Moment Formulas For The Quasi-Nilpotent DT-Operator

Let T be the quasi-nilpotent DT-operator. By use of Voiculescu's amalgamated R-transform we compute the moments of $(T-\lambda 1)^*(T-\lambda 1)$, where $\lambda \in \mathbb C$, and the Brown-measure of $T+\sqrt{\epsilon} Y$, where Y is a circular element *-free from T for $\epsilon>0$. Moreover we give a new proof of \'Sniady's formula for the moments $\tau(((T^*)^k T^k)^n)$ for $k,n\in \mathbb N$.