On existence of hyperinvariant subspaces for quasinilpotent operators with a nonsymmetry in the growth of the resolvent

Let T be a quasinilpotent operator on a Banach space. Under assumptions of a certain nonsymmetry in the growth of the resolvent of T, it is proved that every operator in the commutant of T is not unicellular. In particular, T has nontrivial hyperinvariant subspaces. The proof is based on a modification of the reasoning of [S].