Corollary

a) Let Ω be the unbounded connected component of $\mathbf{C}-$ Sp$_{\omega}(u)$. We have $\Omega \cap$ Sp($u$)$\subset$ Sp$_s(u)$;

b) Any point adherent to Sp$_s(u)$ which does not belong to Sp$_s(u)$ belongs to Sp$_{\omega}(u)$.

Assertion a) is a direct consequence of assertion b) of th. 1. Let $\lambda$ be a point adherent to Sp$_s(u)$ which does not belong to Sp$_s(u)$. It belongs to the spectrum of $u$, since the latter is closed. It does not belong to any of the sets Sp$_n(u)$, for $n\in \overline{\mathbf{Z}}$, since these are open (loc. cit.) and disjoint from Sp$_s(u)$. We therefore have $\lambda \in$ Sp$_{\omega}(u)$, whence b).