On total incomparability of mixed Tsirelson spaces

Abstract

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T[(M_k,{\theta }_k{)}_{k=1}^{\ell }]$ with index $i(M_k)$ 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 $i(M_k)$ and the parameter ${\theta }_k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T[(A_k,{\theta }_k{)}_{k=1}^{\infty }]$ in terms of the asymptotic behaviour of the sequence $\Vert {\sum }_{i=1}^n{e}_i\Vert$ where $(e_i)$ is the canonical basis.

13 pages. To be published in Czech. Math. Jour