Characteristic function of a pure contractive tuple

Abstract

A theorem of Sz.-Nagy and Foias [9] shows that the characteristic function \theta_{T}(z)=-T+zD_{T^{*}}(1_{\mathcal{\mathcal{H}}}-zT^{*})^{-1}D_{T} of a completly non-uniraty 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].