Ken Dykema and Uffe Haagerup
The DT--operators are introduced, one for every pair $(\mu,c)$ consisting of a compactly supported Borel probability measure $\mu$ 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 II$_1$--factor, whose isomorphism class depends only on the number and sizes of atoms of $\mu$. 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 $T$.