UNITARY COLLIGATIONS, REPRODUCING KERNEL-HILBERT SPACES, AND NEVANLINNA-PICK INTERPOLATION IN SEVERAL VARIABLES

Recently J. Agler studied the class S-d of scalar-valued, analytic fun ctions of d complex variables f for which f(T-1,...,T-d) has norm at m ost 1 for any collection of d commuting contractions (T-1,...,T-d) 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-variabl e linear system, as well as a Nevanlinna-Pick interpolation theorem fo r this class of Functions. In this note we examine the system theory a spects and uniqueness of the transfer function representation, and giv e a simpler proof of the Nevanlinna-Pick interpolation theorem for the class S-d and obtain a d-variable version of the Toeplitz corona theo rem. By using ideas of Arov and Grossman introduced for I-variable pro blems, as a bonus we obtain a collection of linear fractional maps whi ch parametrize the set of all S-d solutions of an interpolation proble m. (C) 1998 Academic Press.