The norm of products of free random variables

Abstract

Let X i denote free identically-distributed random variables. This paper investigates how the norm of products Πn= X 1X2⋯Xn behaves as n approaches infinity. In addition, for positive X i it studies the asymptotic behavior of the norm of Yn= X1̂ X2̂ ⋯ ̂ Xn where ̂ denotes the symmetric product of two positive operators: A ̂ B=:A1/2BA1/2. It is proved that if EX i = 1, then ||Yn|| is between c1√n and c 2 n for certain constant c 1 and c 2. For ||Πn|| it is proved that the limit of n-1log ||Πn|| exists and equals log √E(Xi*Xi). Finally, if π is a cyclic representation of the algebra generated by X i , and if ξ is a cyclic vector, then n-1 log || π (Πn) ξ || = log √E(X i*Xi) for all n. These results are significantly different from analogous results for commuting random variables. © 2006 Springer-Verlag.