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Generalized de Branges-Rovnyak spaces

Aleman, Alexandru LU and Dahlin, Frej LU (2025) In Journal of Functional Analysis 288(11).
Abstract

Given the reproducing kernel k of the Hilbert space Hk we study spaces Hk(b) whose reproducing kernel has the form k(1−bb⁎), where b is a row-contraction on Hk. In terms of reproducing kernels this is the most far-reaching generalization of the classical de Branges-Rovnyaks spaces, as well as their very recent generalization to several variables. This includes the so called sub-Bergman spaces [31] in one or several variables. We study some general properties of Hk(b) e.g. when the inclusion map into Hk is compact. Our main result provides a model for Hk(b) reminiscent of the Sz.-Nagy-Foiaş model for contractions (see also [7]). As an application we obtain... (More)

Given the reproducing kernel k of the Hilbert space Hk we study spaces Hk(b) whose reproducing kernel has the form k(1−bb⁎), where b is a row-contraction on Hk. In terms of reproducing kernels this is the most far-reaching generalization of the classical de Branges-Rovnyaks spaces, as well as their very recent generalization to several variables. This includes the so called sub-Bergman spaces [31] in one or several variables. We study some general properties of Hk(b) e.g. when the inclusion map into Hk is compact. Our main result provides a model for Hk(b) reminiscent of the Sz.-Nagy-Foiaş model for contractions (see also [7]). As an application we obtain sufficient conditions for the containment and density of the linear span of {kw:w∈X} in Hk(b). In the standard cases this reduces to containment and density of polynomials. These methods resolve a very recent conjecture [13] regarding polynomial approximation in spaces with kernel [Formula presented],1≤m<β,m∈N.

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author
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Approximations, Compact embedding, De Branges-Rovnyak spaces, Sub-Bergman spaces
in
Journal of Functional Analysis
volume
288
issue
11
article number
110860
publisher
Academic Press
external identifiers
  • scopus:85217934133
ISSN
0022-1236
DOI
10.1016/j.jfa.2025.110860
language
English
LU publication?
yes
additional info
Publisher Copyright: © 2025 The Authors
id
8477e9a3-f7c2-4172-96e3-3b834fb3c0b7
date added to LUP
2026-06-11 10:56:57
date last changed
2026-06-11 10:57:38
@article{8477e9a3-f7c2-4172-96e3-3b834fb3c0b7,
  abstract     = {{<p>Given the reproducing kernel k of the Hilbert space H<sub>k</sub> we study spaces H<sub>k</sub>(b) whose reproducing kernel has the form k(1−bb<sup>⁎</sup>), where b is a row-contraction on H<sub>k</sub>. In terms of reproducing kernels this is the most far-reaching generalization of the classical de Branges-Rovnyaks spaces, as well as their very recent generalization to several variables. This includes the so called sub-Bergman spaces [31] in one or several variables. We study some general properties of H<sub>k</sub>(b) e.g. when the inclusion map into H<sub>k</sub> is compact. Our main result provides a model for H<sub>k</sub>(b) reminiscent of the Sz.-Nagy-Foiaş model for contractions (see also [7]). As an application we obtain sufficient conditions for the containment and density of the linear span of {k<sub>w</sub>:w∈X} in H<sub>k</sub>(b). In the standard cases this reduces to containment and density of polynomials. These methods resolve a very recent conjecture [13] regarding polynomial approximation in spaces with kernel [Formula presented],1≤m&lt;β,m∈N.</p>}},
  author       = {{Aleman, Alexandru and Dahlin, Frej}},
  issn         = {{0022-1236}},
  keywords     = {{Approximations; Compact embedding; De Branges-Rovnyak spaces; Sub-Bergman spaces}},
  language     = {{eng}},
  month        = {{06}},
  number       = {{11}},
  publisher    = {{Academic Press}},
  series       = {{Journal of Functional Analysis}},
  title        = {{Generalized de Branges-Rovnyak spaces}},
  url          = {{http://dx.doi.org/10.1016/j.jfa.2025.110860}},
  doi          = {{10.1016/j.jfa.2025.110860}},
  volume       = {{288}},
  year         = {{2025}},
}