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In this paper, we study the cyclicity problem with respect to the forward shift operator <italic>Sb</italic> acting on the de Branges–Rovnyak space ℋ (<italic>b</italic>) associated to a function <italic>b</italic> in the closed unit ball of <italic>H</italic>∞ and satisfying log(1− |<italic>b</italic>| ∈ <italic>L</italic>1(𝕋). We present a characterisation of cyclic vectors for <italic>Sb</italic> when <italic>b</italic> is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of <italic>Sb</italic> in this case, but we provide here an elementary proof. We also study the situation where <italic>b</italic> has the form <italic>b</italic> = (1+ <italic>I</italic>)/2, where <italic>I</italic> is a non-constant inner function such that the associated model space <italic>K</italic>I = ℋ (<italic>I</italic>) has an orthonormal basis of reproducing kernels.
</abstract>

In this paper, we study the cyclicity problem with respect to the forward shift operator Sb acting on the de Branges–Rovnyak space ℋ (b) associated to a function b in the closed unit ball of H∞ and satisfying log(1− |b| ∈ L1(𝕋). We present a characterisation of cyclic vectors for Sb when b is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of Sb in this case, but we provide here an elementary proof. We also study the situation where b has the form b = (1+ I)/2, where I is a non-constant inner function such that the associated model space KI = ℋ (I) has an orthonormal basis of reproducing kernels.

Cyclicity in de Branges–Rovnyak spaces

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Cyclicity in de Branges\u2013Rovnyak spaces"}

In this paper, we study the cyclicity problem with respect to the forward shift operator Sb acting on the de Branges–Rovnyak space ℋ (b) associated to a...

Cyclicity in de Branges–Rovnyak spaces

Published Online: Jun 07, 2023

Page range: 216 - 237

Received: Sep 23, 2022

Accepted: Nov 22, 2022

DOI: https://doi.org/10.2478/mjpaa-2023-0016

Keywords
Cyclicity, de Branges–Rovnyak spaces, forward shift operator

© 2023 Emmanuel Fricain et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Cyclicity in de Branges–Rovnyak spaces

Published Online: Jun 07, 2023

Page range: 216 - 237

Received: Sep 23, 2022

Accepted: Nov 22, 2022

DOI: https://doi.org/10.2478/mjpaa-2023-0016

KeywordsCyclicity, de Branges–Rovnyak spaces, forward shift operator

© 2023 Emmanuel Fricain et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Cyclicity, de Branges–Rovnyak spaces, forward shift operator