Equivalence of block sequences in Schreier spaces and their duals

We prove that any normalized block sequence in a Schreier space $X_\xi$, of arbitrary order $\xi<\omega_1$, admits a subsequence equivalent to a subsequence of the canonical basis of some Schreier space. The analogous result is proved for dual spaces to Schreier spaces. Basing on these results, we examine the structure of strictly singular operators on Schreier spaces and show that there are $2^\mathfrak{c}$ many closed operator ideals on a Schreier space of any order, its dual and bidual space.Comment: Corrected citatio

On the hereditary proximity to ι1\iota _{1}

AbstractIn the first part of the paper we present and discuss concepts of local and asymptotic hereditary proximity to ℓ1. The second part is devoted to a complete separation of the hereditary local proximity to ℓ1 from the asymptotic one. More precisely for every countable ordinal ξ we construct a separable Hereditarily Indecomposable reflexive space Xξ such that every infinite-dimensional subspace of it has Bourgain ℓ1-index greater than ωξ and the space itself has no ℓ1-spreading model