Two Types Of Operators On The Weighted Dirichlet Space

In this dissertation,we mainly study Toeplitz operators and k-order slant Toeplitz op-erators on the Dirichlet space in the unit disc. We mainly discuss the (semi-)commutativity, finite rank and compactness of Toeplitz operators;By the methods of function theory and operator theory, we study the commutativity and spectrum of k-order slant Toeplitz op-erators,and the compactness of k-order slant Toeplitz operators.In chapter one, we introduce some related research background about Toeplitz op-erators and k-order slant Toeplitz operators.In chapter two,a direct sum decomposition of Sobolev space W1,2(D) is obtained.we study the (semi-)commutativity of Toeplitz operators on the Dirichlet space V, which gets a necessary and sufficient condition.Then a necessary and sufficient condition is ob-tained about that the commutator TφTψ-TψTφ or the semi-commutator TφTψ-TψTφ has finite rank.Finally we prove a Toeplitz operator on the Dirichlet space V is a compact operator if and only if it is a zero operator.In chapter three,on the weighted Dirichlet space Da.firstly we introduce the opera-tor Wk some properties and its integral representation. We get a sufficient and necessary condition for the commutativity of two k-order slant Toeplitz operators. Then we give a result about spectra of k-order slant Toeplitz operator. Finally,we prove the k-order slant Toeplitz operator on the Dirichlet space D is compact only when it is zero.