The function $z\mapsto R_H(x, z)$ is holomorphic in $\mathbf{C}-$ H and in a neighbourhood of $\{\mu\}$ (Prop. 17, b)), hence

\frac{1}{2i\pi}\int_{\Gamma}R_H(x, z)dz= 0

(VAR, R2, p. 48, 11.2.5). The function $z\mapsto R_{\{\lambda\}}(x, z)$ is the resolvent of the element $j_{\{\mu\}}xj_{\{\mu\}}$ of the unital algebra $A_{\{\mu\}}$ . One then has

j_{\{\mu\}}=\frac{1}{2i\pi}\int_{\Gamma}R_{\{\mu\}}(x, z)dz

by Prop. 9 of I, p. 75 applied to $A_{\{\mu\}}$ and to the constant function 1 in a neighbourhood of $\Delta \cup \Gamma$ . The corollary follows.

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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