On some properties of free commutators with semicircular variables

TL;DR

This paper explores properties of free commutators involving semicircular variables, showing their sum satisfies free convolution identities and that certain polynomials are freely infinitely divisible.

Contribution

It demonstrates that the sum of a semicircular variable and a free commutator satisfies a free convolution identity and establishes conditions for free infinite divisibility of related polynomials.

Findings

Sum of semicircular and commutator satisfies free additive convolution

Polynomial x + i[x, s] is freely infinitely divisible if x is

Provides new insights into free probability structures

Abstract

We investigate commutators of free variables of the form \( i[x, s] \), where \( s \) is a semicircular element. We show that although \( s \) and \( i[x, s] \) are not free, their sum nevertheless satisfies the free additive convolution identity \[ \mu_{s + i[x, s]} = \mu_s \boxplus \mu_{i[x, s]}. \] Furthermore, we prove that the polynomial \( x + i[x, s] \) is freely infinitely divisible whenever \( x \) itself is freely infinitely divisible.