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 $\sum \Vert f_n{\Vert }^2<\infty$, then there exists a contractive column multiplier $({\phi }_n)$ of $\mathcal{H}$ and a cyclic vector $F\in \mathcal{H}$ so that ${\phi }_{n}F=f_n$ for all $n$. The space of weak products $\mathcal{H}\odot \mathcal{H}$ is the set of functions of the form $h={\sum }_{i=1}^{\infty }{f}_i{g}_i$ with $f_i,g_i\in \mathcal{H}$ and ${\sum }_{i=1}^{\infty }\Vert f_i\Vert \Vert g_i\Vert <\infty$. 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 $H_d^2$ 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 $f,g\in \mathcal{H}$.