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On isometric embeddability of $S_q^m$ into $S_p^n$ as non-commutative quasi-Banach spaces

Part of: Basic linear algebra Normed linear spaces and Banach spaces; Banach lattices General theory of linear operators Selfadjoint operator algebras

Published online by Cambridge University Press:  22 June 2023

Arup Chattopadhyay ,
Guixiang Hong ,
Chandan Pradhan  and
Samya Kumar Ray
Arup Chattopadhyay
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India (arupchatt@iitg.ac.in, 2003arupchattopadhyay@gmail.com)
Guixiang Hong
Affiliation:
Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China (gxhong@hit.edu.cn)
Chandan Pradhan
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India (chandan.math@iitg.ac.in, chandan.pradhan2108@gmail.com)
Samya Kumar Ray
Affiliation:
Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, India (samyaray7777@gmail.com)
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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

isometric embeddingnon-commutative Lp-spacesSchatten-p classKato–Rellich theoremmultiple operator integraloperator derivatives

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46L51: Noncommutative measure and integration 46L52: Noncommutative function spaces 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A55: Perturbation theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 24
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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