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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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

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

Received: 2 June 2016
Accepted: 29 October 2016
First available in Project Euclid: 22 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Sobolev disk algebra finite Blaschke product multiplication operator commutant


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.

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  • [1] J. B. Conway, Functions of One Complex Variable, II, Grad. Texts in Math. 159, Springer, New York, 1995.
  • [2] Z. Cuckovic, Commutants of Toeplitz operators on the Bergman space, Pacific J. Math. 162 (1994), no. 2, 277–285.
  • [3] J. Dixmier and C. Foiaş, “Sur le spectre ponctuel d’un opérateur” in Hilbert Space Operators and Operator Algebras (Tihany, 1970), Colloq. Math. 5, North-Holland, Amsterdam, 1972, 127–133.
  • [4] P. Griffiths, Algebraic Curves (in Chinese), Publishing house of Beijing Univ., Beijing, 1985.
  • [5] D. A. Herrero, T. J. Taylor, and Z. Y. Wang, “Variation of the point spectrum under compact perturbations” in Topics in Operator Theory, Oper. Theory Adv. Appl. 32, Birkhäuser, Basel, 1988, 113–158.
  • [6] C. L. Jiang and Z. Y. Wang, Structure of Hilbert Space Operators, World Scientific, Hackensack, NJ, 2006.
  • [7] B. Khani Robati and S. M. Vaezpour, On the commutant of operators of multiplication by univalent functions, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2379–2383.
  • [8] Y. Q. Liu and Z. Y. Wang, The commutant of the multiplication operators on Sobolev disk algebra, J. Anal. Appl. 2 (2004), no. 2, 65–86.
  • [9] J. Mashreghi, Derivatives of inner functions, Fields Inst. Monogr. 31, Springer, New York, 2013.
  • [10] Z. Y. Wang, R. F. Zhao, and Y. F. Jin, Finite Blaschke product and the multiplication operators on Sobolev disk algebra, Sci. China Ser. A 52 (2009), no. 1, 142–146.