On some properties of free commutators with semicircular variables

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.