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On certain subspaces of \({\ell _p}\) for \({0<p\le 1}\) and their applications to conditional quasi-greedy bases in p-Banach spaces

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Abstract

We construct for each \(0<p\le 1\) an infinite collection of subspaces of \(\ell _p\) that extend the example of Lindenstrauss (Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12, 539–542, 1964) of a subspace of \(\ell _{1}\) with no unconditional basis. The structure of this new class of p-Banach spaces is analyzed and some applications to the general theory of \({\mathcal {L}}_{p}\)-spaces for \(0<p<1\) are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in p-Banach spaces for \(p<1\). Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces \(\ell _{p}\) for \(p\le 1\) and the careful examination of the conditionality constants of the “natural basis” of these spaces.

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Authors and Affiliations

  1. Mathematics Department-InaMat, Public University of Navarre, Pamplona, 31006, Spain

    Fernando Albiac

  2. Department of Mathematics and Computer Sciences, Universidad de La Rioja, Logroño, 26004, Spain

    José Luis Ansorena

  3. Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warszawa, Poland

    Przemysław Wojtaszczyk

Authors
  1. Fernando Albiac
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  2. José Luis Ansorena
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  3. Przemysław Wojtaszczyk
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Correspondence to Fernando Albiac.

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Communicated by Loukas Grafakos.

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F. Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity under Grant MTM2016-76808-P as well as the Spanish Ministry for Science and Innovation under Grant PID2019-1077701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. P. Wojtaszczyk was supported by National Science Centre, Poland Grant UMO-2016/21/B/ST1/00241.

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Albiac, F., Ansorena, J.L. & Wojtaszczyk, P. On certain subspaces of \({\ell _p}\) for \({0<p\le 1}\) and their applications to conditional quasi-greedy bases in p-Banach spaces. Math. Ann. 379, 465–502 (2021). https://doi.org/10.1007/s00208-020-02069-3

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  • DOI: https://doi.org/10.1007/s00208-020-02069-3

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