Proposition 5

Let E be a Banach space and $u_0$ a Riesz endomorphism of E. Let N (resp. I) be the nilspace (resp. the conilspace) of $u_0$. Let us denote by $d$ the dimension of N. There exist an open neighbourhood U of $u_0$ in $\mathscr{L}(E)$ and an analytic mapping $\pi : U\rightarrow \mathscr{L}(E)$ such that

(i) The endomorphism $\pi (u_0)$ is the projector with image N and kernel I;

(ii) For every $u\in U$, the linear mapping $\pi (u)$ is a projector of rank $d$ belonging to the bicommutant of $u$ and, in particular, commuting with $u$. Moreover, $u$ induces an automorphism of Ker($\pi (u)$);

(iii) Every element of U is a Riesz endomorphism whose nilspace has dimension $\leqslant d$.

If K = $\mathbf{C}$, let Sp($u_0$) denote the spectrum of $u_0$ relative to the algebra $\mathscr{L}(E)$. When K = $\mathbf{R}$, let Sp($u_0$) denote the complex spectrum Sp$_{\mathscr{L}(E_{(\mathbf{C})})}((u_0)_{(\mathbf{C})})$ of $u_0($I, p. 85, n$^o13$). By prop. 14 of III, p. 54, the point 0 is isolated in $\{0\} \cup$ Sp($u_0$). Let $r >0$ be a real number such that every element of Sp($u_0$)$-\{0\}$ has modulus $> r$. Let V denote the open set in $\mathbf{C}$ consisting of the complex numbers of absolute value $\not =r$; let $f$ be the holomorphic function on V defined by $f(z) = 0$ if $|z|> r$ and $f(z) = 1$ if $|z|< r$. If $K =\mathbf{C}$ (resp. $K =\mathbf{R}$), let $U'$ denote the set of elements $u$ of $\mathscr{L}(E)$ whose spectrum (resp. whose complex spectrum) is contained in V. The set $U'$ is an open neighbourhood of $u_0$ in $\mathscr{L}(E)$ and the mapping $u\mapsto f(u)$ of $U'$ into $\mathscr{L}(E)$ is holomorphic (I, p. 76, prop. 10 and I, p. 85, n$^o13$).

Let $u\in U'$. The endomorphism $f(u)$ is the spectral projector $e_0(u)$; it commutes with $u$, and $u$ induces by passing to the subspaces an automorphism of Ker($e_0(u)$) $($cf. III, p. 53, n$^o6$).

The projector $e_0(u_0)$ has image N and kernel I (III, p. 54, prop. 14); its rank is $d$. By lemma 1, the set U of elements $u\in U'$ such that $e_0(u)$ is of rank $d$ is an open neighbourhood of $u_0$. The set U and the mapping $\pi :u\mapsto e_0(u)$ of U into $\mathscr{L}(E)$ satisfy conditions (i) and (ii) of the proposition.

Let $u\in U$. Since $u$ induces an automorphism of Ker($e_0(u)$), which is a closed subspace of finite codimension of E, the endomorphism $u$ is a Riesz endomorphism of E (III, p. 48, prop. 8), the nilspace of $u$ is contained in the image of $\pi (u) =e_0(u)$, and is of dimension $\leqslant d$.