Commuting Toeplitz Operators On The Dirichlet Space Of The Polydisk

The title of this article is COMMUTING TOEPLITZ OPERATORS ON THE DIRICHLET SPACE OF THE PLOYDISK.It completely characterizes the normal Toeplitz operator.And it gives the sufficient and necessary condition for the commuting of two Toeplitz operator with conjugate holomorphic or holomorphic symbols on the Dirichlet space of the polydisk.Some algebraic properties acting on the analytic function spaces in the disk has been well studied.And we have got a lot of good results.In this title we will show that two Toeplitz operator with conjugate holomorphic symbols are commute on the Dirichlet space of the poly-disk if and only if two symbols are linearly dependent,and Toeplitz operator with holomorphic symbol and Toplitz operator with conjugate holomorphic symbol are commute if and only if one of the two symbols is a constant.The article is divided into four parts.In the first part,we introduce briefly the object of the article,and some background. It can be taken as a preface.In the next part some important notions and their properties are given. In the third part,some important propositions,which are related to the main resuits, are given.In the last part we prove the main theorems using the propositions in the third part.