Complete Nevanlinna-Pick kernels And The Characteristic Function

Abstract. This note finds a new characterization of irreducible unitarily invariant complete Nevanlinna-Pick kernels on the Euclidean unit ball Bd. The classical theory of Sz.-Nagy and Foias about the characteristic function can be extended to a commuting tuple T of bounded operators satisfying the natural positivity condition of 1/k-contractivity. 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 relates to the McCullough Trent invariant subspace theorem and explains, among other things, why in the literature a characteristic function for a Bergman contraction (1/k-contraction where k is the Bergman kernel) requires a different reproducing kernel Hilbert space as the domain. The properties of the characteristic function that are proved include a complete unitary invariance result for a certain class of 1/k-contractions, viz., the c.n.c. ones. The characteristic function of a non-trivial c.n.c. 1/k-contraction, i.e., a c.n.c. 1/k-contraction which is not unitarily equivalent to the shift (Mz1 , . . . ,Mzd) on a vector valued Hk, is necessarily non-constant. When k is a Dirichlet type kernel, if the characteristic function of a c.n.c. 1/k-contraction T is a polynomial, then T extends to a 2 × 2 block upper triangular operator matrix with the diagonal entries being a shift and a nilpotent operator.