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On Dirichlet-type and n-isometric shifts in finite rank de Branges-Rovnyak spaces

Luo, Shuaibing and Rydhe, Eskil LU orcid (2025) In Science China Mathematics 68(11). p.2675-2694
Abstract

In this paper, we study the function spaces D(μ) by Richter (1991) and Aleman (1993), and Dμ→ by Rydhe (2019). It is known that the forward shift Mz is bounded and expansive on D(μ), and therefore D(μ) coincides with a de Branges-Rovnyak space H[B]. We show that such a B is rational if and only if μ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of D(μ) for finitely atomic μ. Similarly, we characterize the allowable tuples μ→=(|dz|2π,μ1,⋯,μn−1) such that Mz on Dμ→ is expansive with a finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples... (More)

In this paper, we study the function spaces D(μ) by Richter (1991) and Aleman (1993), and Dμ→ by Rydhe (2019). It is known that the forward shift Mz is bounded and expansive on D(μ), and therefore D(μ) coincides with a de Branges-Rovnyak space H[B]. We show that such a B is rational if and only if μ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of D(μ) for finitely atomic μ. Similarly, we characterize the allowable tuples μ→=(|dz|2π,μ1,⋯,μn−1) such that Mz on Dμ→ is expansive with a finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples μ→.

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publication status
published
subject
keywords
46E22, 47B38, de Branges-Rovnyak space, Dirichlet-type operator, n-isometry
in
Science China Mathematics
volume
68
issue
11
pages
20 pages
publisher
Science China Press
external identifiers
  • scopus:105008909226
ISSN
1674-7283
DOI
10.1007/s11425-024-2386-4
language
English
LU publication?
yes
additional info
Publisher Copyright: © The Author(s) 2025.
id
75477244-9b9d-4f7b-b4a4-8958b96a552d
date added to LUP
2025-12-19 15:16:05
date last changed
2025-12-19 15:17:08
@article{75477244-9b9d-4f7b-b4a4-8958b96a552d,
  abstract     = {{<p>In this paper, we study the function spaces D(μ) by Richter (1991) and Aleman (1993), and Dμ→ by Rydhe (2019). It is known that the forward shift M<sub>z</sub> is bounded and expansive on D(μ), and therefore D(μ) coincides with a de Branges-Rovnyak space H[B]. We show that such a B is rational if and only if μ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of D(μ) for finitely atomic μ. Similarly, we characterize the allowable tuples μ→=(|dz|2π,μ1,⋯,μn−1) such that M<sub>z</sub> on Dμ→ is expansive with a finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples μ→.</p>}},
  author       = {{Luo, Shuaibing and Rydhe, Eskil}},
  issn         = {{1674-7283}},
  keywords     = {{46E22; 47B38; de Branges-Rovnyak space; Dirichlet-type operator; n-isometry}},
  language     = {{eng}},
  number       = {{11}},
  pages        = {{2675--2694}},
  publisher    = {{Science China Press}},
  series       = {{Science China Mathematics}},
  title        = {{On Dirichlet-type and n-isometric shifts in finite rank de Branges-Rovnyak spaces}},
  url          = {{http://dx.doi.org/10.1007/s11425-024-2386-4}},
  doi          = {{10.1007/s11425-024-2386-4}},
  volume       = {{68}},
  year         = {{2025}},
}