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Minimality properties of Tsirelson type spaces

Denka Kutzarova, Denny Leung, Antonis Manoussakis, Wee Kee Tang

math.FAarXiv:math/0702210

Abstract

In this paper, we study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis (ek) is said to be subsequentially minimal if for every normalized block basis (xk) of (ek), there is a further block (yk) of (xk) such that (yk) is equivalent to a subsequence of (ek). Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal and connections with Bourgain's 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.

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