## Annals of Functional Analysis

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

#### Abstract

Let $R(\mathbb{D})$ be the algebra generated in the Sobolev space $W^{22}(\mathbb{D})$ 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 $M_{B(z)}$ on $R(\mathbb{D})$ is studied, where $B(z)$ is an n-Blaschke product. We prove that an operator $A\in\mathcal{L}(R(\mathbb{D}))$ is in $\mathcal{A}'(M_{B(z)})$ if and only if $A=\sum_{i=1}^{n}M_{h_{i}}M_{\Delta(z)}^{-1}T_{i}$, where $\{h_{i}\}_{i=1}^{n}\subset R(\mathbb{D})$, and $T_{i}\in\mathcal{L}(R(\mathbb{D}))$ is given by $(T_{i}g)(z)=\sum_{j=1}^{n}(-1)^{i+j}\Delta_{ij}(z)g(G_{j-1}(z))$, $i=1,2,\ldots,n$, $G_{0}(z)\equiv z$.

#### Article information

Source
Ann. Funct. Anal., Volume 8, Number 3 (2017), 366-376.

Dates
Accepted: 29 October 2016
First available in Project Euclid: 22 April 2017

https://projecteuclid.org/euclid.afa/1492826603

Digital Object Identifier
doi:10.1215/20088752-2017-0002

Mathematical Reviews number (MathSciNet)
MR3689999

Zentralblatt MATH identifier
1381.47027

#### Citation

Zhao, Ruifang; Wang, Zongyao; Larson, David R. The commutant of a multiplication operator with a finite Blaschke product symbol on the Sobolev disk algebra. Ann. Funct. Anal. 8 (2017), no. 3, 366--376. doi:10.1215/20088752-2017-0002. https://projecteuclid.org/euclid.afa/1492826603

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