On some properties of free commutators with semicircular variables

Mihai Popa; Kamil Szpojankowski

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arXiv · 2511.13578

On some properties of free commutators with semicircular variables

Mihai Popa·Kamil Szpojankowski

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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.

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BibTeX RIS

Mihai Popa, Kamil Szpojankowski. 2025-11-17. On some properties of free commutators with semicircular variables. https://arxiv.org/abs/2511.13578

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