A note on composition operators on model spaces
Chalendar, Isabelle1; Gumenyuk, Pavel2; McCarthy, John E.3
1Université Gustave Eiffel, LAMA, (UMR 8050), UPEM, UPEC, CNRS, F-77454, Marne-la-Vallée (France)
2Department of Mathematics, Politecnico di Milano, via E. Bonardi 9, 20133 Milan (Italy)
3Department of Mathematics, Washington University, One Brookings Drive, St. Louis, MO 63130 (USA)
Annales de l'Institut Fourier, Online first, 17 p.

Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankhah we consider the following problem: for a given inner function $\phi \notin \mathsf {Aut}(\mathbb{D})$, find a non-constant inner function $\Psi $ satisfying the functional equation $\Psi \circ \phi =\tau \Psi $, where $\tau $ is a unimodular constant. We prove that this problem has a solution if and only if $\phi $ is of positive hyperbolic step. More precisely, if this condition holds, we show that there is an infinite Blaschke product $B$ satisfying the equation for $\tau =1$. If in addition, $\phi $ is parabolic, we prove that the problem has a solution $\Psi $ for any unimodular $\tau $. Finally, we show that if $\phi $ is of zero hyperbolic step, then no non-constant Bloch function $f$ and no unimodular constant $\tau $ satisfy $f\circ \phi =\tau f$.

Motivés par l’étude des opérateurs de composition sur les espaces modèles initiée par Mashreghi et Shabankhah, nous étudions le problème suivant : étant donnée une fonction intérieure $\phi $ qui n’est pas un automorphisme du disque unité, trouver une fonction intérieure non constante $\Psi $ vérifiant l’équation fonctionnelle $\Psi \circ \phi =\tau \Psi $, où $\tau $ est une constante unimodulaire. Nous prouvons que ce problème a une solution si et seulement si $\phi $ est de pas hyperbolique positif. Plus précisément, si cette condition est satisfaite, nous montrons qu’il existe un produit de Blaschke infini $B$ satisfaisant notre équation avec $\tau =1$. De plus, si $\phi $ est parabolique, nous montrons que le problème a une solution $\Psi $ pour tout $\tau $ unimodulaire. Enfin nous prouvons que si $\phi $ est de pas hyperbolique nul alors il n’existe pas de fonction non constante $f$ de Bloch et il n’existe pas de constante unimodulaire $\tau $ vérifiant $f\circ \phi =\tau f$.

Received:
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Accepted:
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DOI: 10.5802/aif.3781
Classification: 30D05, 30J05, 30J10, 30H30
Keywords: inner functions on the unit disc, Blaschke product, positive hyperbolic step, Bloch function, model space, composition operator, Schröder equation
Mots-clés : fonctions intérieures du disque unité, produit de Blaschke, pas hyperbolique positif, fonction de Bloch, espace modèle, opérateur de composition, équation de Schröder

Chalendar, Isabelle  1 ; Gumenyuk, Pavel  2 ; McCarthy, John E.  3

1 Université Gustave Eiffel, LAMA, (UMR 8050), UPEM, UPEC, CNRS, F-77454, Marne-la-Vallée (France)
2 Department of Mathematics, Politecnico di Milano, via E. Bonardi 9, 20133 Milan (Italy)
3 Department of Mathematics, Washington University, One Brookings Drive, St. Louis, MO 63130 (USA) 
Chalendar, Isabelle; Gumenyuk, Pavel; McCarthy, John E. A note on composition operators on model spaces. Annales de l'Institut Fourier, Online first, 17 p.
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