AI 中文总结
本文用符号的局部平均值刻画 Bergman 空间上 Toeplitz 算子的紧性，证明紧性等价于 Bergman 圆盘平均值趋于零，并给出 Carleson 盒子平均的等价判据及反例。
AI 中文摘要
设 \varphi\in L^{\infty}(\D)。我们研究 Bergman 空间 A^{2}(\D) 上 Toeplitz 算子 T_\varphi 的紧性判据。Axler 和 Zheng~\cite{AZ1998} 建立了用 Berezin 变换表示的紧性的充分必要条件。然而，对于一般的有界可测函数 \varphi，其 Berezin 变换 \tilde{\varphi} 并不能直接揭示 \varphi 的内在性质。受用符号本身进行刻画的启发，Zhu~\cite{ZhuSlides} 提出了关于紧 Toeplitz 算子的一个猜想。在本文中，我们用符号的局部平均值来刻画无加权 Bergman 空间上 T_\varphi 的紧性。我们证明紧性等价于在任意给定半径的 Bergman 圆盘上的平均值趋于零。我们还建立了用 Carleson 盒子平均值在角变量上一致趋于零来刻画的等价判据。最后，我们构造了一个非负有界符号，其 Carleson 盒子平均值在每个固定角度都趋于零，但相应的 Toeplitz 算子并不紧。
英文摘要
Let $φ\in L^\infty(\D)$. We study compactness criteria for \(T_φ\) on the Bergman space $A^2(\D)$. Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function $φ$, however, its Berezin transform $\tildeφ$ does not readily reveal the intrinsic properties of $φ$. Motivated by a characterization in terms of the symbol itself, Zhu~\cite{ZhuSlides} proposed a conjecture on compact Toeplitz operators. In this paper, we characterize compactness of $T_φ$ on the unweighted Bergman space in terms of local averages of the symbol. We prove that compactness is equivalent to the vanishing of averages over Bergman disks of any prescribed fixed radius. We also establish an equivalent criterion in terms of Carleson box averages that tend to zero uniformly in the angular variable. Finally, we construct a nonnegative bounded symbol whose Carleson box averages tend to zero at every fixed angle, although the associated Toeplitz operator is not compact.
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