Toeplitz Operators On The Dirichlet Space

In this thesis we study the hyponormality of Toeplitz operators and the bounded-ness of the product of Toeplitz operators on the Dirichlet space.The contents are as follows:In chapter 1, we review some related research background information and some preliminary knowledge of Toeplitz operators, and we expatiate the significance of the study.In chapter 2, we prove that the necessary and sufficient conditions for a Toeplitz operator Tu on the Dirichlet space to be hyponormal is that the symbol u is a constant function respectively in the case that the projection of u in the Dirichlet space is a polynomial and in the case that u is a class of special symbols. We also prove that a Toeplitz operator with harmonic polynomial symbol on the harmonic Dirichlet space is hyponormal if and only if its symbol is a constant.In chapter 3, we consider the analytic functions f and g in the Dirichlet space, discussing under what condition the densely defined products TfTg can be bounded on the Dirichlet space. Furthermore, we obtain a necessary condition for the products TfTg can be bounded on the Dirichlet space.