Complete Nevanlinna-Pick kernels and the characteristic function

Abstract

This note finds a new characterization of complete Nevanlinna-Pick kernels on the Euclidean unit ball.The classical theory of Sz.-Nagy and Foias about the characteristic function is extended in this note to a commuting tuple T of bounded operators satisfying the natural positivity condition of 1/k-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characteristic function is a multiplier from Hk⊗E to Hk⊗F, factoring a certain positive operator, for suitable Hilbert spaces E and F depending on T. There is a converse, which roughly says that if a kernel k admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction (1/k-contraction where k is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain.