DT-operators and decomposability of Voiculescu's circular operator

Abstract

The DT-operators are introduced, one for every pair (µ, c) consisting of a compactly supported Borel probability measure µ on the complex plane and a constant c > 0. These are operators on Hilbert space that are defined as limits in * -moments of certain upper triangular random matrices. The DT-operators include Voiculescu's circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT-operator is strongly decomposable. We also show that a DT-operator generates a II 1 -factor, whose isomorphism class depends only on the number and sizes of atoms of µ. Those DT-operators that are also R-diagonal are identified. For a quasi-nilpotent DT-operator T, we find the distribution of T * T and a recursion formula for general * -moments of T.

On a Limiting Distribution of Singular Values of Random Band Matrices

“…To conclude the section we note that the results of Theorem 1.3 for ν < ∞, in particular those for the triangular random matrices generalize in part various results of works [5,6,8,10] obtained for matrices with independent entries by various methods. In Section 3 we outline the proof of Theorem 1.4 treating the case of independent entries under condition (1.22), applicable for both finite and infinite ν and based on the scheme developed in [15] to find the limiting eigenvalue distribution of a wide variety of random matrices.…”

“…The asymptotics (1.25) for the lower triangular matrices seems the most singular among the known so far. It follows from the results of [6] that for the matrices M (q)…”

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

“…It was computed in a number of special cases in [9], [2], [5], and [1]. In particular, it was proven in [9, Theorem 4.5] that if T ∈ M is R-diagonal in the sense of Nica and Speicher [16], then μ T can be determined from the S-transform of the distribution μ |T | 2 .…”

Hyperinvariant subspaces for some B–circular operators

“…Hence, [5], the operator T has trivial kernel (in fact, the distribution of T * T was explicitly determined there). Fixing any θ ∈ (0, d − c), we get s θ d−c (T ) = 0, and…”

“…The quasinilpotent DT-operator T measure is contained in the spectrum of the operator, quasinilpotent operators in II 1 -factors are of special interest. The quasinilpotent DT-operator T in the free group factor L(F 2 ), from the family of operators defined in [5], was a particularly compelling example to study. The operator T can be realized as a limit in * -moments of strictly upper triangular random matrices with i.i.d.…”