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.