Proposition 9

Let E be a separated locally convex space and let $u,v$ be permutable elements of $\mathscr{L}(E)$. The following conditions are equivalent:

(i) The endomorphisms $u$ and $v$ are Riesz endomorphisms;

(ii) The endomorphism $u\circ v$ is a Riesz endomorphism.

Suppose that $u$ and $v$ are Riesz endomorphisms; let $u'$ and $v'$ denote their canonical quasi-inverses. The endomorphisms $u,v,u'$ and $v'$ of E commute (prop. 6, d)). The endomorphism $v'\circ u'$ is a quasi-inverse of $u\circ v$ which commutes with $u\circ v$, hence $u\circ v$ is a Riesz endomorphism (prop. 8).

Suppose conversely that $u\circ v$ is a Riesz endomorphism, and let $w$ be its canonical quasi-inverse. Since $u$ commutes with $u\circ v$, the endomorphisms $u$ and $w$ commute by prop. 6, d). Analogously $v$ and $w$ commute. Hence the endomorphisms $u,v$ and $w$ of E commute; it follows that the endomorphism $v\circ w$ of E is a quasi-inverse of $u$ and that $w\circ u$ is a quasi-inverse of $v$. By prop. 8$,u$ and $v$ are Riesz endomorphisms.

Let $u$ be a Riesz endomorphism of E. If $v$ is an automorphism of E, the endomorphism $u\circ v$ of E is not necessarily a Riesz endomorphism of E, even if $u\circ v-v\circ u$ is of finite rank. If $h$ is an endomorphism of finite rank of E, the endomorphism $u+h$ is not necessarily a Riesz endomorphism (cf. III, p. 120, exercise 3).