Abstract
We provide a new separation-based proof of the domination theorem for (q, 1)-summing operators. This result gives the celebrated factorization theorem of Pisier for (q, 1)-summing operators acting in C(K)-spaces. As far as we know, none of the known versions of the proof uses the separation argument presented here, which is essentially the same that proves Pietsch Domination Theorem for p-summing operators. Based on this proof, we propose an equivalent formulation of the main summability properties for operators, which allows to consider a broad class of summability properties in Banach spaces. As a consequence, we are able to show new versions of the Dvoretzky–Rogers Theorem involving other notions of summability, and analyze some weighted extensions of the q-Orlicz property.
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The authors would like to thank the referee for her/his careful reading of the manuscript and for her/his suggestions. The authors declare that there is no data associated with this paper.
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Dedicated to our esteemed Professor Andreas Defant, who long ago posed the problem that originated this work as a question to the second author.
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Both authors were supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación (Spain) and FEDER, the first author under project PGC2018-095366-B-100 and the second under project MTM2016-77054-C2-1-P.
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Calabuig, J.M., Sánchez Pérez, E.A. Absolutely (q, 1)-summing operators acting in C(K)-spaces and the weighted Orlicz property for Banach spaces. Positivity 25, 1199–1214 (2021). https://doi.org/10.1007/s11117-021-00811-y
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DOI: https://doi.org/10.1007/s11117-021-00811-y