Dt–operators and Decomposability of Voiculescu’s Circular Operator

The DT–operators are introduced, one for every pair (μ, c) consisting of a compactly supported Borel probability measure μ on the complex plane and a constant c > 0. These are operators on Hilbert space that are defined as limits in ∗– moments of certain upper triangular random matrices. The DT–operators include Voiculescu’s circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT–operator is strongly decomposable. We also show that a DT–operator generates a II1–factor, whose isomorphism class depends only on the number and sizes of atoms of μ. Those DT–operators that are also R–diagonal are identified. For a quasi–nilpotent DT–operator T , we find the distribution of T ∗T and a recursion formula for general ∗–moments of T .