Characteristic function for polynomially contractive commuting tuples

Abstract

In this note, we develop the theory of characteristic function as an invariant for n-tuples of operators. The operator tuple has a certain contractivity condition put on it. This condition and the class of domains in C n that we consider are intimately related. A typical example of such a domain is the open Euclidean unit ball. Given a polynomial P in C [ z 1 , z 2 , … , z n ] whose constant term is zero, all the coefficients are nonnegative and the coefficients of the linear terms are nonzero, one can naturally associate a Reinhardt domain with it, which we call the P-ball (Definition 1.1). Using the reproducing kernel Hilbert space H P ( C ) associated with this Reinhardt domain in C n , S. Pott constructed the dilation for a polynomially contractive commuting tuple (Definition 1.2) [S. Pott, Standard models under polynomial positivity conditions, J. Operator Theory 41 (1999) 365–389. MR 2000j:47019]. Given any polynomially contractive commuting tuple T we define its characteristic function θ T which is a multiplier. We construct a functional model using the characteristic function. Exploiting the model, we show that the characteristic function is a complete unitary invariant when the tuple is pure. The characteristic function gives newer and simpler proofs of a couple of known results: one of them is the invariance of the curvature invariant and the other is a Beurling theorem for the canonical operator tuple on H P ( C ) . It is natural to study the boundary behaviour of θ T in the case when the domain is the Euclidean unit ball. We do that and here essential differences with the single operator situation are brought out.