Invariant subspaces of the quasinilpotent DT-operator

Abstract. In [4] we introduced the class of DT{operators, which are modeled by certain upper triangular random matrices, and showed that if the spectrum of a DT{operator is not reduced to a single point, then it has a nontrivial, closed, hyperinvariant subspace. In this paper, we prove that also every DT{operator whose spectrum is concentrated on a single point has a nontrivial, closed, hyperinvariant subspace. In fact, each such operator has a one{parameter family of them. It follows that every DT{operator generates the von Neumann algebra L(F2) of the free group on two generators. 1