Some minimization problems for the free analogue of the Fisher information

We consider the free non-commutative analogue Φ, introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value of Φ(a, a), if a is a non-commutative random variable subject to the constraint that the distribution of aa is prescribed. More generally, we obtain the minimal possible value of Φ( {aij , aij}1≤i,j≤d ), if {aij}1≤i,j≤d is a family of noncommutative random variables such that the distribution of AA is prescribed, where A is the matrix (aij) d i,j=1. The d× d-generalization is obtained from the case d = 1 via a result of independent interest, concerning the minimal value of Φ({aij , aij}1≤i,j≤d) when the matrix A = (aij) d i,j=1 and its adjoint have a given joint distribution. (A version of this result describes the minimal value of Φ({bij}1≤i,j≤d) when the matrix B = (bij) d i,j=1 is selfadjoint and has a given distribution.) We then show how the minimization results obtained for Φ lead to maximization results concerning the free entropy χ, also defined by Voiculescu. Research supported by a grant from the Natural Sciences and Engineering Research Council, Canada. Supported in part by a Graduate Research Fellowship of the National Science Foundation, USA. Supported by a Heisenberg Fellowship of the DFG, Germany.