Exercise 3, § 1

We denote by A the subring of K formed by the elements $x\in K$ such that $|x|\leqslant 1$; it is a local ring whose maximal ideal is formed by the $x\in A$ such that $|x|<1$.

Let I be a set and E the Banach space $\mathscr{C}_0(I) ($cf. question c) of Exercise 1). For $i\in I$, denote by $\varphi_i\in E$ the characteristic function of $\{i\}$. Let $u\in \mathscr{L}^c(E)$ be such that $\|u\|\leqslant 1$.

a) The subspace $E_0$ of the elements $x\in E$ such that $\|x\|\leqslant 1$ is stable by $u$. b) For every nonzero ideal $\mathfrak{a}$ of A, the linear mapping $u$ defines by passing to the quotients an endomorphism $u_{\mathfrak{a}}$ of the $A/\mathfrak{a}$-module $E_{\mathfrak{a}}= E_0/\mathfrak{a}E_0$. This $A/\mathfrak{a}$-module is free and the image of $u_{\mathfrak{a}}$ is contained in a finitely generated submodule.

c) There exists a unique formal series $f_u\in A[[t]]$ such that, for every nonzero ideal $\mathfrak{a}$ of A, the image of $f_u$ in $(A/\mathfrak{a})[[t]]$ is equal to the polynomial det(1 $-tu_{\mathfrak{a}}$) defined in A, VIII, p. 455.

d) Let $v\in \mathscr{L}^c$(E), and let $c\in K^*$ be such that $\|cv\|\leqslant 1$. The formal series $f_{cv}$ is independent of the choice of $c$ verifying this property.

We shall denote det(1$-tv$) the formal series $f_v$ for every $c\in K^*$ such that $\|cv\|\leqslant 1$.