Weak products of complete Pick spaces

Let H be the Drury-Arveson or Dirichlet space of unit ball Cd. The weak product ? is collection all functions h that can written as =??n=1 fngn, where ??n=1 ||fn|| ||gn|| < ?. We show contained in Smirnov class H; is, every function a quotient two multipliers H, denominator chosen to cyclic . As consequence, we map N ? closH ?H establishes one-to-one and onto correspondence between multiplier invariant subspaces results hold for many weighted Besov spaces Cd provided reproducing kernel has complete Pick property. One our main technical lemmas states that, satisfy what call inclusion condition, any bounded column multiplication operator ??n=1 induces row For Hd2 this leads an alternate proof characterization interpolating sequences terms separation Carleson measure conditions. (Less)