Integral Equations and Operator Theory

, Volume 89, Issue 1, pp 69–88 | Cite as

Absolutely \(\varvec{(r,q)}\)-Summing Operators on Vector-Valued Function Spaces

Article

Abstract

Let X and Y be Banach spaces and let \(\Omega \) be a compact Hausdorff space. In 1973, Swartz, in his by now classical theorem, characterized the absolute summability of an operator U from \({\mathcal {C}}(\Omega ,X)\) to Y in terms of its associated operator \(U^{\#}\) and of its representing measure m. We study the interplay between U, \(U^{\#}\), and m in the context of absolutely (rq)-summing operators, considering the spaces \({\mathcal {C}}_{p}(\Omega , X)\) of p-continuous functions on \(\Omega \), \(1\le p\le \infty \), instead of \({\mathcal {C}}(\Omega ,X) = {\mathcal {C}}_{\infty }(\Omega , X)\). This encompasses the Swartz theorem together with its existing extensions on absolutely (rq)-summing operators, providing, among others, an improvement even to the Swartz theorem. Counterexamples are exhibited to indicate sharpness of our results.

Keywords

Banach spaces Absolutely \((r , q)\)- and absolutely p-summing operators Operator-valued measures p-Continuous vector-valued functions r-Variation 

Mathematics Subject Classification

Primary 47B10 Secondary 46B25 46B28 46E40 46G10 47B38 47L20 

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Notes

Acknowledgements

The authors thank the referee for helpful suggestions that improved the exposition. The research of Eve Oja was partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The research of Cándido Piñeiro and Fernando Muñoz was partially supported by the Junta de Andalucía P.A.I. FQM-276.

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Departamento de Ciencias Integradas, Facultad de Ciencias ExperimentalesUniversidad de Huelva Campus Universitario de El CarmenHuelvaSpain
  2. 2.Institute of Mathematics and StatisticsUniversity of TartuTartuEstonia
  3. 3.Estonian Academy of SciencesTallinnEstonia

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