Isometric Embeddability of $S_q^m$ into $S_p^n$

TL;DR

This paper investigates the conditions under which non-commutative Schatten spaces can be isometrically embedded into each other, revealing that such embeddings are highly restricted and mostly occur only when the spaces are associated with $q=2$.

Contribution

The work generalizes classical results by establishing new restrictions on isometric embeddings between Schatten spaces, especially for finite-dimensional cases, using advanced operator perturbation techniques.

Findings

Isometric embeddings between $S_q^m$ and $S_p^n$ only occur when $q=2$ under specified conditions.

The results extend classical theorems of Lyubich and Vaserstein to broader settings.

The paper employs novel methods involving perturbation theory, operator integrals, and orthogonality concepts.

Abstract

In this paper, we study existence of isometric embedding of $S_q^m$ into $S_p^n,$ where $1\leq p\neq q\leq \infty$ and $n\geq m\geq 2.$ We show that for all $n\geq m\geq 2$ if there exists a linear isometry from $S_q^m$ into $S_p^n$, where $(q,p)\in(1,\infty]\times(1,\infty) \cup(1,\infty)\setminus\{3\}\times\{1,\infty\}$ and $p\neq q,$ then we must have $q=2.$ This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever $S_q$ embeds isometrically into $S_p$ for $(q,p)\in \left(1,\infty\right)\times\left[2,\infty \right)\cup[4,\infty)\times\{1\} \cup\{\infty\}\times\left( 1,\infty\right)\cup[2,\infty)\times\{\infty\}$ with $p\neq q,$ we must have $q=2.$ Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative $L_p$-spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for $m\geq 2$ and $1<q<2,$ $S_q^m$ embeds isometrically into $S_\infty^n$, was left open in \textit{Bull. London Math. Soc.} 52 (2020) 437-447.