In this paper, we study multiplication operators on weighted Dirichlet spaces \(\mathcal{D}_{\beta}\) (β ∈ ℝ). Let n be a positive integer and β ∈ 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 ≥ 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 β = 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 ≥ 1), and the unitary equivalence of the restrictions of \(M_{z^{n}}\) to different invariant subspaces Sj (0 ≤ 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 β ≠ 0 is different from the class of self-adjoint operators (unitary operators) in \(\mathcal{A}'(M_{z^{n}})\) when β = 0.
