2017/10/13 by Mingchu Gao, Gao, Mingchu
Mathematics · #46L54 #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1710.05087
openalex publication_date 2017/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the partial bi-free S-transform of a pair (a,b) of random variables, and the S-transform of the 2× 2 matrix-valued random variable (\beginmatrixa&0 0&b\endmatrix) associated with (a,b) when restricted to upper triangular 2× 2 matrices. We first derive an explicit expression of bi-free multiplicative convolution (of probability measures on the bi-unit-sphere \mathbbT2 of ℂ2, or on ℝ2+ in ℂ2) from a subordination equation for bi-free multiplicative convolution. We then show that, when (a1, b1) and (a2,b2) are bi-free, the S-transforms of X1=(\beginmatrixa1&0 0&b1\endmatrix), X2=(\beginmatrixa2&0 0&b2\endmatrix) satisfy Dykema's twisted multiplicative equation for free operator-valued random variables if and only if at least one of the two partial bi-free S-transforms of the pairs of random variables is the constant function 1 in a neighborhood of (0,0). This is the case if and only if one of the two pairs, say (a1,b1), has factoring two-band moments (that is, φ(a1mb1n)=φ(a1m)φ(b1n), for all m,n=1, 2, ⋯). We thus find tons of bi-free pairs of random variables to which the S-transforms of the corresponding matrix-value random variables do not satisfy Dykema's twisted multiplicative formula. Finally, if both (a1,b1) and (a2,b2) have factoring two-band moments, we prove that the Ψ-transforms of X1, X2, and X1X2 satisfy a subordination equation.