发表机构
Clemson University（克莱姆森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文构造紧算子反例，证明高阶 Berezin 变换在算子范数下不收敛于原算子，利用移动矩阵边缘和 Pascal–Hankel 定理揭示其本质障碍。
AI 中文摘要
设 $A^2(\mathbb D)$ 为无加权 Bergman 空间，并记 $Q_m(S)=T_{B_m(S)}$ 为第 $m$ 阶高阶 Berezin 变换诱导的映射。Suárez 在 2005 年提出如下问题：对于全 Bergman Toeplitz 代数中的每个 $S$，$Q_m(S)$ 是否在算子范数下收敛到 $S$。我们以强形式否定回答该问题：存在一个紧算子 $S$ 使得 \\[ \sup_{m\geq 0}\\|Q_m(S)\\|=\infty. \\] 特别地，沿严格递增序列 $(m_n)$ 有 \\[ \\|Q_{m_n}(S)-S\\|\longrightarrow\infty. \\] 障碍是一个移动的矩阵边缘。证明使用一个移动的秩一测试算子族，每个算子支撑在第 $m$ 个矩阵列中。在相应边缘附近，这些测试算子被映射为加权 Hankel 矩阵，极限系数构成一个显式的 Pascal 核。我们证明一个独立的 Pascal–Hankel 定理，表明相关的加权 Hankel 变换不能有界地将 $\ell^2$ 映射到迹类。有限截面收敛、迹对偶性和一致有界原理将这些不稳定性转移到精确的 Bergman 映射上。
英文摘要
Let $A^2(\mathbb D)$ be the unweighted Bergman space and write $Q_m(S)=T_{B_m(S)}$ for the map induced by the $m$th higher-order Berezin transform. Suárez asked in 2005 whether $Q_m(S)$ converges to $S$ in operator norm for every $S$ in the full Bergman Toeplitz algebra. We answer this question negatively in a strong form: there is a compact operator $S$ such that \[ \sup_{m\geq 0}\|Q_m(S)\|=\infty. \] In particular, along a strictly increasing sequence $(m_n)$ one has \[ \|Q_{m_n}(S)-S\|\longrightarrow\infty. \] The obstruction is a moving matrix edge. The proof uses a moving family of rank-one test operators, each supported in the $m$th matrix column. Near the corresponding edge, these test operators are sent to weighted Hankel matrices, and the limiting coefficients form an explicit Pascal kernel. We prove a standalone Pascal--Hankel theorem showing that the associated weighted Hankel transformation fails to map $\ell^2$ boundedly into trace class. Finite-section convergence, trace duality, and the Uniform Boundedness Principle then transfer this instability to the exact Bergman maps.
