On Surjective Max-Phase Isometries of Schatten p-Classes (0<p<1)
Abstract
We investigate max-phase isometries on Schatten quasi-classes in the range 0<p<1. Motivated by the rigidity theory for smooth normed spaces [\emph{Results Math.} \textbf{80} (2025), Article 126], we establish its operator-theoretic counterpart in the quasi-Banach setting. We prove that every surjective max-phase isometry on a finite-dimensional Schatten class is a standard real-linear Schatten isometry multiplied by an even sign-valued function, without any continuity assumption. This extends the scope of the classical Schatten isometry classification [\emph{Proc. Edinburgh Math. Soc.} \textbf{37} (1994), 239--242] to nonlinear mappings preserving only unsigned two-point metric data. Our approach combines support orthogonality, a symmetrized rank-one spectral formula, and a max-profile uniqueness principle. For arbitrary Hilbert spaces, we obtain the corresponding classification on finite-rank operators, while continuity yields the full-space result. We also establish preservation of rank, singular values, and two-sided support orthogonality.
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BibTeX
@misc{zhang2026surjective,
title = {{On Surjective Max-Phase Isometries of Schatten p-Classes (0<p<1)}},
author = {Wenwen Zhang and Qi Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13271},
eprint = {2610.13271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13271},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Surjective Max-Phase Isometries of Schatten p-Classes (0<p<1)
AU - Wenwen Zhang
AU - Qi Liu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13271
UR - https://arxiv.org/abs/2610.13271
AB - We investigate max-phase isometries on Schatten quasi-classes in the range 0<p<1. Motivated by the rigidity theory for smooth normed spaces [\emph{Results Math.} \textbf{80} (2025), Article 126], we establish its operator-theoretic counterpart in the quasi-Banach setting. We prove that every surjective max-phase isometry on a finite-dimensional Schatten class is a standard real-linear Schatten isometry multiplied by an even sign-valued function, without any continuity assumption. This extends the scope of the classical Schatten isometry classification [\emph{Proc. Edinburgh Math. Soc.} \textbf{37} (1994), 239--242] to nonlinear mappings preserving only unsigned two-point metric data. Our approach combines support orthogonality, a symmetrized rank-one spectral formula, and a max-profile uniqueness principle. For arbitrary Hilbert spaces, we obtain the corresponding classification on finite-rank operators, while continuity yields the full-space result. We also establish preservation of rank, singular values, and two-sided support orthogonality.
ER -