Characterization of weighted Hardy spaces on which all composition operators are bounded

Lefèvre, Pascal; Li, Daniel; Queffélec, Hervé; Rodríguez-Piazza, Luis

doi:10.2140/apde.2025.18.1921

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Characterization of weighted Hardy spaces on which all composition operators are bounded

Pascal Lefèvre, Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza

Vol. 18 (2025), No. 8, 1921–1954

DOI: 10.2140/apde.2025.18.1921

Abstract

We give a complete characterization of the sequences $β = (β_{n})$ of positive numbers for which all composition operators on $H^{2} (β)$ are bounded, where $H^{2} (β)$ is the space of analytic functions $f$ on the unit disk $𝔻$ such that $\sum_{n = 0}^{\infty} | a_{n} |^{2} β_{n} < + \infty$ if $f (z) = \sum_{n = 0}^{\infty} a_{n} z^{n}$ . We prove that all composition operators are bounded on $H^{2} (β)$ if and only if $β$ is essentially decreasing and slowly oscillating. We also prove that every automorphism of the unit disk induces a bounded composition operator on $H^{2} (β)$ if and only if $β$ is slowly oscillating. We give applications of our results.

Keywords

composition operator, weighted Hardy space, slowly oscillating sequence, automorphism of the unit disk

Mathematical Subject Classification

Primary: 47B33

Secondary: 30H10

Milestones

Received: 23 October 2023

Revised: 30 July 2024

Accepted: 20 September 2024

Published: 25 July 2025

Authors

Pascal Lefèvre
Université d’Artois UR 2462, Laboratoire de Mathématiques de Lens (LML) F-62300 Lens France

Daniel Li
Université d’Artois UR 2462, Laboratoire de Mathématiques de Lens (LML) F-62300 Lens France

Hervé Queffélec
Université de Lille CNRS, UMR 8524 – Laboratoire Paul Painlevé F-59000 Lille France

Luis Rodríguez-Piazza
Dpto. de Análisis Matemático & IMUS Facultad de Matemáticas Universidad de Sevilla Sevilla Spain

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Vol. 18, No. 8, 2025