DT-operators and decomposability of Voiculescu's circular operator

Invariant subspaces of Voiculescu's circular operator

We show that Voiculescu's circular operator and, more generally, each circular free Poisson operator has a continuous family of invariant subspaces relative to the von Neumann algebra it generates. The proof relies on upper triangular random matrix models and consequent realizations of these operators as upper triangular matrices of free random variables.

Decomposibility and norm convergence properties in finite von Neumann algebras

We study Schur-type upper triangular forms for elements, T, of von Neumann algebras equipped with faithful, normal, tracial states. These were introduced in a paper of Dykema, Sukochev and Zanin; they are based on Haagerup-Schultz projections. We investigate when the s.o.t.-quasinilpotent part of this decomposition of T is actually quasinilpotent. We prove implications involving decomposability and strong decomposability of T. We show this is related to norm convergence properties of the sequence |T^n|^{1/n} which, by a result of Haagerup and Schultz, is known to converge in strong operator to…

Moment Formulas For The Quasi-Nilpotent DT-Operator

Let T be the quasi-nilpotent DT-operator. By use of Voiculescu's amalgamated R-transform we compute the moments of $(T-λ1)^*(T-λ1)$, where $λ\in \mathbb C$, and the Brown-measure of $T+\sqrtε Y$, where Y is a circular element *-free from T for $ε>0$. Moreover we give a new proof of Śniady's formula for the moments $τ(((T^*)^k T^k)^n)$ for $k,n\in \mathbb N$.

Random matrices, continuous circular systems and the triangular operator

We present a Hilbert space approach to the limit joint *-distributions of complex independent Gaussian random matrices. For that purpose, we use a suitably defined family of creation and annihilation operators living in some direct integral of Hilbert spaces. These operators are decomposed in terms of continuous circular systems of operators acting between the fibers of the considered Hilbert space direct integral. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain t…

$R$-diagonal pairs - a common approach to Haar unitaries and circular elements

In the free probability theory of Voiculescu two of the most frequently used *-distributions are those of a Haar unitary and of a circular element. We define an $R$-diagonal pair as a generalization of these distributions by the requirement that their two-dimensional $R$-transform (or free cumulants) have a special diagonal form. We show that the class of such $R$-diagonal pairs has an absorption property under nested multiplication of free pairs. This implies that in the polar decomposition of such an element the polar part and the absolute value are free. Our calculations are based on combin…

两个自由随机变量之和的 Brown 测度，其中之一为三角椭圆

The Brown measure of a sum of two free random variables, one of which is triangular elliptic

本文研究了三角椭圆算子与自由随机变量之和的 Brown 测度，通过前推测度公式证明其密度绝对连续且有界，并将算子值从属函数推广至无界算子。