Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems

Let M(d)(C) be the space of d x d complex matrices, and X(d) be the subspace of hermitian matrices. We study the Segal-Bargmann transform of functions on X(d), with values in M(d)(C), which are given by functional calculus. We show that when d --> infinity, the transform of such a map becomes close to the space of holomorphic functional calculus on M(d)(C), and that this yields, in the limit, an isometry between the L(2) space of Wigner's semi-circle distribution and the Hardy space of the disk. We relate this isometry to Voiculescu's theory of circular and semi-circular systems, and we study its analogue when the Segal-Bargmann transform is replaced by the Hall transform on unitary groups of large dimensions. (C) 1997 Academic Press.