Boundary Rigidity and Classification of Spectral Transformations Preserving Frame Generators of Normal Diagonal Operator Orbits

Abstract

Let $\mathcal C$ be the class of Carleson sequences in the unit disk $\mathbb D$. We study arbitrary maps $\Phi:\mathbb D\to\mathbb D$ satisfying, for every sequence $\Lambda=\{\lambda_n\}_{n\ge1}\subset\mathbb D$, both $\Lambda\in\mathcal C\Longleftrightarrow\Phi(\Lambda)=\{\Phi(\lambda_n)\}_{n\ge1}\in\mathcal C$ and $1-|\Phi(z)|^2\asymp1-|z|^2$ for $z\in\mathbb D$. No continuity, measurability, or analyticity is assumed. These conditions arise exactly from universal preservation of frame-generator sets for single orbits of normal diagonal operators. We prove that every such map has a canonical radial boundary trace $h_\Phi(\zeta)=\lim_{r\to1^-}\Phi(r\zeta)$, $\zeta\in\mathbb T$, with uniform convergence, and that $h_\Phi\in\operatorname{BiLip}(\mathbb T)$. For the resulting preserver class $\mathcal P=\mathcal P_1$, let $\mathcal K=\{\Psi\in\mathcal P:h_\Psi=\operatorname{id}_{\mathbb T}\}$ be its boundary-shadow kernel. Every $\Phi\in\mathcal P$ has the unique kernel-angular factorization $\Phi=\Psi\circ E_{h_\Phi}$, where $\Psi\in\mathcal K$, $E_h(0)=0$, and $E_h(r\zeta)=rh(\zeta)$. Hence $\mathcal P\cong\mathcal K\rtimes\operatorname{BiLip}(\mathbb T)$ as a split semidirect product. For every fixed $m\in\mathbb N^+$, the universal preservation class for frames generated by $m$ operator orbits equals the single-orbit class: $\mathcal P_m=\mathcal P$. Its holomorphic members are precisely the automorphisms of $\mathbb D$. For the countable class, $\operatorname{Aut}(\mathbb D)\subseteq\mathcal P_\omega\subseteq\mathcal P$, and every element of $\mathcal P_\omega$ is a pseudohyperbolically uniform homeomorphism of $\mathbb D$. This motivates the conjecture $\mathcal P_\omega=\operatorname{Aut}(\mathbb D)$.