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

Joseph Cima; Raymond Mortini

Studia Mathematica (1995)

  • Volume: 113, Issue: 3, page 211-221
  • ISSN: 0039-3223

Abstract

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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 .

How to cite

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Cima, Joseph, and Mortini, Raymond. "The Bourgain algebra of the disk algebra A(𝔻) and the algebra QA." Studia Mathematica 113.3 (1995): 211-221. <http://eudml.org/doc/216171>.

@article{Cima1995,
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 $A()_b$.},
author = {Cima, Joseph, Mortini, Raymond},
journal = {Studia Mathematica},
keywords = {Bourgain algebra; disk algebra; algebra generated by the Blaschke products having only a finite number of singularities},
language = {eng},
number = {3},
pages = {211-221},
title = {The Bourgain algebra of the disk algebra A(𝔻) and the algebra QA},
url = {http://eudml.org/doc/216171},
volume = {113},
year = {1995},
}

TY - JOUR
AU - Cima, Joseph
AU - Mortini, Raymond
TI - The Bourgain algebra of the disk algebra A(𝔻) and the algebra QA
JO - Studia Mathematica
PY - 1995
VL - 113
IS - 3
SP - 211
EP - 221
AB - 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 $A()_b$.
LA - eng
KW - Bourgain algebra; disk algebra; algebra generated by the Blaschke products having only a finite number of singularities
UR - http://eudml.org/doc/216171
ER -

References

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  1. [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. [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. [3] J. Cima, K. Stroethoff and K. Yale, Bourgain algebras on the unit disk, Pacific J. Math. 160 (1993), 27-41. Zbl0816.46046
  4. [4] J. Cima, S. Janson and K. Yale, The Bourgain algebra of the disk algebra, Proc. Roy. Irish Acad. 94A (1994), 19-23. Zbl0804.46065
  5. [5] J. Cima and R. Timoney, The Dunford-Pettis property for certain planar uniform algebras, Michigan Math. J. 34 (1987), 99-104. Zbl0617.46058
  6. [6] K. F. Clancey and J. A. Gosselin, The local theory of Toeplitz operators, Illinois J. Math. 22 (1978), 449-458. Zbl0384.47022
  7. [7] A. Davie, T. W. Gamelin and J. B. Garnett, Distance estimates and pointwise bounded density, Trans. Amer. Math. Soc. 175 (1973), 37-68. Zbl0263.30033
  8. [8] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. Zbl0469.30024
  9. [9] I. M. Gelfand, D. A. Raikov and G. E. Shilov, Kommutative normierte Algebren, Deutscher Verlag Wiss., Berlin, 1964. 
  10. [10] P. Gorkin and K. Izuchi, Bourgain algebras on the maximal ideal space of H ∞ , Rocky Mountain J. Math., to appear. Zbl0843.46037
  11. [11] P. Gorkin, K. Izuchi and R. Mortini, Bourgain algebras of Douglas algebras, Canad. J. Math. 44 (1992), 797-804. Zbl0763.46046
  12. [12] P. Helmer and J. Pym, Approximation by functions with finitely many discontinuities, Quart. J. Math. Oxford 43 (1992), 223-226. Zbl0766.54011
  13. [13] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, N.J., 1962. Zbl0117.34001
  14. [14] K. Izuchi, Bourgain algebras of the disk, polydisk, and ball algebras, Duke Math. J. 66 (1992), 503-520. Zbl0806.46061
  15. [15] K. Izuchi, K. Stroethoff and K. Yale, Bourgain algebras of spaces of harmonic functions, Michigan Math. J. 41 (1994), 309-321. Zbl0916.46043
  16. [16] D. E. Marshall and K. Stephenson, Inner divisors and composition operators, J. Funct. Anal. 46 (1982), 131-148. Zbl0485.30037
  17. [17] R. Mortini and M. von Renteln, Strong extreme points and ideals in uniform algebras, Arch. Math. (Basel) 52 (1989), 465-470. Zbl0639.46046
  18. [18] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. Zbl0319.42006
  19. [19] C. Sundberg and T. H. Wolff, Interpolating sequences for Q A B , ibid. 276 (1983), 551-581. Zbl0536.30025
  20. [20] T. H. Wolff, Some theorems of vanishing mean oscillation, thesis, Univ. of California, Berkeley, 1979. 

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