Segal–Bargmann Transform, Functional Calculus on Matrix Spaces and the Theory of Semi-circular and Circular Systems

LetMd(C) be the space ofd×dcomplex matrices, and Xdbe the subspace of hermitian matrices. We study the Segal–Bargmann transform of functions on Xd, with values inMd(C), which are given by functional calculus. We show that whend→∞, the transform of such a map becomes close to the space of holomorphic functional calculus onMd(C), and that this yields, in the limit, an isometry between theL2space of Wignerʹs semi-circle distribution and the Hardy space of the disk. We relate this isometry to Voiculescuʹs 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.