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Cesàro sums and algebra homomorphisms of bounded operators

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Abstract

Let X be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra l 1(N0) and fractional versions of Cesàro sums of a linear operator T ∈ B(X) is established. In particular, we show that every (C, α)-bounded operator T induces an algebra homomorphism — and it is in fact characterized by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the (C, α)-boundedness of the resolvent operator for temperated a-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.

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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

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

    Carlos Lizama

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

    Pedro J. Miana

  4. Centro Universitario de la Defensa, Instituto Universitario de Matemáticas y Aplicaciones, Instituto de Matemática Interdisciplinar, 50090, Zaragoza, Spain

    M. Pilar Velasco

Authors
  1. Luciano Abadias
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  2. Carlos Lizama
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  3. Pedro J. Miana
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  4. M. Pilar Velasco
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Corresponding author

Correspondence to Luciano Abadias.

Additional information

L. Abadias, P. J. Miana and M.P. Velasco have been partially supported by Project MTM2013-42105-P and MTM2016-77710-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragón, and Project UZCUD2014-CIE-09, Universidad de Zaragoza. C. Lizama has been partially supported by DICYT, Universidad de Santiago de Chile; Project CONICYT-PIA ACT1112 Stochastic Analysis Research Network; FONDECYT 1140258 and Ministerio de Educación CEI Iberus (Spain).

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Abadias, L., Lizama, C., Miana, P.J. et al. Cesàro sums and algebra homomorphisms of bounded operators. Isr. J. Math. 216, 471–505 (2016). https://doi.org/10.1007/s11856-016-1417-3

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  • DOI: https://doi.org/10.1007/s11856-016-1417-3

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