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

Abstract

Maps $Φ:D\to D$ 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 $1-|Φ(z)|^2\asymp1-|z|^2$ uniformly. Denote this class by $\mathcal P$. Equality of frame generator sets forces the boundary estimate, since every frame generator $f=(f_j)$ satisfies $|f_j|^2\asymp 1-|λ_j|^2$. 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 $Φ(z)$. 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 $Φ\in\mathcal P$ has a canonical bilipschitz radial trace $h_Φ(ζ)=\lim_{r\to1^-}Φ(rζ)$ on $T=\partial D$, with uniform convergence. We give an intrinsic geometric characterization of $\mathcal K=\{Ψ\in\mathcal P:h_Ψ=\mathrm{id}\}$ and prove the unique factorization $Φ=Ψ\circ E_{h_Φ}$, $Ψ\in\mathcal K$, where $E_h(rζ)=rh(ζ)$ and $E_h(0)=0$; thus $\mathcal P\cong\mathcal K\rtimes\mathrm{BiLip}(T)$ as monoids. For $m$ orbits, $\mathcal P_m=\mathcal P$ for every finite $m\ge1$, whereas $\mathrm{Aut}(D)\subsetneq\mathcal P_ω\subsetneq\mathcal P$. Countable-orbit preservers in $\mathcal P_ω$ are uniform pseudohyperbolic homeomorphisms. If $φ\in\mathrm{Aut}(D)$ and $|Φ(z)-φ(z)|=o(1-|z|)$ uniformly as $|z|\to1$, then $Φ\in\mathcal P_ω$ exactly when $Φ$ is a disk homeomorphism. Two-sided tests characterize $\mathcal P_ω$. Holomorphic members of these classes are exactly disk automorphisms.