On similarity and commutant of a class of multiplication operators on the Dirichlet space

Let D be the Dirichlet space on the unit disc D and B(z) be the Blaschke product with n zero, and we prove that multiplication operator M-B on D is similar to ?M-n(1)z on ?D-n(1) by a crucial decomposition of D.D. Two applications of our similarity theorem are presented. First, we characterize the intersection of the commutant of multiplication operator MB on the Dirichlet space setting from the techniques in operator theory combined with matrix manipulations. Second, we give a necessary and sufficient condition for the similarity of two multiplication operators Mf(1) and Mf(2) on D,D, where f(1) and f(2) are analytic functions on the closure of the unit disc D.