Strictly singular operators in Tsirelson like spaces

Abstract

For each $n\in \mathbb{N} $ a Banach space $\mathfrak{X}_{_{^{0,1}}}^{n}$ is constructed having the property that every normalized weakly null sequence generates either a $c_{0}$ or $\ell_{1}$ spreading models and every subspace has weakly null sequences generating both $c_{0}$ and $\ell_{1}$ spreading models. The space $\mathfrak{X}_{_{^{0,1}}}^{n}$ is also quasiminimal and for every infinite dimensional closed subspace $Y$ of $\mathfrak{X}_{_{^{0,1}}}^{n}$, for every $S_{1},S_{2},\ldots,S_{n+1}$ strictly singular operators on $Y$, the operator $S_{1}S_{2}\cdots S_{n+1}$ is compact. Moreover, for every subspace $Y$ as above, there exist $S_{1},S_{2},\ldots,S_{n}$ strictly singular operators on $Y$, such that the operator $S_{1}S_{2}\cdots S_{n}$ is non-compact.