$S_q^m$在$S_p^n$中的等距嵌入
Isometric Embeddability of $S_q^m$ into $S_p^n$
摘要：
本文研究了$S_q^m$到$S_p^n,$的等距嵌入的存在性，其中$1\leq p\neq q\leq \infty$和$n\geq m\geq 2.$证明了对所有的$n\geq m\geq 2$，如果存在一个从$S_q^m$到$S_p^n$的线性等距嵌入，其中$(q,p)\in(1,\infty]\times(1,\infty) \cup(1,\infty)\setminus\{3\}\times\{1,\infty\}$和$p\neq q,$，则必有$q=2.$。这主要推广了Lyubich和Vaserstein的一个经典结果。我们还表明，每当$S_q$等距嵌入到$S_p$为$(q,p)\in \left(1,\infty\right)\times\left[2,\infty \right)\cup[4,\infty)\times\{1\} \cup\{\infty\}\times\left( 1,\infty\right)\cup[2,\infty)\times\{\infty\}$与$p\neq q,$，我们必须有$q=2.$因此，我们的工作补充荣格，Parcet，徐和其他人的等距和几乎等距嵌入理论的非交换$L_p$ -空间。我们的方法依赖于几个新的成分有关的扰动理论的线性算子，即加藤-Rellich定理，理论的多个算子积分和Birkhoff-James正交性，其次是彻底和仔细的情况下，由个案分析。对于$m\geq 2$和$1<q<2,$，$S_q^m$是否等距嵌入$S_\infty^n$的问题在\textit{Bull. London Math. Soc.} 52（2020）437-447中尚未解决。
英文摘要：
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.
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