Characteristic Function of a Pure Commuting Contractive Tuple

Abstract.

A theorem of Sz.-Nagy and Foias [9] shows that the characteristic function $\theta _T (z) = - T + zD_{T^*} (1_\mathcal{H} - zT^*)^{ - 1} D_T$ 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 $\mathbb{C}^n .$ This function is related to the curvature invariant introduced by Arveson [3].