Some minimization problems for the free
نویسنده
چکیده
We consider the free non-commutative analogue , introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value of (a; a), if a is a non-commutative random variable subject to the constraint that the distribution of a a is prescribed. More generally, we obtain the minimal possible value of (fa ij ; a ij g 1i;jd), if fa ij g 1i;jd is a family of non-commutative random variables such that the distribution of A A is prescribed, where A is the matrix (a ij) 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 (fa ij ; a ij g 1i;jd) when the matrix A = (a ij) d i;j=1 and its adjoint have a given joint distribution. (A version of this result describes the minimal value of (fb ij g 1i;jd) when the matrix B = (b ij) d i;j=1 is selfadjoint and has a given distribution.) We then show how the minimization results obtained for lead to maximization results concerning the free entropy , also deened by Voiculescu.
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