On isometric embeddability of Sminto Snpas non-commutative quasi-Banach spaces

Article Type

Research Article

Publication Title

Proceedings of the Royal Society of Edinburgh Section A Mathematics

Abstract

The existence of isometric embedding of Sm into Snp, where 1 ≤p ≠q≤ ∞ and m, n ≥ 2, has been recently studied in. In this article, we extend the study of isometric embeddability beyond the above-mentioned range of p and q. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space lmq(R) into lnp (R), where (q, p) ∈ (0,∞) × (0, 1) and p ≠ q. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of Smq into Snp, where (q, p) ∈ (0, 2) \ {1} × (0, 1) ∪{1} × (0, 1) \{1/n : n ∈ N} ∪ {∞} × (0, 1) \{1/n : n ∈ N} and p ≠ q. Moreover, in some restrictive cases, we also show that there is no isometric embedding of Smq into Snp, where (q, p) ∈ [2,∞) × (0, 1). A new tool in our paper is the non-commutative Clarkson's inequality for Schatten class operators. Other tools involved are the Kato-Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

First Page

1180

Last Page

1203

DOI

10.1017/prm.2023.54

Publication Date

8-1-2024

Link to Full Text

Share

COinS