Proposition 10

Let E and F be locally convex spaces, $p\in \mathscr{L}(E; F)$ and $q\in \mathscr{L}(F; E)$. Put $u= 1_E-q\circ p$ and $v= 1_F-p\circ q$.

a) Let $n\in \mathbf{N}$. The mapping $p$ defines by restriction an isomorphism of topological vector spaces of Ker($u^n$) onto Ker($v^n$) and defines by passing to the quotients an isomorphism of topological vector spaces of Coker($u^n$) onto Coker($v^n$);

b) If the image of $u$ is closed in E, that of $v$ is closed in F;

c) If Ker($u$) has a topological complement in E, then Ker($v$) has one in F. If Im($u$) has a topological complement in E, then Im($v$) has one in F;

d) If $u$ is a strict morphism, then $v$ is a strict morphism;

e) If $u$ is an automorphism of E, then $v$ is an automorphism of F;

f) If $u$ is a Fredholm endomorphism of E, then $v$ is a Fredholm endomorphism of F, and one has ind($u$) $=$ ind($v$);

g) Suppose E and F separated. If $u$ is a Riesz endomorphism of E, then $v$ is a Riesz endomorphism of F.

First of all, let us note the formulas

We then prove two lemmas.