Corollary 1

Let V be a neighbourhood of Sp$_e(u)$.

a) There exists a decomposition of E into a topological direct sum $F\oplus$ G such that F is of finite dimension, that F and G are stable under every endomorphism commuting with $u$, and that the spectrum of the endomorphism of G induced by $u$ is contained in V;

b) Suppose V nonempty. There exists an element $v$ of the bicommutant of $u$ in $\mathscr{L}(E)$ whose spectrum is contained in V and such that $v-u$ is of finite rank.

Let H = Sp($u$)$\cap (\mathbf{C}-\mathring{V})$. The set H is contained in Sp$_s(u)$, hence is discrete; since it is closed in the compact space Sp($u$), it is finite. One can take for F and G the image and the kernel of the spectral projector $e_H(u)$ (Prop. 2).

Suppose that V is nonempty. Let $\mu\in V$ and let $v$ denote the endomorphism of E which agrees with the homothety of ratio $\mu$ in F and with $u$ in G. Its spectrum is contained in $\{\mu\} \cup$ (Sp($u$) - H), hence in V, and $v-u$ is of finite rank since F is finite-dimensional.

For every compact subset S of $\mathbf{C}$, let $\mathscr{O}(S)$ denote the algebra of germs of holomorphic functions in a neighbourhood of S with values in $\mathbf{C}$ (I, p. 49, §4, No. $1$).