Weak products of complete pick spaces

Let H be the Drury-Arveson or Dirichlet space of the unit ball of C^d. The weak product H ☉ H of H is the collection of all functions h that can be written as h =∑^∞_n=1 f_ng_n, where ∑^∞_n=₁ ||f_n|| ||g_n|| < ∞. We show that H ☉ H is contained in the Smirnov class of H; that is, every function in H ☉ H is a quotient of two multipliers of H, where the function in the denominator can be chosen to be cyclic in H . As a consequence, we show that the map N → clos_{H ☉H} N establishes a one-to-one and onto correspondence between the multiplier invariant subspaces of H and of H ☉ H . The results hold for many weighted Besov spaces H in the unit ball of C^d provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that, for weighted Besov spaces H that satisfy what we call the multiplier inclusion condition, any bounded column multiplication operator H → ⊕^∞_n=1 H induces a bounded row multiplication operator ⊕^∞_n=1 H → H . For the Drury-Arveson space H_d² this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.