Weak products of complete pick spaces
Research output: Contribution to journal › Article
Abstract
Let H be the DruryArveson 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_{n}g_{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 onetoone 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 DruryArveson space H_{d}^{2} this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.
Details
Authors  

Organisations  
External organisations 

Research areas and keywords  Subject classification (UKÄ) – MANDATORY
Keywords

Original language  English 

Pages (fromto)  325352 
Number of pages  28 
Journal  Indiana University Mathematics Journal 
Volume  70 
Issue number  1 
Publication status  Published  2021 
Publication category  Research 
Peerreviewed  Yes 