Brown measures of unbounded operators affiliated with a finite von Neumann algebra

Abstract

In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite von Neumann algebra. Moreover, we compute the Brown measure of all unbounded R-diagonal operators in this class. As a particular case, we determine the Brown measure z = xy -1, where (x, y) is a circular system in the sense of Voiculescu, and we prove that for all n ∈ N, zn ∈ Lp (ℳ, τ) if and only if 0 < p < 2/n+1.