Some remarks on James–Schreier spaces

The James–Schreier spaces Vp, where 1 6 p < ∞, were recently introduced by Bird and Laustsen [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 [5]. Specifically, we prove that (i) the standard Schauder basis for the first James–Schreier space V1 is shrinking, and (ii) any two Schreier or James–Schreier spaces with distinct indices are non-isomorphic. The former of these results implies that V1 does not have Pełczyński’s property (u) and hence does not embed in any Banach space with an unconditional Schauder basis. 2000 Mathematics Subject Classification: primary 46B03, 46B45; secondary 46B15.