Embeddings between Lorenz sequence spaces are strictly singular

TL;DR

This paper proves that the natural embedding between certain Lorenz sequence spaces is always strictly singular, meaning it cannot be inverted on any infinite-dimensional subspace, but it is not finitely strictly singular.

Contribution

It establishes a new property of embeddings between Lorenz sequence spaces, showing they are strictly singular but not finitely strictly singular, which was previously unknown.

Findings

Embedding is strictly singular for specified parameters.

Embedding is not finitely strictly singular.

Provides new insights into the structure of Lorenz sequence spaces.

Abstract

Given $0<p,q, r<\infty $ and $ q<r\le \infty$ we consider the natural embedding $\ell_{p,q}\hookrightarrow \ell_{p,r}$ between Lorenz sequence spaces. We prove that this non-compact embedding is always strictly singular but not finitely strictly singular.