Article contents
On isometric embeddability of $S_q^m$
into $S_p^n$
as non-commutative quasi-Banach spaces
Published online by Cambridge University Press: 22 June 2023
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 [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 $\ell _q^m(\mathbb {R})$
into $\ell _p^n(\mathbb {R})$
, 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 $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$
$\cup \, \{1\}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$
$\cup \, \{\infty \}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$
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.
Keywords
- Type
- Research Article
- Information
- Copyright
- Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
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