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Decay rates for approximation numbers of composition operators

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Abstract

A general method for estimating the approximation numbers of composition operators on the Hardy space H 2, using finite-dimensional model subspaces, is studied and applied in the case when the symbol of the operator maps the unit disc to a domain whose boundary meets the unit circle at just one point. The exact rate of decay of the approximation numbers is identified when this map is sufficiently smooth at the point of tangency; it follows that a composition operator with any prescribed slow decay of its approximation numbers can be explicitly constructed. Similarly, an asymptotic expression for the approximation numbers is found when the mapping has a sharp cusp at the distinguished boundary point. Precise asymptotic estimates in the intermediate cases, including that of maps with a corner at the distinguished boundary point, are also established.

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Authors and Affiliations

  1. USTL, Laboratoire Paul Painlev é U. M. R. CNRS 8524, Université Lille Nord de France, F-59 655, Villeneuve d’Ascq Cedex, France

    Hervé Queffélec

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway

    Kristian Seip

Authors
  1. Hervé Queffélec
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  2. Kristian Seip
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Correspondence to Hervé Queffélec.

Additional information

This paper was written while the authors participated in the research program Operator Related Function Theory and Time-Frequency Analysis at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during 2012–2013.

The second author is supported by the Research Council of Norway grant 227768.

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Queffélec, H., Seip, K. Decay rates for approximation numbers of composition operators. JAMA 125, 371–399 (2015). https://doi.org/10.1007/s11854-015-0012-6

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  • DOI: https://doi.org/10.1007/s11854-015-0012-6

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