Lemma 5

Let E be a real Banach space, C a convex cone in E, and $u\in \mathscr{L}(E)$ such that $u(C-\{0\})\subset \mathring{C}$. Let $\ell \in C^{\circ}$ be an eigenvector of $^tu$. Then the kernel of $\ell$ is stable under $u$ and does not meet C $-\{0\}$.

Let $\lambda \in \mathbf{R}$ be such that $^tu(\ell ) =\lambda \ell$. For every $x\in$ Ker($\ell$ ), one has

therefore Ker($\ell$ ) is stable under $u$.

Let $x$ be a nonzero element of Ker($\ell$ ). For every element $y$ of E such that $\langle y, \ell \rangle <0$, one has $\langle u(x) +y, \ell \rangle <0$, whence it follows that $u(x) +$ $y \notin C$ since $\ell \in C^{\circ}$. It follows that $u(x)$ does not belong to $\mathring{C}$. Since $u(C-\{0\})\subset \mathring{C}$, this implies that $x \notin C$.