On Bi-R-Diagonal Pairs of Operators

TL;DR

This paper explores bi-R-diagonal pairs of operators in bi-free probability, demonstrating their stability under powers and certain unitary transformations, and providing new characterizations of these pairs.

Contribution

It introduces the concept of bi-R-diagonal pairs, analyzes their properties under multiplication and powers, and develops new equivalent characterizations in bi-free probability.

Findings

Powers of bi-R-diagonal pairs are also bi-R-diagonal.

Joint $*$-distribution remains invariant under multiplication by bi-Haar unitaries.

New characterizations of bi-R-diagonal pairs are established.

Abstract

We study the properties of the analogue of R-diagonal operators in the setting of bi-free probability. Products of bi-R-diagonal pairs of operators that are $*$-bi-free are studied and powers of such pairs are found to also be bi-R-diagonal. It is moreover shown that the joint $*$-distribution of a bi-R-diagonal pair of operators remains invariant under the multiplication by a $*$-bi-free bi-Haar unitary pair and equivalent characterizations of bi-R-diagonal pairs are developed.