Lemma 2

Let E be a complex Hilbert space and let $u$ be a normal endomorphism of E. Let $(E_i)_{i\in I}$ be a finite family of closed subspaces of E, stable under $u$ and pairwise orthogonal, such that $E =\bigoplus_{i\in I}E_i$. For every $i\in I$, let $u_i$ denote the endomorphism of $E_i$ induced by $u$. One has Sp($u$) $=\bigcup_{i\in I}$ Sp($u_i$), and for every $f\in \mathscr{C}$(Sp($u$)), the endomorphism $f(u)$ stabilises the spaces $E_i$, and $f(u)$ coincides with $f(u_i)$ on $E_i$.

The proof follows that of lemma 1 of I, p. 128, using the remark of 6 of I, p. 110 and prop. 8 of I, p. 112.