发表机构
College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对径向$\widehat{\mathcal{D}}$-加权Bergman空间，证明了Toeplitz算子紧性等价于其Berezin变换在边界处趋于零，同时构造反例说明该刻画不适用于Toeplitz算子乘积及对应代数。
AI 中文摘要
设$\omega$为径向$\widehat{\mathcal{D}}$-权，$u$为单位圆盘$\mathbb{D}$上的有界函数。我们证明Toeplitz算子$T_{\omega,u}$在$A_\omega^2$上是紧的当且仅当其Berezin变换在边界处趋于零。我们的方法基于$A_\omega^2$的多项式框架，以及对所得$T_{\omega,u}$的无限矩阵表示的详细局部化分析。即使在无加权Bergman空间$A^2$中，我们的论证也是全新的，且不依赖经典的平移算子。我们进一步表明，这种Axler--Zheng紧性刻画通常不能推广到$\widehat{\mathcal{D}}$-加权Bergman空间上的Toeplitz算子乘积，因此也不能推广到由有界符号生成的相应Toeplitz代数。更确切地说，我们构造了一个径向对数次调和$\widehat{\mathcal{D}}$-权$\omega$以及有界符号$u,v$，使得乘积$T_{\omega,v}T_{\omega,u}$是非紧的，而其Berezin变换在边界处趋于零。
英文摘要
Let $ω$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{ω,u}\) is compact on \(A_ω^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_ω^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{ω,u}$. Even in the unweighted Bergman space \(A^2\), our argument is new and does not rely on the classical translation operators. We further show that this Axler--Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on \(\widehat{\mathcal D}\)-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic \(\widehat{\mathcal D}\)-weight \(ω\) and bounded symbols \(u,v\) such that the product \(T_{ω,v}T_{ω,u}\) is noncompact, whereas its Berezin transform vanishes at the boundary.
Comments45 pages. We have added a new example concerning a product of Toeplitz operators. Accordingly, we have revised the title and abstract
