Complete Nevanlinna-Pick kernels And The Characteristic Function

TL;DR

This paper characterizes complete Nevanlinna-Pick kernels on the unit ball and extends the classical theory of characteristic functions to certain operator tuples, revealing new structural insights.

Contribution

It provides a new characterization of complete Nevanlinna-Pick kernels and extends the characteristic function theory to operator tuples satisfying specific positivity conditions.

Findings

Characterization of complete Nevanlinna-Pick kernels on the unit ball

Extension of characteristic function theory to operator tuples

Explanation of differences in kernel choices for Bergman contractions

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 $\bfT$ 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 $H_k \otimes \cE$ to $H_k \otimes \cF$, {\em factoring} a certain positive operator, for suitable Hilbert spaces $\cE$ and $\cF$ depending on $\bfT$. There is a converse, which roughly says that if a kernel $k$ {\em 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.