Products of Nevanlinna-Pick kernels and operator colligations

Abstract

The Schur classS _d of analytic functionsf ofd complex variables for whichf(T ₁,T ₂,...,T _d ) has norm at most one for everyd-tuple of commuting strict contractions is characterized by J. Agler [1] in terms of a Szego kernel factorization property and in terms of a transfer function of a certain type ofd-variable unitary colligation. Replacing the Szego kernel by positive kernels whose reciprocal has one positive square we can define a new Schur class in terms of a kernel factorization property. By using, in part, the approach found in Ball and Trent [3], we characterize this Schur class in terms of a transfer function of a certain type of unitary colligation. Further, with these results we establish a Nevanlinna-Pick interpolation theorem for our Schur class. Such interpolation theorems already exist in the literature for two special cases where [1] all kernels are taken to be the Szego kernel, and [2] d=1.