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 [ μ{s + i[x, s]} = μ_s \boxplus μ{i[x, s]}. ] Furthermore, we prove that the polynomial ( x + i[x, s] ) is freely infinitely divisible whenever ( x ) itself is freely infinitely divisible.