1997/04/21 by Argyros, Spiros A., Deliyanni, Irene, Kutzarova, Denka +1
#46B20 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.math/9704215
We study the modified and boundedly modified mixed Tsirelson spaces TM[(\cal Fkn,θn)n=1∞ ] and TM(s)[(\cal Fkn,θn)n=1∞ ] respectively, defined by a subsequence (\cal Fkn) of the sequence of Schreier families (\cal Fn). These are reflexive asymptotic ℓ1 spaces with an unconditio- nal basis (ei)i having the property that every sequence \ xi\i=1n of normalized disjointly supported vectors contained in ⟨ ei⟩i=n∞ is equivalent to the basis of ℓ1n. We show that if limθn1/n=1 then the space T[(\cal Fn,θn) n=1∞ ] and its modified variations are totally incomparable by proving that c0 is finitely disjointly representable in every block subspace of T[(\cal Fn, θn)n=1∞ ]. Next, we present an example of a boundedly modified mixed Tsirelson space XM(1),u which is arbitrarily distortable. Finally, we construct a variation of the space XM(1),u which is hereditarily indecomposable.