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 $$T\left[ {\left( {\mathcal{M}_k ,\theta _k } \right)_{k = 1}^l } \right]$$ with index $$i\left( {\mathcal{M}_k } \right)$$ 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 $$i\left( {\mathcal{M}_k } \right)$$ and the parameter θ k . Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $$T\left[ {\left( {\mathcal{A}_k ,\theta _k } \right)_{k = 1}^\infty } \right]$$ in terms of the asymptotic behaviour of the sequence $$\left\| {\sum\limits_{j = 1}^n {e_i } } \right\|$$ where (e i is the canonical basis.