Weak products of complete Pick spaces

Let $\mathcal H$ be the Drury-Arveson or Dirichlet space of the unit ball of $\mathbb C^d$. The weak product $\mathcal H\odot\mathcal H$ of $\mathcal H$ is the collection of all functions $h$ that can be written as $h=\sum_{n=1}^\infty f_n g_n$, where $\sum_{n=1}^\infty \|f_n\|\|g_n\|<\infty$. We show that $\mathcal H\odot\mathcal H$ is contained in the Smirnov class of $\mathcal H$, i.e. every function in $\mathcal H\odot\mathcal H$ is a quotient of two multipliers of $\mathcal H$, where the function in the denominator can be chosen to be cyclic in $\mathcal H$. As a consequence we show that the map $\mathcal N \to clos_{\mathcal H\odot\mathcal H} \mathcal N$ establishes a 1-1 and onto correspondence between the multiplier invariant subspaces of $\mathcal H$ and of $\mathcal H\odot\mathcal H$. The results hold for many weighted Besov spaces $\mathcal H$ in the unit ball of $\mathbb C^d$ provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that for weighted Besov spaces $\mathcal H$ that satisfy what we call the multiplier inclusion condition any bounded column multiplication operator $\mathcal H \to \oplus_{n=1}^\infty \mathcal H$ induces a bounded row multiplication operator $\oplus_{n=1}^\infty \mathcal H \to \mathcal H$. For the Drury-Arveson space $H^2_d$ this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.

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