On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach space

TL;DR

This paper investigates the conditions under which isometric embeddings between non-commutative Schatten classes and quasi-Banach spaces exist, extending previous results to new parameter ranges and introducing non-commutative Clarkson's inequality as a key tool.

Contribution

It extends the study of isometric embeddability to broader parameter ranges for non-commutative Schatten classes and develops new analytical tools like non-commutative Clarkson's inequality.

Findings

No isometric embedding of $oldsymbol{ extit{ extbf{ell}}}_q^m(oldsymbol{ extbf{R}})$ into $oldsymbol{ extit{ extbf{ell}}}_p^n(oldsymbol{ extbf{R}})$ for certain $(q,p)$.

No isometric embedding of $S_q^m$ into $S_p^n$ for specific $(q,p)$ ranges involving $(0,2)$ and $(0,1)$.

In some cases, no isometric embedding exists for $S_q^m$ into $S_p^n$ with $p o ext{large}$ and $q o 2$.

Abstract

The existence of isometric embedding of $S_q^m$ into $S_p^n$, where $1\leq p\neq q\leq \infty$ and $m,n\geq 2$ has been recently studied in \cite{JFA22}. 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 $\ell_q^m(\R)$ into $\ell_p^n(\R)$, where $(q,p)\in (0,\infty)\times (0,1)$ and $p\neq q$. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$ $\cup\, \{1\}\times (0,1)\setminus \{\frac{1}{n}:n\in\mathbb{N}\}$ $\cup\, \{\infty\}\times (0,1)\setminus \{\frac{1}{n}:n\in\mathbb{N}\}$ and $p\neq q$. Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in [2, \infty)\times (0,1)$. 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.