The norm of products of free random variables

Abstract

Let Xi denote free identically-distributed random variables. This paper investigates how the norm of products \({\Pi_{n} = X_{1}X_{2}\cdots X_{n}}\) behaves as n approaches infinity. In addition, for positive Xi it studies the asymptotic behavior of the norm of \({Y_{n} = X_{1}\circ X_{2}\circ \cdots \circ X_{n},}\) where \({\circ}\) denotes the symmetric product of two positive operators: \({A\circ B=:A^{1/2}BA^{1/2}}\) . It is proved that if EXi = 1, then \({\left\Vert Y_{n}\right\Vert }\) is between \({c_{1}\sqrt{n}}\) and c2n for certain constant c1 and c2. For \({\left\Vert \Pi_{n}\right\Vert ,}\) it is proved that the limit of \({n^{-1}\log \left\Vert \Pi _{n}\right\Vert }\) exists and equals \({\log \sqrt{E\left( X_{i}^{\ast }X_{i}\right) }.}\) Finally, if π is a cyclic representation of the algebra generated by Xi, and if ξ is a cyclic vector, then \({n^{-1}\log \left\Vert \pi \left( \Pi _{n}\right) \xi \right\Vert = \log \sqrt{E\left( X_{i}^{\ast }X_{i}\right) }}\) for all n. These results are significantly different from analogous results for commuting random variables.