Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-29T18:04:44.965Z Has data issue: false hasContentIssue false
  • English
  • Français

A distance formula and Bourgain algebras

Published online by Cambridge University Press:  24 October 2008

R. Younis  and
D. Zheng
R. Younis
Affiliation:
Department of Mathematics and Computer Science, United Arab Emirates University, P.O. Box 17551, Al Ain, United Arab Emirates
D. Zheng
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824†
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, a distance formula to a Douglas algebra is established. We use this distance formula to show that the distance of an L∞ function to the intersection of arbitrary Douglas algebras is equivalent to the supremum of the distance of that function to these algebras. As an application, we prove that the Bourgain algebra of the intersection of two Douglas algebras is equal to the intersection of the Bourgain algebras of these two algebras.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 4 , November 1996 , pp. 631 - 641
Copyright
Copyright © Cambridge Philosophical Society 1996

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

REFERENCES

[1]Bourgain, J.. New Banach space properties of the disc algebra and H ∞. Acta Math. 152 (1984), 1–48.CrossRefGoogle Scholar
[2]Cima, J. A. and Timoney, R.. The Dunford-Pettis property for certain planar uniform algebras. Michigan Math. J. 34 (1987), 99–104.CrossRefGoogle Scholar
[3]Garnett, J. B.. Bounded analytic functions (Academic Press, 1981).Google Scholar
[4]Gorkin, P., Izuchi, K. and Mortini, R.. Bourgain algebras of Douglas algebras. Can. J. Math. 44 (1992), 797–804.CrossRefGoogle Scholar
[5]Gorkin, P. and Zheng, D.. Essentially commuting Toeplitz operators, preprint.Google Scholar
[6]Hoffman, K.. Bounded analytic functions and Gleason parts. Ann. of Math. 86 (1967), 74–111.CrossRefGoogle Scholar
[7]Izuchi, K.. Countably generated Douglas algebras. Trans. Amer. Math. Soc. 299 (1987), 171–192.CrossRefGoogle Scholar
[8]Mortini, R. and Younis, R.. Douglas algebras which are invariant under the Bourgain map. Arch. Math. 59 (1992), 371–378.Google Scholar
[9]Nikolskh, N. K.. Treatise on the shift operator (Springer-Verlag, 1985).Google Scholar
[10]Sundberg, C. and Wolff, T.. Interpolating sequences for QA B. Trans. Amer. Math. Soc. 276 (1983), 551–581.Google Scholar
[11]Younis, R.. Distance estimates and products of Toeplitz operators. Mich. J. Math. 31 (1984), 49–54.CrossRefGoogle Scholar
[12]Zheng, D.. The distribution function inequality and products of Toeplitz operators and Hankel operators, to appear in J. Functional Analysis.Google Scholar