Weak products of complete pick spaces
Research output: Contribution to journal › Article
Let H be the Drury-Arveson or Dirichlet space of the unit ball of Cd. The weak product H ☉ H of H is the collection of all functions h that can be written as h =∑∞n=1 fngn, where ∑∞n=1 ||fn|| ||gn|| < ∞. 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 → closH ☉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 Cd 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 Hd2 this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.
|Research areas and keywords||
Subject classification (UKÄ) – MANDATORY
|Number of pages||28|
|Journal||Indiana University Mathematics Journal|
|Publication status||Published - 2021|