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Co-Isometric Weighted Composition Operators on Hilbert Spaces of Analytic Functions

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Abstract

We obtain a necessary and sufficient condition for a weighted composition operator to be co-isometric on a general weighted Hardy space of analytic functions in the unit disk whose reproducing kernel has the usual natural form. This turns out to be equivalent to the property of being unitary. The result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that support non-trivial operators of this kind.

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Acknowledgements

The first author was supported by the Viera y Clavijo program (2020/0000311) of ULL, Spain, and by the research project PID2019-106093GB-I00, from MICINN, Spain. The third author received partial support from the Erwin Schrödinger Institute, Vienna, during the workshop “Operator Related Function Theory” held in April of 2019. All authors acknowledge support from PID2019-106870GB-I00 from MICINN, Spain.

The authors would like to thank Michael Hartz, José A. Peláez, and Nina Zorboska for some useful comments and references and Iason Efraimidis for pointing out some misprints in an earlier version. We are particularly grateful to the referee for pointing out several imprecise and incomplete details in an earlier version of the manuscript, for a number of useful comments and also for suggesting some alternative ideas of proofs.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, 38200, San Cristóbal de La Laguna, Santa Cruz de Tenerife, Spain

    María J. Martín

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Alejandro Mas & Dragan Vukotić

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  1. María J. Martín
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  2. Alejandro Mas
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  3. Dragan Vukotić
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Correspondence to Dragan Vukotić.

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Martín, M.J., Mas, A. & Vukotić, D. Co-Isometric Weighted Composition Operators on Hilbert Spaces of Analytic Functions. Results Math 75, 128 (2020). https://doi.org/10.1007/s00025-020-01253-w

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