## Annals of Functional Analysis

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

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

Ruifang Zhao, Zongyao Wang, and David R. Larson

#### 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 $\left({T}_{i}g\right)\left(z\right)={\sum}_{j=1}^{n}(-1{)}^{i+j}{\Delta}_{ij}(z\left)g\right({G}_{j-1}\left(z\right))$, $i=1,2,\dots ,n$, ${G}_{0}\left(z\right)\equiv z$.

#### Article information

**Source**

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

**Dates**

Received: 2 June 2016

Accepted: 29 October 2016

First available in Project Euclid: 22 April 2017

**Permanent link to this document**

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

**Subjects**

Primary: 47B38: Operators on function spaces (general)

Secondary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 46E20: Hilbert spaces of continuous, differentiable or analytic functions

**Keywords**

Sobolev disk algebra finite Blaschke product multiplication operator commutant

#### 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