Proposition 6

Let E be a separated locally convex space and let $u$ be a Riesz endomorphism of E. Let N denote the nilspace of $u$ and I its conilspace.

a) The subspaces N and I of E are closed and stable under $u$, and the space E is their topological direct sum;

b) The endomorphism $u$ defines by restriction an automorphism $u_I$ of I;

c) The vector space N is finite-dimensional and $u$ defines by restriction a nilpotent endomorphism $u_N$ of N;

d) Let $v$ be the element of $\mathscr{L}(E)$ which is zero on N and coincides with $u^{-1}_I$ on I. It is a Riesz endomorphism of E and a quasi-inverse of $u$ which commutes with $u$. The endomorphism $1_E-u\circ v$ of E is the projector with image N and kernel I. Every element of $\mathscr{L}(E)$ which commutes with $u$ stabilizes N and I, and commutes with $v$.

Let $n$ be a natural number such that Ker($u^n$) $= N$ and Im($u^n$) $= I$. Since E is separated and $u$ is continuous, N = Ker($u^n$) is closed in E. Since $u$ is a Fredholm endomorphism, the same is true of $u^n$, and I = Im($u^n$) is closed (III, p. 42, prop. 2). By lemma 2, the space E is the direct sum of N and I and assertion a) then follows from prop. 3 of EVT, I, p. 15.

Since I is closed, of finite codimension in E and stable under $u$, the mapping $u_I: I\rightarrow I$ deduced from $u$ is a Fredholm endomorphism (III, p. 44, prop. 3); it is also bijective, hence it is an automorphism of I (III, p. 42, prop. 2). This proves b), and the remaining assertions follow from lemma 2.

The endomorphism $v$ defined in assertion d) of the proposition is a quasi-inverse of $u$, called the canonical quasi-inverse of $u$.

Riesz endomorphisms have certain of the stability properties of Fredholm mappings.