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 $M_{z^n} (n > 1)$ on $D$ if and only if $\phi = az^n$ for some unimodular constant $a$.