We study the unique solution m of the Dyson equation− m (z)− 1= z1− a+ S [m (z)] on a von
Neumann algebra A with the constraint Im m≥ 0. Here, z lies in the complex upper half …

We consider general self-adjoint polynomials in several independent random matrices
whose entries are centered and have the same variance. We show that under certain …

We provide a concrete formula for a dual system as well as for a conjugate system of q-
Gaussians represented on the q-deformed Fock space. Moreover, using this formula, we …

This paper contributes to the current studies on regularity properties of noncommutative
distributions in free probability theory. More precisely, we consider evaluations of selfadjoint …

One of the main applications of free probability is to show that for appropriately chosen
independent copies of d random matrix models, any noncommutative polynomial in these d …

We provide bounds of Berry-Esseen type for fundamental limit theorems in operator-valued
free probability theory such as the operator-valued free Central Limit Theorem and the …

Abstract Let X 1,…, X n be operators in a finite von Neumann algebra and consider their
division closure in the affiliated unbounded operators. We address the question when this …

We calculate the Fuglede-Kadison determinant of arbitrary matrix-valued semicircular
operators in terms of the capacity of the corresponding covariance mapping. We also …

The so-called density of states is a Borel probability measure on the real line associated with
the solution of the Dyson equation which we set up, on any fixed $ C^\ast $-probability …

This thesis is devoted to some problems on non-commutative rational functions in non-
commutative random variables that come from free probability theory and from random …