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Transactions of the American Mathematical Society

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Some examples of composition operators and their approximation numbers on the Hardy space of the bidisk


Authors: Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza
Journal: Trans. Amer. Math. Soc. 372 (2019), 2631-2658
MSC (2010): Primary 47B33; Secondary 30H10, 30H20, 31B15, 32A35, 32U20, 41A35, 46B28
DOI: https://doi.org/10.1090/tran/7692
Published electronically: November 21, 2018
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Abstract: We give examples of composition operators $ C_\Phi $ on $ H^2 ({\mathbb{D}}^2)$ showing that the condition $ \Vert\Phi \Vert _\infty = 1$ is not sufficient for their approximation numbers $ a_n (C_\Phi )$ to satisfy $ \lim _{n \to \infty } [a_n (C_\Phi ) ]^{1/\sqrt {n}} = 1$, contrary to the
$ 1$-dimensional case. We also give a situation where this implication holds. We make a link with the Monge-Ampère capacity of the image of $ \Phi $.


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Additional Information

Daniel Li
Affiliation: Laboratoire de Mathématiques de Lens (LML) & Fédération CNRS Nord-Pas-de-Calais, Université d’Artois, EA 2462, FR 2956, F-62300 Lens, France
Email: daniel.li@euler.univ-artois.fr

Hervé Queffélec
Affiliation: Laboratoire Paul Painlevé U.M.R. CNRS 8524 & Fédération CNRS Nord-Pas-de-Calais, Université Lille Nord de France USTL, FR 2956 F-59655 Villeneuve d’Ascq Cedex, France
Email: Herve.Queffelec@univ-lille1.fr

Luis Rodríguez-Piazza
Affiliation: Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, 41080 Sevilla, Spain
Email: piazza@us.es

DOI: https://doi.org/10.1090/tran/7692
Keywords: Approximation numbers, Bergman space, bidisk, composition operator, Green capacity, Hardy space, Monge-Amp\`ere capacity, weighted composition operator
Received by editor(s): June 12, 2017
Received by editor(s) in revised form: February 27, 2018, and June 11, 2018
Published electronically: November 21, 2018
Additional Notes: The third-named author is partially supported by the project MTM2015-63699-P (Spanish MINECO and FEDER funds)
Article copyright: © Copyright 2018 American Mathematical Society