# Analytic subordination for bi-free convolution

## TL;DR

This paper explores the analytic properties of bi-free additive convolution, deriving simpler formulas through Voiculescu's subordination functions and analyzing the atomic structure of these convolutions.

## Contribution

It introduces new formulas for bi-free convolutions based on subordination functions and applies them to study atoms in bi-free additive convolutions.

## Key findings

- Derived simpler formulas for bi-free convolutions
- Proved results about the atomic structure of bi-free additive convolutions
- Extended properties of Voiculescu's subordination functions to bi-free setting

## Abstract

In this paper we study some analytic properties of bi-free additive convolution, both scalar and operator-valued. We show that using properties of Voiculescu's subordination functions associated to free additive convolution of operator-valued distributions, simpler formulas for bi-free convolutions can be derived. We use these formulas in order to prove a result about atoms of bi-free additive convolutions.