Corollary

Let $u\in \mathscr{L}(E)$. If $u$ is normal, then Sp$_0(u)$ is empty, and if $u$ is hermitian, then Sp$_e(u)$ coincides with the spectrum of $\pi (u)$ relative to the algebra $\mathscr{C}$alk(E).

Both assertions result from the proposition and from the definition of the sets Sp$_n(u)$ for $n\in \overline{\mathbf{Z}}$ and of Sp$_{\omega}(u)$, which form a partition of the essential spectrum of $u$ (def. 2 of III, p. 83).