Brown's Spectral Distribution Measure for R-Diagonal Elements in Finite von Neumann Algebras

Abstract

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 {. In this paper we compute Brown's spectral distribution measure in case T has a polar decomposition T=UH where U is a Haar unitary and U and H are V-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 + T is expressed explicitly in terms of the S-transform of the distribution + T *T of the positive operator T *T. In case T is a circular element, i.e., T=(X 1 +iX 2 )Â-2 where (X 1 , X 2 ) is a free semicircular system, then sp T=D , the closed unit disk, and + T has constant density 1Â? on D . Academic Press

Invariant subspaces of Voiculescu's circular operator

“…We will rely on the result of Haagerup and Larsen [6,Example 5.2] that the spectrum of a circular free Poisson element of parameter c is {z ∈ C | √ c − 1 ≤ |z| ≤ √ c}. …”

On spectra and Brown's spectral measures of elements in free products of matrix algebras

“…. .. By Lemma 4.3 of [8], for λ ∈ C such that |λ| ≥ r(Z − 1), log ((Z − 1) − λ) = log |λ|. By the uniqueness of harmonic functions, we have log ((Z − 1) − λ) = log |λ| for λ ∈ B(λ 0 , δ).…”

“…If T is an R-diagonal operator and S is * -free with T , then both ST and T S are R-diagonal operators (see [12]). If T is an R-diagonal operator and n ∈ N, then T n is also an R-diagonal operator (see [8], [11]). For other important properties of R-diagonal operators, we refer to [8], [11], [12], [13].…”

Sample variance in free probability

“…This fact is indeed a consequence if we put A = [ 0 1 1 0 ] in the preceding proposition. In fact it was shown in [14] that every R-diagonal element can be written as a product of two free even selfadjoint elements. It is an interesting question what would be a natural factorization of R-cyclic matrices.…”