Reducing subspaces of multiplication operators on the Dirichlet space

Abstract

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