正算子的有限生成动力框架的幂稳定性
Power Stability of Finitely Generated Dynamical Frames for Positive Operators
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中文总结 AI 辅助
该研究通过分析Carleson序列有限并的幂集的Hartmann型簇分解与迹空间恒等式，证明正算子的有限生成动力框架具有幂稳定性，即框架经算子正幂次采样后仍为框架，且正性条件不可替换为正规性。
中文摘要 AI 辅助
设\\( Λ=\{λ_j\}_{j\in\mathbb N}\subset[0,1) \\)是Carleson序列的有限并，且对\\(p>0\\)，记\\( Λ^p=\{λ_j^p\}_{j\in\mathbb N} \\)。我们证明\\(Λ\\)和\\(Λ^p\\)容许相容的Hartmann型簇分解，且在自然坐标等同下，\\( H^2(Λ)=H^2(Λ^p) \\)。考虑归一化权重的变化后，该迹空间恒等式给出相关加权赋值算子满足\\( \operatorname{Ran}(T_Λ) = \operatorname{Ran}(T_{Λ^p}) \\)。随后，正规算子的有限生成动力框架的刻画表明，对每个正算子\\(A\in\mathcal B(H)\\)、每个有限指标集\\(I\\)以及每个\\(p>0\\)，\\( \{A^nf_i\}_{n\in\mathbb N,\\,i\in I} \text{ 是 }H\text{ 的框架}\\)当且仅当\\( \{A^{pn}f_i\}_{n\in\mathbb N,\\,i\in I} \text{ 是 }H\text{ 的框架}\\)。特别地，对每个\\(q\in\mathbb N^+\\)，子族\\( \{A^{qn}f_i\}_{n\in\mathbb N,\\,i\in I} \\)仍是框架，因此每个这类动力框架都具有无限冗余度。一个例子表明正性假设一般不能替换为正规性。
英文摘要
Let \( Λ=\{λ_j\}_{j\in\mathbb N}\subset[0,1) \) be a finite union of Carleson sequences, and let \( Λ^p=\{λ_j^p\}_{j\in\mathbb N} \) for \(p>0\). We prove that \(Λ\) and \(Λ^p\) admit compatible Hartmann-type cluster decompositions and that, under the natural coordinate identification, \( H^2(Λ)=H^2(Λ^p). \) After taking account of the change in the normalization weights, this trace-space identity gives \( \operatorname{Ran}(T_Λ) = \operatorname{Ran}(T_{Λ^p}) \) for the associated weighted evaluation operators. The characterization of finitely generated dynamical frames for normal operators then implies that, for every positive operator \(A\in\mathcal B(H)\), every finite index set \(I\), and every \(p>0\), \( \{A^nf_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H\) if and only if \( \{A^{pn}f_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H. \) In particular, for every \(q\in\mathbb N^+\), the subfamily \( \{A^{qn}f_i\}_{n\in\mathbb N,\,i\in I} \) is still a frame, and hence every such dynamical frame has infinite excess. An example shows that the positivity assumption cannot in general be replaced by normality.
