Minimality properties of Tsirelson type spaces
Volume 187 / 2008
Studia Mathematica 187 (2008), 233-263
MSC: Primary 46B20; Secondary 46B45.
DOI: 10.4064/sm187-3-3
Abstract
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.
