单位多圆柱上Bergman空间中的分别准齐次Toeplitz算子

Separately quasihomogeneous Toeplitz operators on the Bergman space of the polydisk

本文研究了单位多圆柱上Bergman空间中以分别准齐次函数为记号的Toeplitz算子的代数性质.我们首先得到了两个以分别准齐次函数为记号的Toeplitz算子可以写成一个Toeplitz算子的充分必要条件,然后利用L2(Dn,dV)的一个极分解式证明了,只要其中有一个Toeplitz算子是分别准齐次的,则其零乘积问题只有平凡解.最后我们研究了分别径向Toeplitz算子的交换性问题.

The paper is devoted to the study of Toeplitz operators with separately quasihomogeneous symbols on the Bergman space of the polydisk. First, we obtain necessary and sufficient conditions for the product of two Toeplitz operators with separately quasihomogeneous symbols to be a Toeplitz operator. Next, we provide a decomposition of L2（Dn, dV）. Then we use this to show that the zero product of two Toeplitz operators has only a trivial solution if one of the symbols is separately quasihomogeneous and the other is arbitrary. Also, the corresponding commuting problem of Toeplitz operators is studied.