Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion
Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion
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DOI：
10.1016/j.aim.2021.107890
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发表时间：
2021-09
影响因子：
1.7
通讯作者：
Yi Wang;Jingbo Xia
中科院分区：
文献类型：
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作者：
Yi Wang;Jingbo Xia
考虑Cn中闭单位球的开邻域的一类解析子集M ∈ N.这样一个M在单位球面S <$Cn上产生哈代模H2（S）的一个子模R和一个商模Q。我们证明了，正如几何Arveson-Douglas猜想所预言的，商模Q是p-本质正规的，其中p> d= dim C M.我们进一步证明了，更有趣的是，商模Q表现出一种仅在Bergman空间和Fock空间上发现的行为：Q上Toeplitz代数中的算子A是紧的当且仅当它的Berezin变换在M S附近为零。
We consider a class of analytic subsets M˜ of an open neighborhood of the closed unit ball in C n. Such an M˜ gives rise to a submodule R and a quotient module Q of the Hardy module H 2 (S) on the unit sphere S⊂ C n. We show that, as predicted by the geometric Arveson-Douglas conjecture, the quotient module Q is p-essentially normal for p> d= dim C M˜. We further show that, more interestingly, the quotient module Q exhibits a behavior that is only found on the Bergman space and the Fock space: an operator A in the Toeplitz algebra on Q is compact if and only if its Berezin transform vanishes near M˜∩ S.