Cyclicity in de Branges-Rovnyak spaces

Fricain, Emmanuel; Grivaux, Sophie

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.2478/mjpaa-2023-0016

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]

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

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 >

Language :

Anglais

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :