Operator-valued free Fisher information of random matrices

Abstract

We study the operator-valued free Fisher information of random matrices in an operator-valued noncommutative probability space. We obtain a formula for $\Phi^\ast_{M_2(\mb)}(A,A^\ast,M_2(\mb),\eta)$, where $A\in M_2(\mb)$ is a $2\times 2$ operator matrix on $\mb$, and $\eta$ is linear operators on $M_2(\mb)$. Then we consider a special setting: $A$ is an operator-valued semicircular matrix with conditional expectation covariance, and find that $\Phi_\mb^\ast(c,c^\ast:\mb,id)=2Index(E)$, where $E$ is a conditional expectation of $\mb$ onto $\md$ and $c$ is a circular variable with covariance $E$.