Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T11:16:46.092Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of Hankel operators in the essential commutant of quasicontinuous Toeplitz operators

Part of: Function theory on the disc Commutative Banach algebras and commutative topological algebras Special classes of linear operators

Published online by Cambridge University Press:  09 February 2021

Yi Yan
Yi Yan*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS66045, USA
*
e-mail: yiyan@ku.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.

Keywords

Hankel operatorToeplitz operatorquasicontinuousDouglas algebrainterpolating Blaschke productsingular inner function

MSC classification

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators
Secondary: 46J10: Banach algebras of continuous functions, function algebras 30J05: Inner functions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 44 - 51
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barría, J., On Hankel operators not in the Toeplitz algebra. Proc. Amer. Math. Soc. 124(1996), 1507–1511.CrossRefGoogle Scholar
Barría, J. and Halmos, P. R., Asymptotic Toeplitz operators. Trans. Amer. Math. Soc. 273(1982), 621–630.10.2307/1999932CrossRefGoogle Scholar
Böttcher, A. and Silbermann, B., Analysis of Toeplitz operators. 2nd ed. Springer-Verlag, Berlin, Germany, 2006.Google Scholar
Chen, X. and Chen, F., Hankel operators in the set of essential Toeplitz operators. Acta Math. Sinica 6(1990), 354–363.Google Scholar
Chen, X., Guo, K., Izuchi, K., and Zheng, D., Compact perturbations of Hankel operators. J. Reine Angew. Math. 578(2005), 1–48.CrossRefGoogle Scholar
Davidson, K., On operators commuting with Toeplitz operators modulo the compact operators. J. Funct. Anal. 24(1977), 291–302.CrossRefGoogle Scholar
Garnett, J. B., Bounded analytic functions. Academic Press, New York, 1981.Google Scholar
Gorkin, P. and Mortini, R., Interpolating Blaschke products and factorization in Douglas algebras. Michigan Math. J. 38(1991), 147–160.CrossRefGoogle Scholar
Gorkin, P. and Zheng, D., Essentially commuting Toeplitz operators. Pacific J. Math. 190(1999), 87–109.CrossRefGoogle Scholar
Guillory, C. and Sarason, D., The algebra of quasicontinuous functions. Proc. Roy. Irish Acad. Sect. A 84(1984), 57–67.Google Scholar
Guo, K. and Zheng, D., Essentially commuting Hankel and Toeplitz operators. J. Funct. Anal. 201(2003), 121–147.10.1016/S0022-1236(03)00100-9CrossRefGoogle Scholar
Hoffman, K., Banach spaces of analytic functions. Prentice-Hall, Englewood Cliffs, NJ, 1962.Google Scholar
Power, S. C., Hankel operators with PQC symbols and singular integral operators . Proc. Lond. Math. Soc. (3) 41(1980), 45–65.10.1112/plms/s3-41.1.45CrossRefGoogle Scholar
Sarason, D., Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207(1975), 391–405.CrossRefGoogle Scholar
Sarason, D., Teoplitz operators with piecewise quasicontinuous symbols. Indiana Univ. Math. J. 26(1977), 817–838.CrossRefGoogle Scholar
Xia, J., On the essential commutant of $\mathcal {T}(QC)$ . Trans. Amer. Math. Soc. 360(2008), 1089–1102.10.1090/S0002-9947-07-04345-0CrossRefGoogle Scholar
Yan, Y., On compact perturbations of Hankel operators and commutators of Toeplitz and Hankel operators. Integr. Equ. Oper. Theory 92(2020), no. 2.CrossRefGoogle Scholar