On similarity and reducing subspaces of a class of operator on the Dirichlet space

Let $Y_{p}$ be the multiplication operator $M_{p}$ plus the Volterra operator $V_{p}$ induced by $p(z)$, where $p$ is a polynomial. Under a mild condition, we prove that $Y_{p}$ acting on the Dirichlet space $\mathfrak{D}$ is similar to multiplication operator $M_{p}$ acting on a subspace $S(\mathbb{D})$ of $\mathfrak{D}$. Furthermore, it shows that $T_{z^n}\,(n\geq2)$ has exactly $2^{n}$ reducing subspaces on $\mathfrak{D}$. KCI Citation Count: 0