Reducing Subspaces of Multiplication Operators on the Dirichlet Space

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product phi on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of M-phi are orthogonal. When the order n of phi is 2 or 3, we show that M-phi is reducible on D if and only if phi is equivalent to z(n). When the order of phi is 4, we determine the reducing subspaces for M-phi, and we see that in this case M-phi can be reducible on D when phi is not equivalent to z(4). The same phenomenon happens when the order n of phi is not a prime number. Furthermore, we show that M-phi is unitarily equivalent to M-zn (n > 1) on D if and only if phi = az(n) for some unimodular constant a.