Free convolution operators and free Hall transform
Résumé
We define an extension of the polynomial calculus on a W*-probability space by introducing an algebra C{X-i: i is an element of I} which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free convolution. It also permits us to characterize the free Segal-Bargmann transform and the free Hall transform introduced by Biane, in a manner which is closer to classical definitions. Finally, we use this extension of polynomial calculus to prove two asymptotic results on random matrices: the convergence for each fixed time, as N tends to infinity, of the *-distribution of the Brownian motion on the linear group GL(N)(C) to the *-distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U (N) to the free Hall transform.