Lorentz spaces and embeddings induced by almost greedy bases in superreflexive Banach spaces

TL;DR

This paper demonstrates that almost greedy bases in superreflexive Banach spaces lead to tighter embeddings, enabling the space to be closely squeezed between two similar superreflexive Lorentz sequence spaces.

Contribution

It establishes that almost greedy bases induce more refined embeddings in superreflexive Banach spaces than in general spaces, connecting them with Lorentz sequence spaces.

Findings

Almost greedy bases induce tighter embeddings in superreflexive spaces.

Superreflexive Banach spaces can be squeezed between similar Lorentz sequence spaces.

Embeddings are characterized by the fundamental function of the spaces.

Abstract

The aim of this paper is to show that almost greedy bases induce tighter embeddings in superreflexive Banach spaces than in general Banach spaces. More specifically, we show that an almost greedy basis in a superreflexive Banach space $\mathbb{X}$ induces embeddings that allow squeezing $\mathbb{X}$ between two superreflexive Lorentz sequence spaces that are close to each other in the sense that they have the same fundamental function.