Exercise 11, § 6

Let E be a complete normable space over $\mathbf{C}$, $v \in \mathscr{L}(E)$ and $g = \exp(v)$. Suppose that $\mathrm{Sp}(v) \cap 2i\pi(\mathbf{Z} - \{0\}) = \emptyset$. Let $E_1, E_2$ be closed vector subspaces of E which are stable under v, such that $E_2 \subset E_1$. Suppose that the automorphism of $E_1/E_2$ derived from g is the identity. Then $v(E_1) \subset E_2$.