Garling sequence spaces

Abstract

By generalizing a construction of Garling, for each $1⩽p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0\setminus {\ell }_1$ we exhibit an ${\ell }_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$. Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when $w=(n^{-\theta }{)}_{n=1}^{\infty }$ for some $0<\theta <1$, it does not admit a symmetric basis. We then discuss some additional properties of $g(w,p)$ related to uniform convexity and superreflexivity.

the paper has been replaced by the paper "On Garling Sequence spaces" by Albiac/Ansorena/Wallis: arXiv.org > math > arXiv:1703.07772