Some minimization problems for the free analogue of the Fisher information

We consider the free non-commutative analogue Phi*, introduced by D. Voicul escu, of the concept of Fisher information for random variables. We determi ne the minimal possible value of Phi*(a, a*), if a is a non-commutative ran dom variable subject to the constraint that the distribution of a*a is pres cribed. More generally, we obtain the minimal possible value of Phi*({a(ij) , a(ij)*}(1 less than or equal to i,j less than or equal to d)), if {a(ij)} (1 less than or equal to i,j less than or equal to d) is a family of non-co mmutative random variables such that the distribution of A*A is prescribed. where A is the matrix (a(ij))(i,j=1)(d). The d x d-generalization is obtai ned from the case d = 1 via a result of independent interest, concerning th e minimal value of Phi*({a(ij), a(ij)*}(1 less than or equal to i,j less th an or equal to d)) when the matrix A = (a(ij))(i,j=1)(d) and its adjoint ha ve a given joint distribution. (A version of this result describes the mini mal value of Phi*({b(ij)}(1 less than or equal to i,j less than or equal to d)) when the matrix B = (b(ij))(i,j=1)(d) is selfadjoint and has a given d istribution.) We then show how the minimization results obtained for Phi* lead to maximiz ation results concerning the free entropy chi*, also defined by Voiculescu. (C) 1999 Academic Press.