Gelfand numbers of Schatten class embeddings in the quasi-Banach range
Abstract
We prove Conjecture 1.4 of Prochno and Strzelecki [J. Approx. Theory 277 (2022), 105736] on Gelfand numbers of embeddings between finite-dimensional Schatten classes. For 0<p≤1 and indices bounded away from the ambient dimension by a fixed proportion, we obtain the conjectured lower bound, uniformly in the target exponent 2≤ q≤∞. Together with the known upper estimate obtained by Hinrichs, Prochno and Vybíral [Math. Ann., 380(3-4):1563-1593, 2021], this gives the exact asymptotic order in the previously unresolved range. As a consequence, we obtain a complete all-index description by combining our result with the estimates of Hinrichs, Prochno and Vybíral and of Prochno and Strzelecki. The proof is deterministic and is based on Hilbert--Schmidt projections on rectangular matrix spaces.
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BibTeX
@misc{prochno2026gelfand,
title = {{Gelfand numbers of Schatten class embeddings in the quasi-Banach range}},
author = {Joscha Prochno},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08448},
eprint = {2611.08448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08448},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gelfand numbers of Schatten class embeddings in the quasi-Banach range
AU - Joscha Prochno
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08448
UR - https://arxiv.org/abs/2611.08448
AB - We prove Conjecture 1.4 of Prochno and Strzelecki [J. Approx. Theory 277 (2022), 105736] on Gelfand numbers of embeddings between finite-dimensional Schatten classes. For 0<p≤1 and indices bounded away from the ambient dimension by a fixed proportion, we obtain the conjectured lower bound, uniformly in the target exponent 2≤ q≤∞. Together with the known upper estimate obtained by Hinrichs, Prochno and Vybíral [Math. Ann., 380(3-4):1563-1593, 2021], this gives the exact asymptotic order in the previously unresolved range. As a consequence, we obtain a complete all-index description by combining our result with the estimates of Hinrichs, Prochno and Vybíral and of Prochno and Strzelecki. The proof is deterministic and is based on Hilbert--Schmidt projections on rectangular matrix spaces.
ER -