On quasinilpotent operators

In this note we modify a new technique of Enflo for producing hyperinvariant subspaces to obtain a much improved version of his "two sequences" theorem with a somewhat simpler proof. As a corollary we get a proof of the "best" theorem (due to V. Lomonosov) known about hyperinvariant subspaces for quasinilpotent operators that uses neither the Schauder-Tychonoff fixed point theorem nor the more recent techniques of Lomonosov.