GEOMETRIC AND DIFFERENTIAL INVARIANTS OF NUMERICAL RANGES UNDER UNITARY EQUIVALENCE

We establish a theory of geometric and differential invariants for numerical ranges under unitary equivalence. Our principal result demonstrates that if operators and satisfy for a unitary operator , then not only is , but the entire geometric structure is preserved: the extreme point set , the boundary curvature function for all , and the geometric multiplicity at every boundary point. We prove that the boundary stratification into exposed points, flat arcs, curved arcs, and sharp comers constitutes a complete unitary invariant. For the Lipschitz geometry of numerical ranges, we establish the sharp bound with equality characterization. We prove that for normal operators, the pair completely determines the unitary equivalence class. Furthemore, we establish the curvature-multiplicity relation connecting differential geometry to algebraic properties, and prove that the integrated curvature moments form a complete sequence of unitary invariants.