On isometric embeddability of into as 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 Smq into Snp, where 1 ≤ p ≠ q < ∞ and m, n ≥ 2, has been recently studied in [6]. 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 {equation presented}, 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.

DOI

https://10.1017/prm.2023.54

Publication Date

1-1-2023

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