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On coincidence results for summing multilinear operators: interpolation, \(\ell _1\)-spaces and cotype

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Abstract

Grothendieck’s theorem asserts that every continuous linear operator from \(\ell _1\) to \(\ell _2\) is absolutely (1, 1)-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by \(\ell _1\) spaces is also investigated with relation to interpolation of tensor products. In particular, an open problem on the interpolation of m injective tensor products is solved.

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Notes

  1. After this paper was put on ArXiv, D. Popa informed us that he introduced independently this notion in one of his submitted papers.

  2. A version of Theorem 2.5 with restricted conditions on \(p_0\) and \(p_1\) was recently given in an updated version of [16].

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Acknowledgements

The authors are very grateful to the referee for her/his valuable suggestions that improved the final version of this paper.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, Campus des Cézeaux 3, place Vasarely TSA 60026 CS 60026, 63178, Aubière cedex, France

    Frédéric Bayart

  2. Departamento de Matemática, Universidade Federal da Paraíba, CEP: 58.051-900, João Pessoa, PB Coubtry, Brazil

    Daniel Pellegrino

  3. Department d’Anàlisi Matemàtica, Facultat de Matematiques, Universitat de València, C/Dr. Moliner 50, 46100, Burjassot, Valencia, PB Coubtry, Spain

    Pilar Rueda

Authors
  1. Frédéric Bayart
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  2. Daniel Pellegrino
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  3. Pilar Rueda
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Correspondence to Pilar Rueda.

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F. Bayart was partially supported by the Grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). D. Pellegrino was partially supported by the Réseau Franco-Brésilien en Mathématiques. P. Rueda is supported by Ministerio de Economía, Industria y Competitividad and FEDER under Project MTM2016-77054-C2-1-P.

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Bayart, F., Pellegrino, D. & Rueda, P. On coincidence results for summing multilinear operators: interpolation, \(\ell _1\)-spaces and cotype. Collect. Math. 71, 301–318 (2020). https://doi.org/10.1007/s13348-019-00261-6

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  • DOI: https://doi.org/10.1007/s13348-019-00261-6

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