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Types of tightness in spaces with unconditional basis

100%
Manoussakis A., Pelczar-Barwacz A.
Studia Mathematica
|
2014
|
tom 220
|
nr 3
243-264
EN
We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal's list related to Gowers' classification program of Banach spaces, but in contrast to the recently constructed space of type (4), our space is also tight with constants, thus essentially extending the list of known examples in Gowers' program. The space is defined on the basis of a boundedly modified mixed Tsirelson space with the use of a special coding function.
2
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An indecomposable and unconditionally saturated Banach space

100%
Argyros S., Manoussakis A.
Studia Mathematica
|
2003
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tom 159
|
nr 1
1-32
EN
We construct an indecomposable reflexive Banach space $X_{ius}$ such that every infinite-dimensional closed subspace contains an unconditional basic sequence. We also show that every operator $T ∈ ℬ (X_{ius})$ is of the form λI + S with S a strictly singular operator.
3
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Minimality properties of Tsirelson type spaces

81%
Kutzarova D., Leung D., Manoussakis A., Tang W.-K.
Studia Mathematica
|
2008
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tom 187
|
nr 3
233-263
EN
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 ℓ¹-index are established. It is also shown that a large class of mixed Tsirelson spaces fails to be subsequentially minimal in a strong sense.
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