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On Bi-R-Diagonal Pairs of Operators

Georgios Katsimpas

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.

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 bi-R-diagonal pairs are developed.

Paper Structure

This paper contains 8 sections, 28 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. The Lattice of Bi-Non-Crossing Partitions
  4. Bi-Free Independence and Bi-Free Cumulants
  5. Bi-R-Diagonal Pairs of Operators
  6. Operator-Valued Bi-Free Independence and R-cyclic Pairs of Matrices
  7. Operations Involving Bi-R-Diagonal Pairs
  8. Joint $*$-Distributions of Bi-R-Diagonal Pairs

Key Result

Lemma 2.4

Let $n\in\mathbb{N}$ and $\chi\in {\{l,r\}}^n$. Then for all $\tau\in\mathop{\mathrm{BNC}}\nolimits(\chi)$ we have In particular, where $C_n$ denotes the $n$-th Catalan number.

Theorems & Definitions (70)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 60 more