Similarity of multiplication operators on the Sobolev disk algebra (Q1944852)

Let \(\mathbb{D}\) be the unit disk. Introduce the Sobolev space \(W^{22}(\mathbb{D})\), which consists of all functions \(f \in L^2(\mathbb{D})\) whose distributional partial derivatives of the first and second orders belong to \(f \in L^2(\mathbb{D})\) and its subspace \(R(\mathbb{D})\), which is generated by the rational functions with poles outside \(\mathbb{D}\). The main result of the paper states that, given two functions \(f\) and \(g\) being analytic on the closure of the unit disk, the multiplication operator \(M_f\) is similar to \(M_g\) if and only if there exist two finite Blaschke products \(B\) and \(B_1\) of the same order and a function \(h\) analytic on \(\mathbb{D}\) such that \(f = h \circ B\) and \(g = h \circ B_1\).