Operator-Valued Infinitesimal Multiplicative Convolutions

TL;DR

This paper develops operator-valued infinitesimal free, Boolean, and monotone independence theories, introduces corresponding transforms with multiplicative properties, extends t-coefficients, and applies the framework to complex Wishart matrices.

Contribution

It introduces the concept of operator-valued infinitesimal independence and transforms, extending existing theories and providing new tools for analyzing complex random matrices.

Findings

Infinitesimal transforms satisfy multiplicative properties.

Extension of t-coefficients to the infinitesimal setting.

Application to complex Wishart matrices demonstrates practical utility.

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. Additionally, we extend the concept of $t$-coefficients to the infinitesimal framework and investigate its properties. Finally, we present an application involving complex Wishart matrices utilizing our infinitesimal free multiplicative formula.