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arXiv · 2609.37504

Further Results on the Isometric Embeddability of \(S_q^m\) into \(S_p^n\)

Ying Xu·Wenwen Zhang·Qi Liu
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Abstract

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.

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Ying Xu, Wenwen Zhang, Qi Liu. 2026-09-26. Further Results on the Isometric Embeddability of \(S_q^m\) into \(S_p^n\). https://arxiv.org/abs/2609.37504

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