On The Zero Product Problem For Radial Toeplitz Operators And Mellin Transform On The Weighted Bergman Space

The Mellin transform of the operator plays an important role in studying the commutant of the operator. Applying the Mellin transform, people can give the concrete form of some functions such that the product of the operators which induced by these functions is commntant.In this paper, we introduce the definition of convolution of f and g on the weighted Bergman space. Then using the Mellin transform, we obtain the properties of the Mellin trans-form and the convolution of f and g. Let f,g,h? L(0,1), we give a sufficient and necessary condition such that TfTg=Th. At last, we obtain the zero product problem for radial Toeplitz operators and a sufficient and necessary condition such that the product of a class of Toeplitz operators induced by polynomical symbols is commutant on the weighted Bergman space.The content of this article is arranged as follows:In the first part, we mainly introduce several concepts such as the weighted Bergman space and the weighted Mellin transform, then we introduce the definition of the convolution of f and g.In the second part, using the properties of Mellin transform and the definition of the con-volution of f and g, we give a sufficient and necessary condition such that TfTg=Th.In the third part, we introduce the radial operator and the radialization of the operator. We give an example. That is, if f? I1(0,1), then Rad(Tf)=Tf. On the weighted Bergman space, we prove that the zero product problem for radial Toeplitz operators.In the fourth part, using the weighted Mellin transform, we extend Cuckovic and Rao’s re-sult to the weighted Bergman space, we construct a new period function and find the sufficient and necessary condition such that their product is commutant.