Corollary 2

Let $f\in \mathscr{O}$(Sp($u$)) and let $\mu$ be a complex number. Let H be the set of $\lambda \in$ Sp($u$) such that $f(\lambda ) =\mu$. Then $\mu$ is a point of the discrete spectrum of $f(u)$ if and only if H is finite, nonempty, and contained in the discrete spectrum of $u$. Under these conditions, the spectral projector $e_\mu(f(u))$ is equal to the projector $e_H(u)$ associated with $u$ and H. One has

and the spectral multiplicity of $\mu$ for $f(u)$ is the sum of the spectral multiplicities of the elements of H, that is to say

The complex number $\mu$ belong to $f$(Sp($u$)), and hence to the spectrum of $f(u)$ (Prop. 8 of I, p. 75), if and only if H is nonempty. One then has $e_\mu(f(u)) =e_H(u)$ (No. $1$ of I, p. 127). The other assertions are deduced from this by Prop. 2.