Proposition 8

Let E be a nonzero complete normable space over $\mathbf{R}$. Let C be a closed salient convex cone with nonempty interior $\mathring{C}$ in E, and let $u\in \mathscr{L}(E)$ be a compact linear mapping such that $u(C-\{0\})\subset \mathring{C}$.

a) One has $\varrho (u)>0$ and there exists an eigenvector $x_0$ of $u$ in $\mathring{C}$ for the eigenvalue $\varrho (u)$;

b) The spectral projector of $u$ corresponding to $\varrho (u)$ is of rank 1;

c) For every eigenvalue $\lambda \not =\varrho (u)$ of $u_{(\mathbf{C})}$, one has $|\lambda |< \varrho (u)$;

d) Let F be a subspace of E stable under $u$ such that (C $-\{0\}$)$\cap F$ is nonempty. Then $x_0\in F$. In particular, the only eigenvectors of $u$ in C are the multiples of $x_0$.

Let us prove two preliminary lemmas.