Operator Theory On The Dirichlet Space

The thesis mainly concerns the multiplication operators(i.e.,analytic Toeplitz operators)on the Dirichlet space and Toeplitz operators on the harmonic Dirichlet space.The multiplication operator M_z on the Dirichlet space(i.e.,Dirichlet shift)is an important operator.A natural question about the operator M_z is the characteri-zation of its invariant subspaces,which has been studied extensively and deeply.In [43],S.Richter considered the unitary equivalence problem of M_z's invariant sub-spaces,and proved that if two unitarily equivalent invariant subspaces M,N of M_z satisfy one of the following conditions;(â…°)M(?)N,(â…±)M contains an outer func-tion,then they are equal.In chapter one,we show that any two unitarily equivalent invariant subspaces of M_z are equal.In chapter two,we study some unitary equivalence problem of the multiplica-tion operators on the Dirichlet space.In section two,we consider the multiplication operators on the Dirichlet space which are unitarily equivalent to a weighted uni-lateral shift operator of finite multiplicity.The reducing subspaces of multiplication operators defined by a Blaschke product of order two is characterized in section three.In the last section of this chapter,the unitary equivalence of multiplication operator defined by a Blaschke product of order two is studied.All of these results are distinct from the corresponding results in the Hardy space and the Bergman space.In chapter three,algebraic property of Toeplitz operators on the harmonic Dirichlet space is studied.The commutativity and semi-commutativity of Toeplitz operators defined by harmonic functions on the harmonic Dirichlet space are char-acterized completely.