Factorization in weak products of complete Pick spaces

TL;DR

This paper proves a factorization property for functions in weak product spaces of complete Pick spaces, showing they can be expressed as single products under certain conditions, which advances understanding of these function spaces.

Contribution

It establishes that functions in the weak product of complete Pick spaces can be factored as single products, extending previous results and including important spaces like the Drury-Arveson and Dirichlet spaces.

Findings

Functions in weak products can be factored as single products.

The factorization applies to spaces including Drury-Arveson and Dirichlet spaces.

The results connect multiplier theory with weak product space structure.

Abstract

Let $\mathcal H$ be a reproducing kernel Hilbert space with a normalized complete Nevanlinna-Pick (CNP) kernel. We prove that if $(f_n)$ is a sequence of functions in $\mathcal H$ with $\sum\|f_n\|^2<\infty$, then there exists a contractive column multiplier $(\varphi_n)$ of $\mathcal H$ and a cyclic vector $F\in \mathcal H$ so that $\varphi_ n F=f_n$ for all $n$. The space of weak products $\mathcal H\odot\mathcal H$ is the set of functions of the form $h=\sum_{i=1}^\infty f_ig_i$ with $f_i, g_i\in\mathcal H$ and $\sum_{i=1}^\infty \|f_i\|\|g_i\|<\infty$. Using the above result, in combination with a recent result of Aleman, Hartz, McCarthy, and Richter, we show that for a large class of CNP spaces (including the Drury-Arveson spaces $H^2_d$ and the Dirichlet space in the unit disk) every $h\in\mathcal H\odot\mathcal H$ can be factored as a single product $h=fg$ with $f,g\in\mathcal H$.