多圆盘上的Toeplitz算子的紧的乘积

Compact Product of Toeplitz Operators on the Polydisk

对于多圆盘上的有界多重调和函数f,g,我们证明多圆盘Bergman空间上的Toeplitz算子的乘积TfTg是紧算子的充要条件是TfTg是零.这等价于f或g是零.换位子TfTg-TgTf是紧算子当且仅当换位子TfTg-TgTf是零.这等价于对每一个j,存在不全为零的常数αj和βj,使得αjf+βjg关于变量zj(1≤j≤n)是常数.

For bounded pluriharmonic functions f, g on the polydisk, we show that two Toeplitz operators product TfTg on the Bergman space of the polydisk is compact if and only if TfTg = 0, or if and only if f or g is zero; TfTg - TgTf is compact if and only if TfTg - TgTf = 0, or if and only if for every j, there are constants αj and βj, not both zero, such that αjf + βjg is constant in variable zj (1≤j ≤ n).