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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 II 1 -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.

The algebra Mul [[B]] of formal multilinear function series over an algebra B and its quotient SymMul [[B]] are introduced, as well as corresponding operations of formal composition. In the setting of Mul [[B]], the unsymmetrized R-and T-transforms of random variables in B-valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usual R-and T-transforms (the latter being just the reciprocal of the S-transform), but describe all moments of a random variable, not only the symmetric moments. The partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized T-transform.