Minimality properties of Tsirelson type spaces

Tom 187 / 2008

Denka Kutzarova, Denny H. Leung, Antonis Manoussakis, Wee-Kee Tang Studia Mathematica 187 (2008), 233-263 MSC: Primary 46B20; Secondary 46B45. DOI: 10.4064/sm187-3-3


We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be subsequentially minimal if for every normalized block basis $(x_{k})$ of $(e_{k}) ,$ there is a further block basis $(y_{k})$ of $(x_{k}) $ such that $(y_{k})$ is equivalent to a subsequence of $(e_{k}).$ Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain's $\ell^{1}$-index are established. It is also shown that a large class of mixed Tsirelson spaces fails to be subsequentially minimal in a strong sense.


  • Denka KutzarovaInstitute of Mathematics
    Bulgarian Academy of Sciences
    Department of Mathematics
    University of Illinois at Urbana-Champaign
    Urbana, IL 61801, U.S.A.
  • Denny H. LeungDepartment of Mathematics
    National University of Singapore
    2 Science Drive 2
    Singapore 117543
  • Antonis ManoussakisDepartment of Mathematics
    University of Aegean
    Karlovasi, Samos, GR 83200, Greece
  • Wee-Kee TangMathematics and Mathematics Education
    National Institute of Education
    Nanyang Technological University
    1 Nanyang Walk
    Singapore 637616

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