On the complemented subspaces of the Schreier spaces

Abstract

It is shown that for every 1 ≤ ξ < w, two subspaces of the Schreier space Xξ generated by subsequences (eξ ln) and (eξ mn), respectively, of the natural Schauder basis (eξ n) of Xξ are isomorphic if and only if (eξ ln) and (eξ mn) are equivalent. Further, Xξ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of (eξ n). It is also shown that there exists a complemented subspace spanned by a block basis of (eξ n), which is not isomorphic to a subspace generated by a subsequence of (eζ n), for every 0 ≤ ζ ≤ ξ. Finally, an example is given of an uncomplemented subspace of Xξ which is spanned by a block basis of (eξ n).