Advertisement

Revisit to a Theorem of Wogen

  • Hyungwoon Koo
  • Maofa Wang
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

In this note we provide a new proof of a theorem of Wogen on the boundedness criterion for composition operators on Hardy space H2(Un) induced by holomorphic self-maps of the unit ball Un, and then generalize it to more general inducing self-maps.

Keywords

Composition operator smooth map Bergman space Hardy space Carleson measure boundedness Wogen criterion 

Mathematics Subject Classification (2000)

Primary 47B33 Secondary 30D55 46E15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, Springer-Verlag, New York, 2001.zbMATHGoogle Scholar
  2. [2]
    L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math., 76 (1962), 547–559.CrossRefMathSciNetGoogle Scholar
  3. [3]
    J. Cima and W. Wogen, Unbounded composition operators on H 2(B 2), Proc. Amer. Math. Soc., 99(1987), 477–483.zbMATHMathSciNetGoogle Scholar
  4. [4]
    C. Cowen and B. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, 1995.zbMATHGoogle Scholar
  5. [5]
    P. Duren, Theory of H p spaces, Academic Press, New York, 1970.zbMATHGoogle Scholar
  6. [6]
    P. Halmos, Measure Theory, Springer-Verlag, New York, 1974.zbMATHGoogle Scholar
  7. [7]
    H. Koo and W. Smith, Composition operators induced by smooth self maps of the unit ball in C N, J. Math. Anal. Appl. 329(2007), 617–633.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    H. Koo, M. Stessin and K. Zhu, Composition operators on the polydisc induced by smooth symbols, J. Funt. Anal., 254 (2008), 2911–2925.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    H. Koo and M. Wang, Composition operators induced by smooth self-maps on the real or the complex unit balls, J. Funct. Anal., 256 (2009), 2747–2767.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    W. Rudin, Function theory in the unit ball of C n, Springer-Verlag, New York, 1980.Google Scholar
  11. [11]
    W. Wogen, The smooth mappings which preserve the Hardy space HB n/2, Operator Theory: Advances Appl. 35 (1988), 249–267.MathSciNetGoogle Scholar
  12. [12]
    W. Wogen, On geometric properties of smooth maps which preserve HB n 2n, Michigan Math. J., 54 (2006), 301–306.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Hyungwoon Koo
    • 1
  • Maofa Wang
    • 2
  1. 1.Department of MathematicsKorea UniversitySeoulKorea
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanP.R. China

Personalised recommendations