Corollary

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 $u$ be a compact endomorphism of E such that $u(C)\subset C$. Suppose that there exists an integer $k\geqslant 1$ such that $u^k(C-\{0\})\subset \mathring{C}$.

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) The only eigenvectors of $u$ in C are the multiples of $x_0$.

Put $\varrho =\varrho (u)$. We may apply Theorem 4 to $u$ and Proposition 8 to $u^k$. There therefore exists an eigenvector $x_0$ of $u$ in C for the eigenvalue $\varrho$. Since $0< \varrho (u^k) =\varrho^k$ (formula (4) of I, p. 21), we have in particular $\varrho >0$, and since $x_0$ is an eigenvector of $u^k$ for the eigenvalue $\varrho (u^k)$, we have $x_0\in \mathring{C}$.

We have Sp($u^k_{(\mathbf{C})}$) $=$ Sp($u_{(\mathbf{C})}$)$^k$ (remark 4 of I, p. 2), hence every eigenvalue $\lambda \in \mathbf{C}$ of $u_{(\mathbf{C})}$ such that $|\lambda |=\varrho$ satisfies $\lambda^k=\varrho^k$, and since every eigenvector corresponding to $\lambda$ is an eigenvector of $u^k$ for $\varrho^k$, hence proportional to $x_0$, we have $\lambda =\varrho$.

If the spectral projector of $u$ for the eigenvalue $\varrho$ were of rank at least 2, the same would be true of that of $u^k$ for the eigenvalue $\varrho^k($cf. No.$^o2$ of I, p. 129).

Finally, if $x\in C$ is an eigenvector of $u$, it is also one for $u^k$, hence is proportional to $x_0$.

Let $n$ be an integer $\geqslant 1$. The set $\mathbf{R}_+^n$ is a closed pointed salient convex cone in $\mathbf{R}^n$, with nonempty interior equal to $(\mathbf{R}_+^*)^n$.

Let A $= (a_{i,j})$ be a real matrix of type $(n, n)$, and $u_A$ the endomorphism $x\mapsto Ax$ of $\mathbf{R}^n$. Let $\varrho =\varrho (u_A)$ be its spectral radius.