Factorizations induced by complete Nevanlinna-Pick factors

TL;DR

This paper establishes a factorization theorem for reproducing kernel Hilbert spaces with kernels containing a normalized complete Nevanlinna-Pick factor, linking functions to multipliers and enabling new applications.

Contribution

It introduces a novel factorization result for spaces with Nevanlinna-Pick kernels, connecting functions to pointwise multipliers and broadening understanding of these spaces.

Findings

Constructed pluriharmonic majorants for functions in the space

Linked bounded majorants to pointwise multipliers

Applied results to Dirichlet and Drury-Arveson spaces

Abstract

We prove a factorization theorem for reproducing kernel Hilbert spaces whose kernel has a normalized complete Nevanlinna-Pick factor. This result relates the functions in the original space to pointwise multipliers determined by the Nevanlinna-Pick kernel and has a number of interesting applications. For example, for a large class of spaces including Dirichlet and Drury-Arveson spaces, we construct for every function $f$ in the space a pluriharmonic majorant of $|f|^2$ with the property that whenever the majorant is bounded, the corresponding function $f$ is a pointwise multiplier.