Some remarks on James–Schreier spaces

Abstract

The James-Schreier spaces V p , where 1 p < ∞, were recently introduced by Bird and Laustsen (in press) [5] as an amalgamation of James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. The purpose of this note is to answer some questions left open in Bird and Laustsen (in press) [5]. Specifically, we prove that (i) the standard Schauder basis for the first James-Schreier space V 1 is shrinking, and (ii) any two Schreier or James-Schreier spaces with distinct indices are non-isomorphic. The former of these results implies that V 1 does not have Pełczyński's property (u) and hence does not embed in any Banach space with an unconditional Schauder basis.

An amalgamation of the Banach spaces associated with James and Schreier, Part II: Banach-algebra structure

“…Wp < ∞ , and the James-Schreier space, V p := span {e n : n ∈ N} ⊂ W p . We let X p := bip(V p ) with norm • Xp := • bip(Vp), and recall that X p = W p +Cχ N ∼ = V p[2, 4.15], due to the shrinking basis of V p (proved in[2, 4.12] in the case p > 1, and[3] for p = 1). We write • Vp when we consider • Wp restricted to V p .Lemma 4.1.…”