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mixed Tsirelson spaces; totally incomparable spaces
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 \Vert \sum _{i=1}^n e_i\Bigr \Vert $ where $(e_i)$ is the canonical basis.
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