Weak products of complete pick spaces
(2021) In Indiana University Mathematics Journal 70(1). p.325-352- Abstract
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... (More)
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.
(Less)
- author
- Aleman, Alexandru LU ; Hartz, Michael ; McCarthy, John E. and Richter, Stefan
- organization
- publishing date
- 2021
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Besov space, Complete Pick space, Multiplier, Smirnov class
- in
- Indiana University Mathematics Journal
- volume
- 70
- issue
- 1
- pages
- 28 pages
- publisher
- Indiana University
- external identifiers
-
- scopus:85102299143
- ISSN
- 0022-2518
- DOI
- 10.1512/iumj.2021.70.8122
- language
- English
- LU publication?
- yes
- id
- 45680e52-b37a-4f4d-82b3-a36b3cc20960
- date added to LUP
- 2021-03-29 10:37:04
- date last changed
- 2022-04-27 01:04:27
@article{45680e52-b37a-4f4d-82b3-a36b3cc20960, abstract = {{<p>Let H be the Drury-Arveson or Dirichlet space of the unit ball of C<sup>d</sup>. The weak product H ☉ H of H is the collection of all functions h that can be written as h =∑<sup>∞</sup><sub>n=</sub><sub>1</sub> f<sub>n</sub>g<sub>n</sub>, where ∑<sup>∞</sup><sub>n</sub>=<sub>1</sub> ||f<sub>n</sub>|| ||g<sub>n</sub>|| < ∞. 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<sub>H ☉H</sub> 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<sup>d</sup> 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 → ⊕<sup>∞</sup><sub>n=</sub><sub>1</sub> H induces a bounded row multiplication operator ⊕<sup>∞</sup><sub>n=</sub><sub>1</sub> H → H . For the Drury-Arveson space H<sub>d</sub><sup>2</sup> this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.</p>}}, author = {{Aleman, Alexandru and Hartz, Michael and McCarthy, John E. and Richter, Stefan}}, issn = {{0022-2518}}, keywords = {{Besov space; Complete Pick space; Multiplier; Smirnov class}}, language = {{eng}}, number = {{1}}, pages = {{325--352}}, publisher = {{Indiana University}}, series = {{Indiana University Mathematics Journal}}, title = {{Weak products of complete pick spaces}}, url = {{http://dx.doi.org/10.1512/iumj.2021.70.8122}}, doi = {{10.1512/iumj.2021.70.8122}}, volume = {{70}}, year = {{2021}}, }