Brownʹs Spectral Distribution Measure for R-Diagonal Elements in Finite von Neumann Algebras

In 1983 L. G. Brown introduced a spectral distribution measure for non-normal elements in a finite von Neumann algebra M with respect to a fixed normal faithful tracial state τ. In this paper we compute Brownʹs spectral distribution measure in case T has a polar decomposition T=UH where U is a Haar unitary and U and H are *-free. (When Ker T={0} this is equivalent to that (T, T*) is an R-diagonal pair in the sense of Nica and Speicher.) The measure μT is expressed explicitly in terms of the S-transform of the distribution μT*T of the positive operator T*T. In case T is a circular element, i.e., T=(X1+iX2)/2 where (X1, X2) is a free semicircular system, then sp T=D, the closed unit disk, and μT has constant density 1/π on D.