On c.n.c. commuting contractive tuples

The characteristic function has been an important tool for studying completely non unitary contractions on Hilbert spaces. In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space $\text{clh}$. We show that the characteristic function, which is now an operator valued analytic function on the open Euclidean unit ball in ${\mathbb{C}}^n$, is a complete unitary invariant for such a tuple. We prove that the characteristic function satisfies a natural transformation law under biholomorphic mappings of the unit ball. We also characterize all operator-valued analytic functions which arise as characteristic functions of pure commuting contractive tuples.