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On total incomparability of mixed Tsirelson spaces. (English) Zbl 1080.46507
Summary: We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form \(T[(\mathcal M_k,\theta _k)_{k =1}^l]\) with index \(i(\mathcal M_k)\) finite are either \(c_0\) or \(\ell _p\) saturated for some  \(p\) and we characterize when any two spaces of such a form are totally incomparable in terms of the index \(i(\mathcal M_k)\) and the parameter \(\theta _k\). Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form \(T[(\mathcal A_k,\theta _k)_{k = 1}^\infty ]\) in terms of the asymptotic behaviour of the sequence \(\Bigl \| \sum _{i=1}^n e_i\Bigr \| \), where \((e_i)\)  is the canonical basis.
MSC:
46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
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