A similarity invariant and the commutant of some multiplication operators on the Sobolev disk algebra (Q456494)

In this paper, the author studies the properties of the similarity relation of the multiplication operators \(M_g\) defined on \(R(\mathbb{D})\), the algebra generated in the Sobolev space \(W^{2, 2}(\mathbb{D})\) by the rational functions with poles outside the unit disk \(\mathbb{D}\), when \(g\) is univalent analytic on \(\mathbb{D}\) or \(M_g\) is strongly irreducible. In particular, the author shows that, for \(f,g\in R(\mathbb{D})\):NEWLINENEWLINE(i) if \(f,g\) are univalent and analytic on \(\mathbb{D}\), then \(M_f \sim M_g\) if and only if \(f(\mathbb{D})=g(\mathbb{D})\);NEWLINENEWLINE(ii) if \(f\) is univalent and analytic on \(\mathbb{D}\), then \(M_f \sim M_g\) if and only if there exists \(\chi\), an analytic automorphism of the unit disk, such that \(f=g\circ \chi\).NEWLINENEWLINEIn addition, the author determines the commutant algebra of \(M_g\). The author considers the cases when the symbol \(g\) is a nonconstant entire function, a composition of a univalent function belonging to \(R(\mathbb{D})\) and a finite Blaschke product, or a product of a function \(h\in R(\mathbb{D})\), nonvanishing on \(\mathbb{\overline{D}}\), and a finite Blaschke product.