TY - JOUR

T1 - Modified Mixed Tsirelson Spaces

AU - Argyros, S. A.

AU - Deliyanni, I.

AU - Kutzarova, D. N.

AU - Manoussakis, A.

N1 - Funding Information:
We study the modified and boundedly modified mixed Tsirelson spaces TM[(Fkn, %n)n=1] and TM(s)[(Fkn, %n)n=1], respectively, defined by a subsequence (Fkn) of the sequence of Schreier families (Fn). These are reflexive asymptotic l1 spaces with an unconditional basis (ei)i having the property that every sequence [xi]i=1n of normalized disjointly supported vectors contained in (ei)i=n is equiv-alenttothebasisofl1n.Weshowthatiflim%n1 n=1thenthespaceT[(Fn,%n)n=1] and its modified variations TM[(Fn,%n)n=1] or TM(s)[(Fn,%n)n=1] are totally incomparable by proving that c0 is finitely disjointly representable in every block subspaceofT[(Fn,%n)n=1].Next,wepresentanexampleofaboundedlymodified mixed Tsirelson space XM(1),u=TM(1)[(Fkn, %n)n=1] which is arbitrarily distortable. Finally, we construct a variation of the space XM(1),u which is hereditarily indecomposable. 1998 Academic Press * Research partially supported by the Bulgarian Ministry of Education and Science under Contract MM-703 97.

PY - 1998/10/20

Y1 - 1998/10/20

N2 - 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,θ n)∞n=1] which is arbitrarily distortable. Finally, we construct a variation of the spaceXM(1),uwhich is hereditarily indecomposable.

AB - 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,θ n)∞n=1] which is arbitrarily distortable. Finally, we construct a variation of the spaceXM(1),uwhich is hereditarily indecomposable.

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U2 - 10.1006/jfan.1998.3310

DO - 10.1006/jfan.1998.3310

M3 - Article

AN - SCOPUS:0002480051

VL - 159

SP - 43

EP - 109

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -