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$\sum_{n=1}^\infty |f_n||g_n|<\infty$\mathrm{.}Weshowthat$\mathcal H\odot\mathcal H$iscontainedintheSmirnovclassof$\mathcal H$, 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 $\mathcal{N}\to clo{s}_{\mathcal{H}\odot \mathcal{H}}\mathcal{N}$ establishes a 1-1 and onto correspondence between the multiplier invariant subspaces of $\mathcal H$andof$\mathcal H\odot\mathcal H$. 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 $\mathcal H \to \oplus_{n=1}^\infty \mathcal H$inducesaboundedrowmultiplicationoperator$\oplus_{n=1}^\infty \mathcal H \to \mathcal H$\mathrm{.}FortheDrury-Arvesonspace$H^2_d$ this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.