Brown measure of $R$-diagonal operators, revisited

TL;DR

This paper revisits the Brown measure of R-diagonal operators, reformulating existing approaches using subordination functions to simplify the derivation and establish connections between different methods.

Contribution

It introduces a simplified reformulation of the Brown measure for R-diagonal operators using subordination functions, linking previous approaches.

Findings

Simplified the proof of Brown measure for R-diagonal operators.

Connected Haagerup--Schultz and Belinschi-Šniady-Speicher approaches.

Expressed the Brown measure formula in terms of subordination functions.

Abstract

We use subordination functions perspective to reformulate Haagerup--Schultz's approach for the Brown measure of $R$-diagonal operators. This allows us to simplify the original argument and find a connection with the other approach due to Belinschi-\'{S}niady-Speicher. The Brown measure formula can be rewritten in terms of subordination functions.