Cyclicity in de Branges-Rovnyak spaces
Document type :
Compte-rendu et recension critique d'ouvrage
DOI :
Title :
Cyclicity in de Branges-Rovnyak spaces
Author(s) :
Fricain, Emmanuel [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Grivaux, Sophie [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Grivaux, Sophie [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Journal title :
Moroccan Journal of Pure and Applied Analysis
Publisher :
Sidi Mohamed Benabdallah University [Ifrane] Al Akhawyn University
Publication date :
2023
ISSN :
2351-8227
English keyword(s) :
30J05
30H10
46E22
Cyclicity
de Branges–Rovnyak spaces
forward shift operator
30H10
46E22
Cyclicity
de Branges–Rovnyak spaces
forward shift operator
HAL domain(s) :
Mathématiques [math]
English abstract : [en]
In this paper, we study the cyclicity problem with respect to the forward shift operator $S_b$ acting on the de Branges--Rovnyak space $\mathscr{H}(b)$ associated to a function $b$ in the closed unit ball of $H^\infty$ and ...
Show more >In this paper, we study the cyclicity problem with respect to the forward shift operator $S_b$ acting on the de Branges--Rovnyak space $\mathscr{H}(b)$ associated to a function $b$ in the closed unit ball of $H^\infty$ and satisfying $\log(1-|b|)\in L^1(\mathbb T)$. We present a characterisation of cyclic vectors for $S_b$ when $b$ is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [S. Luo, C. Gu, S. Richter, Higher order local Dirichlet integrals and de Branges--Rovnyak spaces, \emph{Adv. Math., \textbf{385} (2021), paper No. 107748, 47], of invariant subspaces of $S_b$ 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 $K_I=\mathscr{H}(I)$ has an orthonormal basis of reproducing kernels.Show less >
Show more >In this paper, we study the cyclicity problem with respect to the forward shift operator $S_b$ acting on the de Branges--Rovnyak space $\mathscr{H}(b)$ associated to a function $b$ in the closed unit ball of $H^\infty$ and satisfying $\log(1-|b|)\in L^1(\mathbb T)$. We present a characterisation of cyclic vectors for $S_b$ when $b$ is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [S. Luo, C. Gu, S. Richter, Higher order local Dirichlet integrals and de Branges--Rovnyak spaces, \emph{Adv. Math., \textbf{385} (2021), paper No. 107748, 47], of invariant subspaces of $S_b$ 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 $K_I=\mathscr{H}(I)$ has an orthonormal basis of reproducing kernels.Show less >
Language :
Anglais
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