发表机构
School of Mathematics and Statistics, Anqing Normal University（安庆师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过迹幂圆周均值与复曲率框架，分类了 \\(\ell_q^2\\) 到 \\(\Sp_p^n\\) 的等距嵌入条件，否定了相关问题的部分猜想，并给出了有限秩畸变界。
AI 中文摘要
我们通过迹幂的圆周均值研究 Schatten 嵌入。在有限维情形中，通过奇异值消失阶数的分解表明，阶数至多为二的每一项都具有非负系数。这给出了分类：\\(\ell_q^2(\C)\hookrightarrow\Sp_p^n\\) 当且仅当 \\(q=p\\) 或 \\(q=2\\)，其中 \\(n\ge2\\) 且 \\(p<\infty\\)。特别地，这为 Chattopadhyay--Pradhan--Skripka（arXiv:2603.07359v2）中问题 3.1 的第 (ii)、(iii) 和 (iv) 部分给出了否定答案。一个正则化曲率估计还排除了 \\(\ell_q^2\\) 嵌入无限维 \\(\Sp_p\\) 的情形，其中 \\(0<p\le2<q\le\infty\\)，并给出了有限秩畸变界。该问题的其他部分未被声称解决。这些结果通过基于复凸性理论（\cite{BR,DGT}）发展复曲率框架，扩展了先前关于 Schatten 嵌入的研究（特别是 \cite{CHPPR, CHPR, CPS}），解决了若干剩余的拟巴拿赫情形，并为等距嵌入产生了新的几何障碍。
英文摘要
We study Schatten embeddings through circular means of trace powers. In finite dimensions, a decomposition by singular-value vanishing orders shows that every term of order at most two has a nonnegative coefficient. This yields the classification $\ell_q^2(\C)\hookrightarrow\Sp_p^n$ if and only if $q=p$ or $q=2$, for $n\ge2$ and $p<\infty$. In particular, it gives negative answers to parts (ii), (iii), and (iv) of Problem 3.1 in Chattopadhyay--Pradhan--Skripka, arXiv:2603.07359v2. A regularized curvature estimate also excludes $\ell_q^2$ from infinite-dimensional $\Sp_p$ for $0<p\le2<q\le\infty$ and gives finite-rank distortion bounds. The other parts of that problem are not claimed to be solved.Our results extend previous studies on Schatten embeddings, particularly those in \cite{CHPPR, CHPR, CPS}, by developing a complex-curvature framework based on complex convexity theories \cite{BR,DGT}, which resolves several remaining quasi-Banach cases and yields new geometric obstructions for isometric embeddings.
