Annals of Functional Analysis

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

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Abstract

Let R(D) be the algebra generated in the Sobolev space W22(D) by the rational functions with poles outside the unit disk D¯. This is called the Sobolev disk algebra. In this article, the commutant of the multiplication operator MB(z) on R(D) is studied, where B(z) is an n-Blaschke product. We prove that an operator AL(R(D)) is in A'(MB(z)) if and only if A=i=1nMhiMΔ(z)1Ti, where {hi}i=1nR(D), and TiL(R(D)) is given by (Tig)(z)=j=1n(1)i+jΔij(z)g(Gj1(z)), i=1,2,,n, G0(z)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


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