On the complemented subspaces of the Schreier spaces

It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space ${\displaystyle X^{\xi }}$ generated by subsequences ${\displaystyle \left(e_{l_n}^{\xi }\right)}$ and ${\displaystyle \left(e_{m_n}^{\xi }\right)}$, respectively, of the natural Schauder basis ${\displaystyle \left(e_n^{\xi }\right)}$ of ${\displaystyle X^{\xi }}$ are isomorphic if and only if ${\displaystyle \left(e_{l_n}^{\xi }\right)}$ and ${\displaystyle \left(e_{m_n}^{\xi }\right)}$ are equivalent. Further, ${\displaystyle X^{\xi }}$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of ${\displaystyle \left(e_n^{\xi }\right)}$. It is also shown that there exists a complemented subspace spanned by a block basis of ${\displaystyle \left(e_n^{\xi }\right)}$, which is not isomorphic to a subspace generated by a subsequence of ${\displaystyle \left(e_n^{\zeta }\right)}$, for every ${\displaystyle 0\le \zeta \le \xi }$. Finally, an example is given of an uncomplemented subspace of ${\displaystyle X^{\xi }}$ which is spanned by a block basis of ${\displaystyle \left(e_n^{\xi }\right)}$.