AI 中文总结
该研究针对加权Bergman空间，构造了具有$L^1$符号的Toeplitz算子，其Berezin变换在边界消失但算子非紧，回答了Bauer与Isralowitz的问题，还在无权重圆盘上验证了相关性质并确定Zorboska假设的最优性。
AI 中文摘要
对于每个$n\geq1$和$\u03b3>-1$，我们构造了属于$L^1(\u03b2_n,dv_\u03b3)$的函数$f$，其Toeplitz形式延拓为加权Bergman空间$A^2_\u03b3(\u03b2_n)$上的有界非紧算子，且其Berezin变换在边界处消失。因此，对于具有可积符号的Toeplitz算子，Berezin变换的边界消失性并不意味着紧性，这与由有界符号生成的Toeplitz代数中的算子形成对比；该结果回答了Bauer和Isralowitz提出的问题。构造过程通过具有光滑紧支符号的Toeplitz算子在范数下逼近秩一算子，并将适当分离的块向边界迁移。在无权重圆盘上，一种对角版本产生了具有相同消失性和非紧性性质的实符号，其中算子是正的，且在$p=3$时满足Zorboska的双边局部化条件，其假设$p>3$被证明是最优的。
英文摘要
For every $n\geq1$ and $γ>-1$, we construct $f\in L^1(\mathbb B_n,dv_γ)$ whose Toeplitz form extends to a bounded, noncompact operator on the weighted Bergman spaces $A^2_γ(\mathbb B_n)$ and whose Berezin transform vanishes at the boundary. Boundary vanishing of the Berezin transform therefore does not imply compactness for Toeplitz operators with integrable symbols, in contrast with operators in the Toeplitz algebra generated by bounded symbols; this answers a question of Bauer and Isralowitz. The construction approximates rank-one operators in norm by Toeplitz operators with smooth, compactly supported symbols and transports suitably separated blocks toward the boundary. On the unweighted disk, a diagonal version produces a real symbol with the same vanishing and noncompactness properties, for which the operator is positive and Zorboska's two-sided localization condition holds at $p=3$. Her hypothesis $p>3$ is shown to be sharp.
Comments27 pages. Answers a question of Bauer and Isralowitz (2012); shows that the exponent $p>3$ in Zorboska's compactness criterion (2003) is sharp. Comments welcome
