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Weak products of complete pick spaces

Aleman, Alexandru LU ; Hartz, Michael ; McCarthy, John E. and Richter, Stefan (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.

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author
; ; and
organization
publishing date
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>|| &lt; ∞. 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}},
}