Download PDF - The Bourgain algebra of the disk algebra A() and the algebra QAAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

The Bourgain algebra of the disk algebra A() and the algebra QA

Authors 1, 2

Affiliations

  1. Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27514, U.S.A.
  2. Mathematisches Institut I, Universität Karlsruhe, 76128 Karlsruhe, Germany

Abstract

It is shown that the Bourgain algebra A()b of the disk algebra A() with respect to H∞() is the algebra generated by the Blaschke products having only a finite number of singularities. It is also proved that, with respect to H∞(), the algebra QA of bounded analytic functions of vanishing mean oscillation is invariant under the Bourgain map as is A()b.

Bibliography

  1. S.-Y. Chang and D. E. Marshall, Some algebras of bounded analytic functions containing the disc algebra, in: Lecture Notes in Math. 604, Springer, 1977, 12-20.
  2. J. Cima, S. Janson and K. Yale, Completely continuous Hankel operators on H∞ and Bourgain algebras, Proc. Amer. Math. Soc. 105 (1989), 121-125.
  3. J. Cima, K. Stroethoff and K. Yale, Bourgain algebras on the unit disk, Pacific J. Math. 160 (1993), 27-41.
  4. J. Cima, S. Janson and K. Yale, The Bourgain algebra of the disk algebra, Proc. Roy. Irish Acad. 94A (1994), 19-23.
  5. J. Cima and R. Timoney, The Dunford-Pettis property for certain planar uniform algebras, Michigan Math. J. 34 (1987), 99-104.
  6. K. F. Clancey and J. A. Gosselin, The local theory of Toeplitz operators, Illinois J. Math. 22 (1978), 449-458.
  7. A. Davie, T. W. Gamelin and J. B. Garnett, Distance estimates and pointwise bounded density, Trans. Amer. Math. Soc. 175 (1973), 37-68.
  8. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
  9. I. M. Gelfand, D. A. Raikov and G. E. Shilov, Kommutative normierte Algebren, Deutscher Verlag Wiss., Berlin, 1964.
  10. P. Gorkin and K. Izuchi, Bourgain algebras on the maximal ideal space of H∞, Rocky Mountain J. Math., to appear.
  11. P. Gorkin, K. Izuchi and R. Mortini, Bourgain algebras of Douglas algebras, Canad. J. Math. 44 (1992), 797-804.
  12. P. Helmer and J. Pym, Approximation by functions with finitely many discontinuities, Quart. J. Math. Oxford 43 (1992), 223-226.
  13. K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, N.J., 1962.
  14. K. Izuchi, Bourgain algebras of the disk, polydisk, and ball algebras, Duke Math. J. 66 (1992), 503-520.
  15. K. Izuchi, K. Stroethoff and K. Yale, Bourgain algebras of spaces of harmonic functions, Michigan Math. J. 41 (1994), 309-321.
  16. D. E. Marshall and K. Stephenson, Inner divisors and composition operators, J. Funct. Anal. 46 (1982), 131-148.
  17. R. Mortini and M. von Renteln, Strong extreme points and ideals in uniform algebras, Arch. Math. (Basel) 52 (1989), 465-470.
  18. D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405.
  19. C. Sundberg and T. H. Wolff, Interpolating sequences for QAB, ibid. 276 (1983), 551-581.
  20. T. H. Wolff, Some theorems of vanishing mean oscillation, thesis, Univ. of California, Berkeley, 1979.
Studia Mathematica
1995/113/3
Export to PBN, Opens in new tab
Pages:
211-221
Main language of publication
English
Published
1995
Exact and natural sciences