Compact Toeplitz operators via the Berezin transform on bounded symmetric domains
Compact Toeplitz operators via the Berezin transform on bounded symmetric domains
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DOI：
10.1007/bf01291836
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发表时间：
1999-12
影响因子：
0.8
通讯作者：
M. Engliš
中科院分区：
文献类型：
--
作者：
M. Engliš
设Ω是亏格p的不可约有界对称整环，h（x，y）是它的Jordan三重行列式，A ν2（Ω）是Ω上全纯函数关于测度h（z，z）ν−pdz平方可积的标准加权Bergman空间.推广了Axler和Zheng最近的结果，对于Ω=D，ν=p=2（单位圆盘上的未加权Bergman空间），我们证明了：如果S是A ν2（Ω）和ν上Toeplitz算子的有限乘积的有限和，则S是紧的当且仅当S的Berezin变换在z接近Ω时趋于零。对于Fock空间也得到了类似的断言。
Let Ω be an irreducible bounded symmetric domain of genusp, h(x, y)its Jordan triple determinant, andAν2(Ω) the standard weighted Bergman space of holomorphic functions on Ω square-integrable with respect to the measureh(z, z)ν−pdz. Extending the recent result of Axler and Zheng for Ω=D, ν=p=2 (the unweighted Bergman space on the unit disc), we show that ifSis a finite sum of finite products of Toeplitz operators onAν2(Ω) and ν is sufficiently large, thenSis compact if and only if the Berezin transformofStends to zero aszapproaches ∂Ω. An analogous assertion for the Fock space is also obtained.