Corollary 2

Let E be a separated locally convex space, $u$ a Riesz endomorphism of E and $h\in \mathscr{L}(E)$ a compact linear mapping. If $u$ and $h$ commute, then $u+h$ is a Riesz endomorphism of E.

Let $v$ be the canonical quasi-inverse of $u$. It commutes with $u$ and with $h($III, p. 47, prop. 6), hence with $u+h$. The endomorphism $1_E-(u+h)\circ v$ of E is the sum of the continuous linear mapping of finite rank $1_E-u\circ v$ and the compact linear mapping $-h\circ v$, hence is compact. It follows that $u+h$ is a Riesz endomorphism of E (corollary 1).