Abstract: In this paper, we study multiplication operators on weighted Dirichlet spaces $\mathcal{D}_{\beta}$ $(\beta\in \mathbb{R})$. Let $n$ be a positive integer and $\beta\in \mathbb{R}$, we show that the multiplication operator $M_{z^{n}}$ on $\mathcal{D}_{\beta}$ is similar to the operator $\oplus_{1}^{n}M_{z}$ on the space $\oplus_{1}^{n}\mathcal{D}_{\beta}$. Moreover, we prove that $M_{z^{n}}$ $(n\geq 2)$ on $\mathcal{D}_{\beta}$ is unitarily equivalent to $\oplus_{1}^{n}M_{z}$ on $\oplus_{1}^{n}\mathcal{D}_{\beta}$ if and only if $\beta=0$. In addition, we completely characterize the unitary equivalence of the restrictions of $M_{z^{n}}$ to different invariant subspaces $z^{k}\mathcal{D}_{\beta}$ $(k\geq 1)$, and the unitary equivalence of the restrictions of $M_{z^{n}}$ to different invariant subspaces $S_{j}$ $(0\leq j<n)$.
Abkar, Cao and Zhu [Complex Anal Oper Theory, 2020, 14: Art 58] pointed out that it is an important, natural, and difficult question in operator theory to identify the commutant of a bounded linear operator. They characterized the commutant $\mathcal{A}'(M_{z^{n}})$ of $M_{z^{n}}$ on a family of analytic function spaces $A_{\alpha}^{2}$ $(\alpha\in \mathbb{R})$ on $\mathbb{D}$ (in fact, the family of spaces $A_{\alpha}^{2}$ $(\alpha\in \mathbb{R})$ is the same with the family of spaces $\mathcal{D}_{\beta}$ $(\beta\in \mathbb{R})$) in terms of the multiplier algebra of the underlying function spaces. In this paper, we give a new characterization of the commutant $\mathcal{A}'(M_{z^{n}})$ of $M_{z^{n}}$ on $\mathcal{D}_{\beta}$, and characterize the self-adjoint operators and unitary operators in $\mathcal{A}'(M_{z^{n}})$. We find that the class of self-adjoint operators (unitary operators) in $\mathcal{A}'(M_{z^{n}})$ when $\beta \neq 0$ is different from the class of self-adjoint operators (unitary operators) in $\mathcal{A}'(M_{z^{n}})$ when $\beta =0$.

Key words: multiplication operator, weighted Dirichlet space, similarity, unitary equivalence, commutant