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Volterra operators and semigroups in weighted Banach spaces of analytic functions

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Abstract

We characterize the boundedness, compactness and weak compactness of Volterra operators \(V_{g}(f)(z) := \int _{0}^{z}f({\zeta })g^{\prime }({\zeta })\,d{\zeta }\) acting between different weighted spaces of type \(H^{\infty }_{v}\) in terms of the symbol function \(g\), for the case when \(v\) is a quasi-normal weight, a notion weaker than normality. Then we apply the characterization of compactness to analyze the behavior of semigroups of composition operators on \(H^{\infty }_{v}\).

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Authors and Affiliations

  1. Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092,  Seville, Spain

    Manuela Basallote, Manuel D. Contreras, Carmen Hernández-Mancera & Pedro J. Paúl

  2. Departamento de Matemáticas (Módulo 17, Edificio de Ciencias), Universidad Autónoma de Madrid, 28049 , Madrid, Spain

    María J. Martín

Authors
  1. Manuela Basallote
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  2. Manuel D. Contreras
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  3. Carmen Hernández-Mancera
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  4. María J. Martín
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  5. Pedro J. Paúl
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Corresponding author

Correspondence to Carmen Hernández-Mancera.

Additional information

M. Basallote, M. D. Contreras and C. Hernández-Mancera was partially supported by the Ministerio de Ciencia e Innovación, Spain, and the European Union (FEDER) project MTM2009-14694-C02-02, by the ESF Networking Programme “Harmonic and Complex Analysis and its Applications”, and by La Consejería de Economía, Innovación y Ciencia de la Junta de Andalucía (research group FQM-133). M. J. Martín was partially supported by grant MTM1009-14694-C02-01, Ministerio de Ciencia e Innovación, Spain, and by the Instituto de Matemáticas de la Universidad de Sevilla (IMUS). P. J. Paúl was partially supported by La Consejería de Economía, Innovación y Ciencia de la Junta de Andalucía (research group FQM-133).

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Basallote, M., Contreras, M.D., Hernández-Mancera, C. et al. Volterra operators and semigroups in weighted Banach spaces of analytic functions. Collect. Math. 65, 233–249 (2014). https://doi.org/10.1007/s13348-013-0092-5

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