Exercise 2, § 5

Let $E$ be a Banach space of infinite dimension and $u\in \mathscr{L}(E)$. We denote by $S(u)$ the set of complex numbers $\lambda$ such that there exist a real number $\delta >0$ and a subspace F of finite codimension of E such that $\|u(x)-\lambda x\|\geqslant \delta \|x\|$ for all $x\in F$.

a) The set $S(u)$ is open in $\mathbf{C}$ and contains the complement in $\mathbf{C}$ of the spectrum of $u$.

b) If $\lambda \in S(u)$, the endomorphism $u-\lambda 1_E$ has a closed image and a kernel of finite dimension.

c) The complement $F(u) =\mathbf{C}-S(u)$ is not empty. (Now prove that an element of Sp$_e(u)$ of maximal modulus belongs to $F(u).$)