Corollary

Suppose that the unbounded connected component of the complement of Sp$_{\omega}(u)$ contains 0. Then $u+h$ is a Riesz endomorphism of E.

One has Sp$_{\omega}(u+h) =$ Sp$_{\omega}(u)$ (Theorem 2), hence 0 belong to the unbounded connected component of $\mathbf{C}-$ Sp$_{\omega}(u+h)$. By the corollary to Theorem 1 of III, p. 87, either 0 does not belong to the spectrum of $u+h$, or else it is a sensitive point of this spectrum. In both cases, $u+h$ is a Riesz endomorphism of E.