Exercise 29, § 6

We take up again the notations of the preceding exercise; in particular, we suppose that C has nonempty interior. We suppose that $u$ is a linear mapping and that conditions (i), (ii) and (iii) are satisfied.

(a) If $u$ is a compact endomorphism of E, then conditions (iv) and (v) are satisfied.

We suppose that conditions (iv) and (vi) are verified. Let $(\lambda_m, x_m)$ be given by the preceding exercise.

(b) For every $\lambda \in \mathbf{R}_+$ such that $\lambda < \lambda_m$, the endomorphism $1_E-\lambda u$ is invertible. (Now prove first that $1_E-\lambda u$ induces by passing to the subspaces a bijection of $\mathring{C}$, cf. questions a) and f) of the preceding exercise; deduce then that $1_E-\lambda u$ is surjective, and then that it is injective by writing $x_1-x_2$ an element of the kernel of $1_E-\lambda u$, with $x_i\in \mathring{C}$, and by verifying that there exists $z\in E$ such that $x_1+z-\lambda u(x_1+z)\in \mathring{C}$.)

(c) Let $y$ in E be such that $y=\lambda_mu(y)$. Then one has $y\in C$. (There exists $c$ such that $cx_m+y\succ_C0$; conclude with the aid of question e) of the preceding exercise.) d) Let $C^{\circ}\subset E'$ be the polar of C. For every $x\in \mathring{C}$ and for every $\xi \in C^{\circ}-\{0\}$, one has $\langle x, \xi \rangle >0$.

(e) The polar $C^{\circ}$ of C is preserved by $^tu: E'\rightarrow E'$; for every $\lambda \in \mathbf{R}_+$ such that $\lambda < \lambda_m$, the endomorphism $1_{E'}-\lambda^tu$ of $E'$ is an isomorphism.