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

The existence of isometric embedding of $S_q^m$ into $S_p^n$, where $1\le p\mathrm{\ne }q\le \mathrm{\infty }$ and $m,n\ge 2$ 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 ${\mathrm{\ell }}_q^m(\mathbb{R})$ into ${\mathrm{\ell }}_p^n(\mathbb{R})$, where $(q,p)\in (0,\mathrm{\infty })\times (0,1)$ and $p\mathrm{\ne }q$. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$ $\cup \{1\}\times (0,1)\setminus \{\frac{1}{n}:n\in \mathbb{N}\}$ $\cup \{\mathrm{\infty }\}\times (0,1)\setminus \{\frac{1}{n}:n\in \mathbb{N}\}$ and $p\mathrm{\ne }q$. Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in [2,\mathrm{\infty })\times (0,1)$. 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.