Characteristic function of a pure commuting contractive tuple

Abstract

A theorem of Sz.-Nagy and Foias[9] shows that the characteristic function θ_T(z)=−T + zD_TT∗(1_H−zT∗)^-1−1D_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 Cⁿ. This function is related to the curvature invariant introduced by Arveson[3].