Unitary Colligations, Reproducing Kernel Hilbert Spaces, and Nevanlinna–Pick Interpolation in Several Variables

Recently J. Agler studied the class Sdof scalar-valued, analytic functions ofdcomplex variablesffor whichf(T1, …, Td) has norm at most 1 for any collection ofdcommuting contractions (T1, …, Td) on a Hilbert space H. Among other results he obtained a characterization of such functions in terms of a positivity property and in terms of a representation as the transfer function of a certain type of d-variable linear system, as well as a Nevanlinna–Pick interpolation theorem for this class of functions. In this note we examine the system theory aspects and uniqueness of the transfer function representation, and give a simpler proof of the Nevanlinna–Pick interpolation theorem for the class Sdand obtain ad-variable version of the Toeplitz corona theorem. By using ideas of Arov and Grossman introduced for 1-variable problems, as a bonus we obtain a collection of linear fractional maps which parametrize the set of all Sdsolutions of an interpolation problem.