Corollary 1

Let $\mu$ be an isolated point of Sp$_A(x)$ and $p$ a strictly positive integer. In order that $\mu$ be a pole of order $p$ of the resolvent of $x($cf. VAR, R1, p. 30, 3.3.9), it is necessary and sufficient that $(x-\mu)^{p-1}j_{\{\mu\}}\not= 0$ and $(x-\mu)^pj_{\{\mu\}}= 0$.