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

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 if and only if 0<p<2/n+1.