Characteristic Function of a Pure Commuting Contractive Tuple

A theorem of Sz.-Nagy and Foias [9] shows that the characteristic function of a completely non-unitary contraction T is a complete unitary invariant for T. In this note we extend this theorem to the case of a pure commuting contractive tuple using a natural generalization of the characteristic function to an operator-valued analytic function defined on the open unit ball of This function is related to the curvature invariant introduced by Arveson [3].