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Poisson equation and discrete one-sided Hilbert transform for (C, α)-bounded operators

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Abstract

We characterize the solutions of the Poisson equation and the domain of its associated one-sided Hilbert transform for (C, α)-bounded operators, α > 0. This extends known results for power bounded operators to the setting of Cesàro bounded operators of fractional order, thus generalizing the results substantially. In passing, we obtain a generalization of the mean ergodic theorem in our framework. Examples are given to illustrate the theory.

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Acknowledgments

The authors wish to thank the anonymous referee of a preliminary version of this paper for their careful reading and suggestions that have notably contributed to improve the final version of the article.

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Authors and Affiliations

  1. Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009, Zaragoza, Spain

    Luciano Abadias & José E. Galé

  2. Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

    Carlos Lizama

Authors
  1. Luciano Abadias
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  2. José E. Galé
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  3. Carlos Lizama
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Corresponding author

Correspondence to Luciano Abadias.

Additional information

The two first-named authors have been partly supported by Project PID2019-105979GB-I00, of Ministry of Science of Spain, and Project E26-17R, D.G. Aragón, Universidad de Zaragoza, Spain. The first author has been also supported by Project JIUZ-2019-CIE-01 for Young Researchers, Fundación Ibercaja and Universidad de Zaragoza, Spain. The third author has been supported by the CONICYT-Chile under FONDECYT grant number 1180041 and DICYT-Universidad de Santiago de Chile, USACH.

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Abadias, L., Galé, J.E. & Lizama, C. Poisson equation and discrete one-sided Hilbert transform for (C, α)-bounded operators. Isr. J. Math. 253, 917–987 (2023). https://doi.org/10.1007/s11856-022-2353-z

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  • DOI: https://doi.org/10.1007/s11856-022-2353-z

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