MPS-RR 2003-3
January 2003

Invariant subspaces of the quasinilpotent DT-operator

Uffe Haagerup

Ken Dykema

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 this paper, we pove that also every DT-operator whose spectrum is concentrated on a single point has a non-trivial, 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 $L(\mathbf{F}_{2})$ of the free group on two generators.

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