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Some Minimization Problems for the Free Analogue of the Fisher Information

✍ Scribed by Alexandru Nica; Dimitri Shlyakhtenko; Roland Speicher

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
274 KB
Volume
141
Category
Article
ISSN
0001-8708

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✦ Synopsis

We consider the free non-commutative analogue 8*, introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value of 8*(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 8*([a ij ,

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 8*([a ij , a ij *] 1 i, j d ) 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 8*([b ij ] 1 i, j d ) when the matrix B=(b ij ) d i, j=1 is selfadjoint and has a given distribution.)

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