Factorization in weak products of complete Pick spaces

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 $f,g\in\mathcal H$.