Corollary

Let E be a complete normable space over $\mathbf{R}$. Let C be a total salient closed convex cone in E and $u\in \mathscr{L}(E)$ a compact linear mapping such that $u(C)\subset C$. If the spectral radius $\varrho (u)$ of $u$ is $>0$, then $\varrho (^tu) =\varrho (u)$ is an eigenvalue of $^tu$, and $^tu$ admits an eigenvector corresponding to $\varrho (u)$ in $C^{\circ}$.

One has $\varrho (^tu) =\varrho (u)$ (prop. 3 of I, p. 131). By Corollary 1 of III, p. 9, the endomorphism $^tu$ of $E'$ is compact. Moreover, one has $^tu(C^{\circ})\subset C^{\circ}$; the assertion then follows from the Krein-Rutman theorem applied to $^tu$, and to the closed convex cone $C^{\circ}$, since the latter is salient (because C is total) and total (because C is salient).