Exercise 12, § 7

Let $u$ be an endomorphism of a Banach space E. We denote by $\sigma_a(u)$ the set of $\lambda \in \mathbf{C}$ such that, for every $\alpha \in \mathbf{R}_+^*$, there exists $x\in E$ such that $\|\lambda x-u(x)\|< \alpha \|x\|$.

a) Exercise prove that $\sigma_a(u)$ is a closed subset of $\sigma (u)$.

b) Exercise prove that $\sigma_a(u)$ contains the boundary of $\sigma (u)$.