Operator-Valued Infinitesimal Multiplicative Convolutions

Abstract

We consider the notions of operator-valued infinitesimal (OVI) free independence, OVI Boolean independence, and OVI monotone independence. For each notion of OVI independence, we introduce the corresponding infinitesimal transforms, and then we show that the transforms satisfy certain multiplicative property. We also provide an application on complex Wishart matrices via our infinitesimal free multiplicative formula.

Asymptotic free independence and entry permutations for Gaussian random matrices

“…This notion, under in the framework free probability of type B, was introduced in [6] and the infinitesimal freeness interpretation was found in [5], see also [11] for combinatorial developments related to infinitesimal freeness. As mentioned above asymptotic freeness connects directly to random matrix theory and several classes of random matrices have been proved to be asymptotically infinitesimally free [17,8,22] and many related properties have been discovered [11,26,27,7,10,12].…”