Classify discontinuous infinite-dimensional surjective max-phase isometries

Extend the classification of surjective max-phase isometries to arbitrary infinite-dimensional Schatten quasi-classes mathcal S_p(H)math for 0<p<1math without assuming continuity, in particular by extending the max-profile uniqueness principle from finite-dimensional or finite-rank operators to arbitrary Schatten operators.

Background

The paper completely classifies surjective max-phase isometries on finite-dimensional Schatten classes and determines their action on the finite-rank ideal of an arbitrary complex Hilbert space. In the infinite-dimensional setting, the authors also prove a full classification under continuity: every continuous surjective max-phase isometry is a constant global sign times a standard real-linear Schatten isometry.

Without continuity, the finite-rank result only determines the map on a dense ideal. Extending this conclusion to infinite-rank compact operators is technically difficult because the finite-dimensional proof differentiates the Schatten p-power in a finite support corner, whereas for an infinite-rank operator the operator math |A|^{p-1} math is unbounded. The unresolved task is therefore to establish a suitable max-profile uniqueness theorem for arbitrary Schatten operators, which would enable classification of all discontinuous surjective max-phase isometries.