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Journal of Operator Theory

Volume 62, Issue 2, Fall 2009  pp. 371-419.

Rectangular random matrices, entropy, and Fisher's information

Authors Florent Benaych-Georges
Author institution: LPMA, UPMC Univ Paris 6, Case courier 188, 4, Place Jussieu, 75252 Paris Cedex 05, France
CMAP, \'Ecole Polytechnique, route de Saclay, 91128 Palaiseau Cedex, France

Summary:  We prove that independent rectangular random matrices, when embedded in an algebra of larger square matrices, are asymptotically free with amalgamation over a commutative finite dimensional subalgebra $\D$ (under an hypothesis of unitary invariance). Then we consider elements of a $W^*$-probability space containing $\D$, which have kernel and range projection in $\D$. We associate to them a free entropy constructed with micro-states given by rectangular matrices. We also associate to them a free Fisher's information with a conjugate variables approach. Both approaches give rise to optimization problems whose solutions involve freeness with amalgamation over $\D$. It could possibly be a first proposition for the study of sets of operators between different Hilbert spaces with the tools of free probability. As an application, we prove a result of freeness with amalgamation between the two parts of the polar decomposition of $R$-diagonal elements with nontrivial kernel.

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