Proposition 17

Let $\mu$ be an isolated point of Sp$_A(x)$. Then

a) For every $\lambda \in \mathbf{C}-$ Sp$_A(x)$, we have

$R(x, \lambda ) = R_{\{\mu\}}(x, \lambda ) + R_{Sp_A(x)-\{\mu\}}(x, \lambda )$ ;

b) The function which associates to $\lambda$ $R_{Sp_A(x)-\{\mu\}}(x, \lambda )$ is holomorphic in $\mathbf{C}-$ Sp$_A(x)$ and in a neighbourhood of $\mu$; moreover, the function which associates to $\lambda$ $R_{\{\mu\}}(x, \lambda )$ is holomorphic in $\mathbf{C}-\{\mu\}$;

c) One has

lim$_{n\rightarrow+\infty}\|(x-\mu)^nj_{\{\mu\}}\|^{1/n}= 0$

and, for $\lambda \in \mathbf{C}-\{\mu\}$, the formula

The foregoing implies assertions a) and b). Let us prove c). Replacing $x$ by $x-\mu$, we are reduced to the case where $\mu= 0$. Put $H =\{0\}$; this is an open and closed subset of Sp$_A(x)$. By formula (18), the spectrum of $x_H$ in $A_H$ is $\{0\}$, hence $x_H$ is quasi-nilpotent, that is to say, $\|x^nj_H\|^{1/n}=\|(xj_H)^n\|^{1/n}$ tends to 0 as $n$ tends to $+\infty$. Moreover, for $\lambda \not= 0$, one has in $A_H$

(Theorem 1 of I, p. 24, d)), whence (20).