  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.
