Sequence Lorentz spaces and their geometric structure

TL;DR

This paper investigates the geometric properties of sequence Lorentz spaces, providing criteria for various structural features, characterizing extreme points, and describing dual spaces, with applications to subspace structure.

Contribution

It offers new complete characterizations of geometric features and duality for sequence Lorentz spaces, especially $oldsymbol{ ext{γ}_{1,w}}$, in the case of pure atomic measures.

Findings

Criteria for order continuity, Fatou property, strict monotonicity, strict convexity.

Full characterization of extreme points of the unit ball in γ_{1,w}.

Description of dual and predual spaces in terms of Marcinkiewicz spaces.

Abstract

This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity and strict convexity in the sequence Lorentz spaces $\gamma_{p,w}$. Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space $\gamma_{1,w}$. We also establish a complete description with an isometry of the dual and predual spaces of the sequence Lorentz spaces $\gamma_{1,w}$ written in terms of the Marcinkiewicz spaces. Finally, we show a fundamental application of geometric structure of $\gamma_{1,w}$ to one-complemented subspaces of $\gamma_{1,w}$.