Lemma 1

Let E be a Banach space and let $u$ be an endomorphism of E. Let $(E_i)_{i\in I}$ be a finite family of closed subspaces of E, stable under $u$, such that $E =\bigoplus_{i\in I}E_i$. For each $i\in I$, let $u_i$ denote the endomorphism of $E_i$ deduced from $u$. Then Sp($u$) $=\bigcup_{i\in I}$ Sp($u_i$), and for every $f\in \mathscr{O}$(Sp($u$)), the endomorphism $f(u)$ stabilises the spaces $E_i$, and $f(u)$ coincides with $f(u_i)$ on $E_i$.

The endomorphism $u$ is an isomorphism if and only if $u_i$ is an isomorphism for every $i\in I$. Applying this property to $u-\lambda 1_E$, we deduce that Sp($u$) is the union of the sets Sp($u_i$) for $i\in I$.

Let $f\in \mathscr{O}$(Sp($u$)). The endomorphism $f(u)$ of E belongs to the bicommutant of $u$ in $\mathscr{L}(E)$ (Theorem 5 of I, p. 74), and therefore commutes with the projectors $p_i$. It therefore stabilizes the spaces $E_i$. Consider the continuous unital morphism $\varpi$ of the product algebra $\prod_{i\in I}\mathscr{L}(E_i)$ into $\mathscr{L}(E)$ defined by $(v_i)_{i\in I}\mapsto \bigoplus_iv_i$. It maps the family $(u_i)$ to $u$. Proposition 7 of I, p. 75 then implies that Sp($u$)$\subset \bigcup_{i\in I}$ Sp($u_i$) and

which concludes the proof of the lemma.