Exercise 4, § 6

Let $E$ be a Banach space of countable type. Let $u\in \mathscr{L}(E)$. Suppose that the sequence $(\|u^k\|)_{k\in\mathbf{N}}$ is bounded.

For every $\lambda \in$ Sp($u$)$\cap \mathbf{U}$, let $x_{\lambda}\in E$ be a proper vector of $u$ of norm 1 relative to $\lambda$.

a) For all $\lambda_1,\lambda_2$ in Sp($u$)$\cap \mathbf{U}$ and every integer $k\in \mathbf{N}$, one has

b) There exists a real number $c >0$ such that $\|x_{\lambda_1}-x_{\lambda_2}\|\geqslant c$ for all $\lambda_1\not =\lambda_2$ in Sp($u$)$\cap \mathbf{U}$.

c) The set Sp($u$)$\cap \mathbf{U}$ is countable (« Jamison theorem »).

d) The Jamison theorem does not hold in general if $E$ is not of countable type.