Garling sequence spaces

TL;DR

This paper introduces a new class of Banach spaces, $g(w,p)$, generalizing Garling's construction, which are $ ext{ell}_p$-saturated, have unique subsymmetric bases, and exhibit specific geometric properties.

Contribution

It constructs and analyzes a new family of Banach spaces $g(w,p)$ related to Lorentz spaces, revealing their basis structure and geometric properties.

Findings

$g(w,p)$ has a unique subsymmetric basis.

When $w=(n^{- heta})$, $g(w,p)$ lacks a symmetric basis.

$g(w,p)$ exhibits properties related to uniform convexity and superreflexivity.

Abstract

By generalizing a construction of Garling, for each $1\leqslant p<\infty$ 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.