保持正规对角算子轨道框架生成元的谱变换的边界刚性与分类
Carleson Interpolation, Boundary Rigidity, and Classification of Spectral Transformations Preserving Diagonal Operator Orbit Frames
AI总结：
该研究刻画了保持单位圆盘Carleson序列且满足伪双曲距离等价条件的映射类，证明其边界迹为双Lipschitz映射，给出核-角分解与半直积结构，明确多轨道保持类与单轨道一致，全纯成员为圆盘自同构，并提出可数情形的相关猜想。
AI中文摘要：
设$\u27c8$为单位圆盘$\u2139$中的Carleson序列类。我们研究满足以下条件的任意映射$Φ:\u2139→\u2139$：对每个序列$Λ=\u27e8λ_n\u27e9_{n≥1}⊂\u2139$，既有$Λ∈\u27c8⟺Φ(Λ)=\u27e8Φ(λ_n)\u27e9_{n≥1}∈\u27c8$，又对$z∈\u2139$有$1-|Φ(z)|^2≍1-|z|^2$。我们不假设映射具有连续性、可测性或解析性。这些条件恰好源于正规对角算子单轨道的框架生成元集的普适保持性。\n我们证明，每个此类映射都存在典范径向边界迹$h_Φ(ζ)=\u27e8_{r→1^-}Φ(rζ)$（$ζ∈\u2164$）且收敛是一致的，同时$h_Φ∈\u212biLip(\u2164)$。对于得到的保持类$\u213b=\u213b_1$，设$\u212a=\u27e8Ψ∈\u213b:h_Ψ=id_{\u2164}\u27e9$为其边界影子核。每个$Φ∈\u213b$都有唯一的核-角分解$Φ=Ψ∘E_{h_Φ}$，其中$Ψ∈\u212a$，$E_h(0)=0$，且$E_h(rζ)=rh(ζ)$。因此$\u213b≅\u212a⋊\u212biLip(\u2164)$，构成分裂半直积。\n对每个固定的$m∈\u2115^+$，由$m$条算子轨道生成的框架的普适保持类与单轨道类相等：$\u213b_m=\u213b$。其全纯成员恰好是$\u2139$的自同构。对于可数类，有$\u212but(\u2139)⊆\u213b_ω⊆\u213b$，且$\u213b_ω$的每个元素都是$\u2139$的伪双曲一致同胚。这引出了$\u213b_ω=\u212but(\u2139)$的猜想。
英文摘要：
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.
