Modified Mixed Tsirelson Spaces

We study the modified and boundedly modified mixed Tsirelson spacesTM[(Fkn, θn)∞n=1] andTM(s)[(Fkn, θn)∞n=1], respectively, defined by a subsequence (Fkn) of the sequence of Schreier families (Fn). These are reflexive asymptotic ℓ1spaces with an unconditional basis (ei)ihaving the property that every sequence {xi}ni=1of normalized disjointly supported vectors contained in ⦠ei⦔∞i=nis equivalent to the basis of ℓn1. We show that if limθ1/nn=1 then the spaceT[(Fn, θn)∞n=1] and its modified variationsTM[(Fn, θn)∞n=1] orTM(s)[(Fn, θn)∞n=1] are totally incomparable by proving thatc0is finitely disjointly representable in every block subspace ofT[(Fn, θn)∞n=1]. Next, we present an example of a boundedly modified mixed Tsirelson spaceXM(1), u=TM(1)[(Fkn, θn)∞n=1] which is arbitrarily distortable. Finally, we construct a variation of the spaceXM(1), uwhich is hereditarily indecomposable.