Definition 2

Tag 0253, permanent. Cite it as [Bourbaki, Tag 0253].

Théories spectrales, chapter I, § 1. Spectres et caractères

Let A be a unital algebra over K and $x\in A$ . For every $\lambda \in K$ - Sp( $x$ ), one sets

R(x, \lambda ) = (\lambda e-x)^{-1}

The mapping of K - Sp( $x$ ) into A given by $\lambda \mapsto R(x, \lambda )$ is called the resolvent of $x$ .

For fixed $x$ , the values of $R(x, \lambda )$ are pairwise permutable. If $\lambda , \mu\in K$ , one has:

(\lambda -\mu)e= (\lambda e-x)-(\mu e-x)

hence, if $\lambda , \mu\in K$ - Sp( $x$ ), one has the relation

(\lambda -\mu)R(x, \lambda )R(x, \mu) = R(x, \mu)-R(x, \lambda ) \tag{1}

If $x, y\in A$ and $\lambda \in K$ , one has:

y-x= (\lambda e-x)-(\lambda e-y)

hence, if $\lambda \in K$ - (Sp( $x$ ) $\cup$ Sp( $y$ )), one has the relation

R(y, \lambda )(y-x)R(x, \lambda ) = R(y, \lambda )-R(x, \lambda ) \tag{2}

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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