Characterising subspaces of Banach spaces with a Schauder basis having the shift property

Abstract

We give an intrinsic characterisation of the separable reflexive Banach spaces that embed into separable reflexive spaces with an unconditional basis all of whose normalised block sequences with the same growth rate are equivalent. This uses methods of E. Odell and T. Schlumprecht. 1. THE SHIFT PROPERTY We consider in this paper a property of Schauder bases that has come up on sev- eral occasions since the first construction of a truly non-classical Banach space by B. S. Tsirelson in 1974 (11). It is a weakening of the property of perfect homogeneity, which replaces the condition all normalised block bases are equivalent with the weaker all normalised block bases with the same growth rate are equivalent, and is satisfied by bases constructed along the lines of the Tsirelson basis, including the standard bases for the Tsirelson space and its dual. To motivate our study and in order to fix ideas, in the following result we sum up a number of conditions that have been studied at various occasions in the literature and that can all be seen to be reformulations of the aforementioned property. Though I know of no single reference for the proof of the equivalence, parts of it are implicit in J. Lindenstrauss and L. Tzafriri's paper (7) and the paper by P. G. Casazza, W. B. Johnson and L. Tzafriri (2). Moreover, any idea needed for the proof can be found in,