A note on composition operators on model spaces
[Une note sur les opérateurs de composition sur les espaces modèles]
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 $f\circ \phi =\tau f$.

Reçu le :
Révisé le :
Accepté le :
Première publication :
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) 
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Chalendar, Isabelle; Gumenyuk, Pavel; McCarthy, John E. A note on composition operators on model spaces. Annales de l'Institut Fourier, Online first, 17 p.

[1] Abate, Marco Holomorphic dynamics on hyperbolic Riemann surfaces, De Gruyter Studies in Mathematics, 89, Walter de Gruyter, 2023, xiii+356 pages | MR | Zbl

[2] Agler, Jim; McCarthy, John E.; Young, Nicholas J. Operator Analysis: Hilbert Space Methods in Complex Analysis, Cambridge Tracts in Mathematics, 219, Cambridge University Press, 2020 | MR | DOI | Zbl

[3] Arosio, Leandro; Bracci, Filippo Canonical models for holomorphic iteration, Trans. Am. Math. Soc., Volume 368 (2016) no. 5, pp. 3305-3339 | DOI | MR | Zbl

[4] Baker, Irvine N.; Pommerenke, Christian On the iteration of analytic functions in a half-plane. II, J. Lond. Math. Soc. (2), Volume 20 (1979) no. 2, pp. 255-258 | DOI | MR | Zbl

[5] Bell, Steven R. The Cauchy transform, potential theory and conformal mapping, Chapman & Hall/CRC, 2016, xii+209 pages | DOI | MR | Zbl

[6] Cassier, Gilles; Chalendar, Isabelle The group of the invariants of a finite Blaschke product, Complex Variables, Theory Appl., Volume 42 (2000) no. 3, pp. 193-206 | DOI | Zbl

[7] Chalendar, Isabelle; Gorkin, Pamela; Partington, Jonathan R. Inner functions and operator theory, North-West. Eur. J. Math., Volume 1 (2015), pp. 9-28 | MR | Zbl

[8] Collingwood, Edward F.; Lohwater, Arthur J. The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, 56, Cambridge University Press, 1966, xi+211 pages | MR | Zbl

[9] Cowen, Carl C. Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Am. Math. Soc., Volume 265 (1981) no. 1, pp. 69-95 | DOI | MR | Zbl

[10] Cowen, Carl C.; MacCluer, Barbara D. Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, 1995 | MR | Zbl

[11] Fricain, Emmanuel; Mashreghi, Javad The theory of ℋ(b) spaces. Volume 1, New Mathematical Monographs, 20, Cambridge University Press, 2016 | DOI | Zbl

[12] Garcia, Stephan Ramon; Mashreghi, Javad; Ross, William T. Introduction to model spaces and their operators, Cambridge Studies in Advanced Mathematics, 148, Cambridge University Press, 2016 | MR | DOI | Zbl

[13] Garnett, John B. Bounded Analytic Functions, Pure and Applied Mathematics, 96, Academic Press Inc., 1981 | MR | Zbl

[14] Julia, Gaston Extension nouvelle d’un lemme de Schwarz, Acta Math., Volume 42 (1920), pp. 349-355 | Zbl | MR | DOI

[15] Lyubarskii, Yurii I.; Malinnikova, Eugenia Composition operators on model spaces, Recent trends in analysis. Proceedings of the conference in honor of Nikolai Nikolski on the occasion of his 70th birthday, Bordeaux, France, August 31 – September 2, 2011 (Theta Series in Advanced Mathematics), Volume 16, The Theta Foundation, 2013, pp. 149-157 | MR | Zbl

[16] Mashreghi, Javad; Shabankhah, Mahmood Composition operators on finite rank model subspaces, Glasg. Math. J., Volume 55 (2013) no. 1, pp. 69-83 | MR | DOI | Zbl

[17] Mashreghi, Javad; Shabankhah, Mahmood Composition of inner functions, Can. J. Math., Volume 66 (2014) no. 2, pp. 387-399 | DOI | Zbl | MR

[18] Pommerenke, Christian Univalent functions, Studia Mathematica/Mathematische Lehrbücher, XXV, Vandenhoeck & Ruprecht, 1975, 376 pages (with a chapter on quadratic differentials by Gerd Jensen) | MR | Zbl

[19] Pommerenke, Christian On the iteration of analytic functions in a halfplane, J. Lond. Math. Soc. (2), Volume 19 (1979) no. 3, pp. 439-447 | DOI | MR | Zbl

[20] Sarason, Donald Algebraic properties of truncated Toeplitz operators, Oper. Matrices, Volume 1 (2007) no. 4, pp. 491-526 | DOI | Zbl | MR

[21] Shapiro, Joel H. The essential norm of a composition operator, Ann. Math. (2), Volume 125 (1987) no. 2, pp. 375-404 | DOI | MR | Zbl

[22] Shapiro, Joel H. Composition operators and classical function theory, Universitext, Springer, 1993 | Zbl | DOI | MR

[23] Shapiro, Joel H. What do composition operators know about inner functions?, Monatsh. Math., Volume 130 (2000) no. 1, pp. 57-70 | DOI | Zbl | MR

[24] Valiron, Georges Fonctions analytiques, Presses Universitaires de France, 1954, 236 pages | MR | Zbl

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