GEOMETRIC AND DIFFERENTIAL INVARIANTS OF NUMERICAL RANGES UNDER UNITARY EQUIVALENCE

Abstract

We establish a theory of geometric and differential invariants for numerical ranges under unitary equivalence. Our principal result demonstrates that if operators $T$ and $S$ satisfy $S=U^* T U$ for a unitary operator $U$, then not only is $W(S)=W(T)$, but the entire geometric structure is preserved: the extreme point set $\mathrm{E}(S)=\mathrm{E}(T)$, the boundary curvature function $\kappa_S(\lambda)=\kappa_T(\lambda)$ for all $\lambda \in \partial W(T)$, and the geometric multiplicity $m_g(\lambda, S)=m_g(\lambda, T)$ at every boundary point. We prove that the boundary stratification $\partial W(T)=\mathrm{E}(T) \bigcup \Phi(T) \bigcup \mathrm{X}(T) \bigcup \Sigma(T)$ 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 $|\langle T x, x\rangle-\langle T y, y\rangle| \leq 2\|T\| \cdot\|x-y\|$ with equality characterization. We prove that for normal operators, the pair $\left(W(T),\left\{m_g(\lambda, T)\right\}_{\lambda, \in \sigma(T)}\right)$ completely determines the unitary equivalence class. Furthemore, we establish the curvature-multiplicity relation $\kappa_T(\lambda)=2 \pi /\left(\mu_K(\lambda) \cdot\left|\gamma_K^{\prime}(\theta)\right|^2\right)$ connecting differential geometry to algebraic properties, and prove that the integrated curvature moments $M_k(T)=\int_{\partial W(T)}{ }^{\kappa_T}(\lambda)^k d s$ form a complete sequence of unitary invariants.

Received: June 10, 2025
Revised: July 2, 2025
Accepted: July 31, 2025