The commutant of a multiplication operator with a finite Blaschke product symbol on the Sobolev disk algebra

Abstract

Let $R\left(\mathbb{D}\right)$ be the algebra generated in the Sobolev space $W^{22}\left(\mathbb{D}\right)$ by the rational functions with poles outside the unit disk $\overline{\mathbb{D}}$. This is called the Sobolev disk algebra. In this article, the commutant of the multiplication operator $M_{B\left(z\right)}$ on $R\left(\mathbb{D}\right)$ is studied, where $B\left(z\right)$ is an n-Blaschke product. We prove that an operator $A\in \mathcal{L}\left(R\right(\mathbb{D}\left)\right)$ is in $\mathcal{A}\text{'}\left(M_{B\left(z\right)}\right)$ if and only if $A={\sum }_{i=1}^n{M}_{h_i}{M}_{\Delta \left(z\right)}^{-1}{T}_i$, where $\{h_i{\}}_{i=1}^n\subset R(\mathbb{D})$, and $T_i\in \mathcal{L}\left(R\right(\mathbb{D}\left)\right)$ is given by $\left(T_{i}g\right)\left(z\right)={\sum }_{j=1}^n(-1{)}^{i+j}{\Delta }_{ij}(z\left)g\right(G_{j-1}\left(z\right))$, $i=1,2,\dots ,n$, $G_0\left(z\right)\equiv z$.