Quasiminimality in mixed Tsirelson spaces

We prove quasiminimality of regular the mixed Tsirelson spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T[(\mathcal {S}_{n},\theta _{n})_{n}]$\end{document} with the sequence \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\big (\frac{\theta _n}{\theta ^n}\big )_n$\end{document} decreasing, where θ = lim n θ 1/ n n , and quasiminimality of all mixed Tsirelson spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T[(\mathcal {A}_{n},\theta _{n})_{n}]$\end{document} . We prove that under certain assumptions on the sequence (θ n ) n the dual spaces are quasiminimal.