Tsirelson superspaces and \(\ell _ p\)
Tsirelson space, T, is a Banach space which contains neither \(c_ 0\) nor \(\ell_ p\)-subspaces. Indeed, T contains no subsymmetric sequence. The main parameter in the construction of T, is the definition of what is an ''admissible'' sequence of finite subsets of N, \((E_ i)^ k_ 1\). In T, the \((E_ i)^ k_ 1\) are required to satisfy min \(E_{i+1}>\max E_ i\) and \(\min (\cup E_ i)\geq k.\) By modifying this ''admissible'' to require \(f(\min (\cup E_ i))\geq k\), for other non-decreasing integer functions f, we obtain a large collection of spaces which include the \(\ell_ p\)-spaces and Tsirelson superspaces S. For these spaces S the ''strict'' and ''modified'' norms are equivalent (2- equivalent for T). Some of these spaces S are not isomorphic to the Cartesian square. Thus techniques different from which have been used to study T are developd. Norming trees turn out to be the main tool. For each S, there is a p so that S satisfies an upper \(\ell_ p\)-estimate and lower \(\ell r\)-estimates for each \(r>p\).
scientific article; zbMATH DE number 3985914