Carleson Interpolation, Boundary Rigidity, and Classification of Spectral Transformations Preserving Diagonal Operator Orbit Frames

Abstract

Maps $\Phi :D\to D$ of the unit disk preserve frame generator sets for single orbits of normal diagonal operators exactly when they preserve Carleson interpolating sequences bidirectionally and satisfy $1-|\Phi (z){|}^2\asymp 1-|z{|}^2$ uniformly. Denote this class by $\mathcal{P}$. Equality of frame generator sets forces the boundary estimate, since every frame generator $f=(f_j)$ satisfies $|f_j{|}^2\asymp 1-|{\lambda }_j{|}^2$. No continuity, measurability, or analyticity is assumed; Carleson preservation alone remains flexible and, to our knowledge, unclassified in general. Equivalently, normalized Szego kernel Riesz sequences are preserved bidirectionally, with uniformly comparable unnormalized kernel norms at $z$ and $\Phi (z)$. Equivalently, we require uniform two-sided Carleson norm bounds for ordinary pushforwards of finite positive discrete measures and bidirectional preservation of pairwise pseudohyperbolic separation. Every $\Phi \in \mathcal{P}$ has a canonical bilipschitz radial trace $h_{\Phi }(\zeta )={\lim }_{r\to 1^{-}}\Phi (r\zeta )$ on $T=\partial D$, with uniform convergence. We give an intrinsic geometric characterization of $\mathcal{K}=\{\Psi \in \mathcal{P}:h_{\Psi }=\mathrm{id}\}$ and prove the unique factorization $\Phi =\Psi \circ E_{h_{\Phi }}$, $\Psi \in \mathcal{K}$, where $E_h(r\zeta )=rh(\zeta )$ and $E_h(0)=0$; thus $\mathcal{P}\cong \mathcal{K}⋊\mathrm{BiLip}(T)$ as monoids. For $m$ orbits, ${\mathcal{P}}_m=\mathcal{P}$ for every finite $m\ge 1$, whereas $\mathrm{Aut}(D)⊊{\mathcal{P}}_{\omega }⊊\mathcal{P}$. Countable-orbit preservers in ${\mathcal{P}}_{\omega }$ are uniform pseudohyperbolic homeomorphisms. If $\phi \in \mathrm{Aut}(D)$ and $|\Phi (z)-\phi (z)|=o(1-|z|)$ uniformly as $|z|\to 1$, then $\Phi \in {\mathcal{P}}_{\omega }$ exactly when $\Phi$ is a disk homeomorphism. Two-sided tests characterize ${\mathcal{P}}_{\omega }$. Holomorphic members of these classes are exactly disk automorphisms.