On Bi-R-Diagonal Pairs of Operators

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 the condition of bi-R-diagonality are developed.