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Constructive Approximation

, Volume 49, Issue 1, pp 103–122 | Cite as

Conditional Quasi-Greedy Bases in Non-superreflexive Banach Spaces

  • Fernando AlbiacEmail author
  • José L. Ansorena
  • Przemysław Wojtaszczyk
Article

Abstract

For a conditional quasi-greedy basis \(\mathcal {B}\) in a Banach space, the associated conditionality constants \(k_{m}[\mathcal {B}]\) verify the estimate \(k_{m}[\mathcal {B}]={\mathcal {O}}(\log m)\). Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies \(k_{m}[\mathcal {B}]=\mathcal O((\log m)^{1-\epsilon })\) for some \(0<\epsilon <1\), and this is optimal. Our first goal in this paper will be to fill the gap between the general case and the superreflexive case and investigate the growth of the conditionality constants in nonsuperreflexive spaces. Roughly speaking, the moral will be that we can guarantee optimal bounds only for quasi-greedy bases in superreflexive spaces. We prove that if a Banach space \(\mathbb {X}\) is not superreflexive, then there is a quasi-greedy basis \(\mathcal {B}\) in a Banach space \(\mathbb {Y}\) finitely representable in \(\mathbb {X}\) with \(k_{m}[\mathcal {B}] \approx \log m\). As a consequence, we obtain that for every \(2<q<\infty \), there is a Banach space \(\mathbb {X}\) of type 2 and cotype q possessing a quasi-greedy basis \(\mathcal {B}\) with \(k_{m}[\mathcal {B}] \approx \log m\). We also tackle the corresponding problem for Schauder bases and show that if a space is nonsuperreflexive, then it possesses a basic sequence \(\mathcal {B}\) with \(k_m[\mathcal {B}]\approx m\).

Keywords

Thresholding greedy algorithm Conditional basis Conditionality constants Quasi-greedy basis Type Cotype Reflexivity Superreflexivity Super property Finite representability Banach spaces 

Mathematics Subject Classification

46B15 41A65 

Notes

Acknowledgements

The authors would like to thank Prof. Thomas Schlumprecht for helpful discussions and for bringing to their attention the paper [14].

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Fernando Albiac
    • 1
    Email author
  • José L. Ansorena
    • 2
  • Przemysław Wojtaszczyk
    • 3
    • 4
  1. 1.Mathematics DepartmentUniversidad Pública de NavarraPamplonaSpain
  2. 2.Department of Mathematics and Computer SciencesUniversidad de La RiojaLogroñoSpain
  3. 3.Interdisciplinary Centre for Mathematical and Computational ModellingUniversity of WarsawWarszawaPoland
  4. 4.Institute of Mathematics of the Polish Academy of SciencesWarszawaPoland

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