Moment Formulas For The Quasi-Nilpotent DT-Operator

Abstract

Let T be the quasi-nilpotent DT-operator. By use of Voiculescu's amalgamated R-transform we compute the moments of $(T-\lambda 1{)}^{\ast }(T-\lambda 1)$, 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 Śniady's formula for the moments $\tau (((T^{\ast }{)}^k{T}^k{)}^n)$ for $k,n\in \mathbb{N}$.

50 pages