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 $\Lambda =\{{\lambda }_n{\}}_{n\ge 1}\subset \mathbb{D}$, both $\Lambda \in \mathcal{C}⟺\Phi (\Lambda )=\{\Phi ({\lambda }_n){\}}_{n\ge 1}\in \mathcal{C}$ and $1-|\Phi (z){|}^2\asymp 1-|z{|}^2$ 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 $h_{\Phi }(\zeta )={\lim }_{r\to 1^{-}}\Phi (r\zeta )$, $\zeta \in \mathbb{T}$, with uniform convergence, and that $h_{\Phi }\in \mathrm{BiLip}(\mathbb{T})$. For the resulting preserver class $\mathcal{P}={\mathcal{P}}_1$, let $\mathcal{K}=\{\Psi \in \mathcal{P}:h_{\Psi }={\mathrm{id}}_{\mathbb{T}}\}$ 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 $\mathcal{P}\cong \mathcal{K}⋊\mathrm{BiLip}(\mathbb{T})$ 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, $\mathrm{Aut}(\mathbb{D})\subseteq {\mathcal{P}}_{\omega }\subseteq \mathcal{P}$, and every element of ${\mathcal{P}}_{\omega }$ is a pseudohyperbolically uniform homeomorphism of $\mathbb{D}$. This motivates the conjecture ${\mathcal{P}}_{\omega }=\mathrm{Aut}(\mathbb{D})$.