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PreprintarXiv licencearXiv

Further Results on the Isometric Embeddability of Sqm into Spn

  • 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 ℓq2(C)↪Sppn if and only if q=p or q=2, for n≥2 and p<∞. 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 ℓq2 from infinite-dimensional Spp for 0<p≤2<q≤∞ 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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Cite

BibTeXRIS
BibTeX
@misc{xu2026further,
  title = {{Further Results on the Isometric Embeddability of \(S_q^m\) into \(S_p^n\)}},
  author = {Ying Xu and Wenwen Zhang and Qi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37504},
  eprint = {2609.37504},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37504},
}
RIS
TY  - UNPB
TI  - Further Results on the Isometric Embeddability of S_q^m into S_p^n
AU  - Ying Xu
AU  - Wenwen Zhang
AU  - Qi Liu
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.37504
UR  - https://arxiv.org/abs/2609.37504
AB  - 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 ℓ_q^2(C)↪Sp_p^n if and only if q=p or q=2, for n≥2 and p<∞. 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 ℓ_q^2 from infinite-dimensional Sp_p for 0<p≤2<q≤∞ 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.
ER  -