Corollary 2 (Lomonosov)

Let E be a separated locally convex space of dimension at least 2 over the field $\mathbf{C}$ and $u$ an endomorphism of E. Suppose that there exists a compact endomorphism $h\not = 0$ of E permutable with $u$. Then there exists a closed vector subspace F of E, distinct from $\{0\}$ and from E, such that $u(F)\subset F$.

In fact, Corollary 1 shows that the set $A =\{u, h\}$ cannot satisfy assumption (i) of Theorem 3.