Hyperinvariant subspaces for a class of quasinilpotent operators

Quasinilpotent operators on Hilbert space are very little understood. Except for the classification, up to similarity, as parts of quasinilpotent backward weighted shifts of infinite multiplicity (Foiaş and Pearcy, 1974), and the recently introduced technique of extremal vectors (see the references), there are few known structure theorems or theorems proving the existence of invariant or hyperinvariant subspaces for such operators. In this paper we use a structure theorem for the class of weakly centered operators (Paulsen et al., 1995) to obtain a structure theorem for a certain subclass of quasinilpotent operators that immediately yields the existence of hyperinvariant subspaces for such operators.