On series of free $R$-diagonal operators

TL;DR

This paper establishes an analogue of the three series theorem for series of free $R$-diagonal operators, linking almost uniform convergence to the convergence of two numerical series.

Contribution

It introduces a new convergence criterion for series of free $R$-diagonal operators, extending classical results to free probability theory.

Findings

Series of free $R$-diagonal operators converge almost uniformly if two associated numerical series converge.

The paper generalizes the classical three series theorem to the setting of free probability.

Provides a new tool for analyzing convergence in free probability contexts.

Abstract

For a series of free $R$-diagonal operators, we prove an analogue of the three series theorem. We show that a series of free $R$-diagonal operators converges almost uniformly if and if two numerical series converge.