Carleson Interpolation, Boundary Rigidity, and Classification of Spectral Transformations Preserving Diagonal Operator Orbit Frames
Abstract
Maps Φ:D→ 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)|2asymp1−|z|2 uniformly. Denote this class by 𝒫. Equality of frame generator sets forces the boundary estimate, since every frame generator f=(fj) satisfies |fj|2asymp 1−|λj|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∈𝒫 has a canonical bilipschitz radial trace hΦ(ζ)=limr→1-Φ(rζ) on T=∂ D, with uniform convergence. We give an intrinsic geometric characterization of 𝒦={Ψ∈𝒫:hΨ=id} and prove the unique factorization Φ=Ψ∘ EhΦ, Ψ∈𝒦, where Eh(rζ)=rh(ζ) and Eh(0)=0; thus 𝒫≅𝒦rtimesBiLip(T) as monoids. For m orbits, 𝒫m=𝒫 for every finite m≥1, whereas Aut(D)subsetneq𝒫ωsubsetneq𝒫. Countable-orbit preservers in 𝒫ω are uniform pseudohyperbolic homeomorphisms. If φ∈Aut(D) and |Φ(z)−φ(z)|=o(1−|z|) uniformly as |z|→1, then Φ∈𝒫ω exactly when Φ is a disk homeomorphism. Two-sided tests characterize 𝒫ω. Holomorphic members of these classes are exactly disk automorphisms.
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BibTeX
@misc{wu2026carleson,
title = {{Carleson Interpolation, Boundary Rigidity, and Classification of Spectral Transformations Preserving Diagonal Operator Orbit Frames}},
author = {Jian Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.07259},
eprint = {2608.07259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07259},
}
RIS
TY - UNPB
TI - Carleson Interpolation, Boundary Rigidity, and Classification of Spectral Transformations Preserving Diagonal Operator Orbit Frames
AU - Jian Wu
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.07259
UR - https://arxiv.org/abs/2608.07259
AB - Maps Φ:D→ 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)|^2asymp1−|z|^2 uniformly. Denote this class by 𝒫. Equality of frame generator sets forces the boundary estimate, since every frame generator f=(f_j) satisfies |f_j|^2asymp 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∈𝒫 has a canonical bilipschitz radial trace h_Φ(ζ)=lim_r→1^-Φ(rζ) on T=∂ D, with uniform convergence. We give an intrinsic geometric characterization of 𝒦={Ψ∈𝒫:h_Ψ=id} and prove the unique factorization Φ=Ψ∘ E_h_Φ, Ψ∈𝒦, where E_h(rζ)=rh(ζ) and E_h(0)=0; thus 𝒫≅𝒦rtimesBiLip(T) as monoids. For m orbits, 𝒫_m=𝒫 for every finite m≥1, whereas Aut(D)subsetneq𝒫_ωsubsetneq𝒫. Countable-orbit preservers in 𝒫_ω are uniform pseudohyperbolic homeomorphisms. If φ∈Aut(D) and |Φ(z)−φ(z)|=o(1−|z|) uniformly as |z|→1, then Φ∈𝒫_ω exactly when Φ is a disk homeomorphism. Two-sided tests characterize 𝒫_ω. Holomorphic members of these classes are exactly disk automorphisms.
ER -