Toeplitz Operators On The Dirichlet Space

In this thesie we mainly study the Toeplitz operators on the Dirichlet space of disk and on the Dirichlet space of ball,focus on normality,self adjointness,hyponormality and commutativity of Toeplitz operators.In chapter1,we give some related research background of the development of Dirichlet space and Toeplitz operators,and some notations and definitions that are necessary in the article.In chapter2,we consider the hyponormality of Toeplitz operators on the Dirich-let space of the unit disk,introduce a class of continuous functions Lθ∞,1(D)={f∈L∞,1(D):for almost every r∈[0,1),f(reiθ) is absolutely continuous on θ∈[0,2π]},give necessary and sufficient conditions for the hyponormality of Tocplitz operators with this class of continuous symbols on Dirichlct space.In chapter3,we consider algebraic properties of Toeplitz operators with conju-gate holomorphic or holomorphic symbols on the Dirichlet space of the ball,give necessary and sufficient conditions for the normality of Toeplitz operators with holomorphic or ant-holomorphic symbols,give necessary and sufficient conditions for the isometry and self adjointness of Toeplitz operators with harmonic symbols,also give some result for commutativity of two Toeplitz operators with holomor-phic and ant-holomorphic symbols.