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Norm Inequalities for Composition Operators on Hardy and Weighted Bergman Spaces

  • Christopher Hammond
  • Linda J. Patton
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

Any analytic self-map of the open unit disk induces a bounded composition operator on the Hardy space H 2 and on the standard weighted Bergman spaces A 2 β. For a particular self-map, it is reasonable to wonder whether there is any meaningful relationship between the norms of the corresponding operators acting on each of these spaces. In this paper, we demonstrate an inequality which, at least to a certain degree, provides an answer to this question.

Keywords

Composition operator operator norm Hardy space weighted Bergman spaces Schur product 

Mathematics Subject Classification (2000)

47B33 

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References

  1. [1]
    M.J. Appel, P.S. Bourdon, and J.J. Thrall, Norms of composition operators on the Hardy space, Experiment. Math. 5 (1996), 111–117.zbMATHMathSciNetGoogle Scholar
  2. [2]
    E.L. Basor and D.Q. Retsek, Extremal non-compactness of composition operators with linear fractional symbol, J. Math. Anal. Appl. 322 (2006), 749–763.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    P.S. Bourdon, E.E. Fry, C. Hammond, and C.H. Spofford, Norms of linear-fractional composition operators, Trans. Amer. Math. Soc. 356 (2004), 2459–2480.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    B.J. Carswell and C. Hammond, Composition operators with maximal norm on weighted Bergman spaces, Proc. Amer. Math Soc. 134 (2006), 2599–2605.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    C.C. Cowen, Linear fractional composition operators on H 2, Integral Equations Operator Theory 11 (1988), 151–160.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    C.C. Cowen, Transferring subnormality of adjoint composition operators, Integral Equations Operator Theory 15 (1992), 167–171.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    C.C. Cowen and T.L. Kriete, Subnormality and composition operators on H 2, J. Funct. Anal. 81 (1988), 298–319.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.zbMATHGoogle Scholar
  9. [9]
    C. Hammond, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged) 69 (2003), 813–829.zbMATHMathSciNetGoogle Scholar
  10. [10]
    C. Hammond, Zeros of hypergeometric functions and the norm of a composition operator, Comput. Methods Funct. Theory 6 (2006), 37–50.zbMATHMathSciNetGoogle Scholar
  11. [11]
    H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Springer-Verlag, New York, 2000.zbMATHGoogle Scholar
  12. [12]
    R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.zbMATHGoogle Scholar
  13. [13]
    R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.zbMATHGoogle Scholar
  14. [14]
    P.R. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel) 68 (1997), 503–513.zbMATHMathSciNetGoogle Scholar
  15. [15]
    J.E. McCarthy, Geometric interpolation between Hilbert spaces, Ark. Mat. 30 (1992), 321–330.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    A.E. Richman, Subnormality and composition operators on the Bergman space, Integral Equations Operator Theory 45 (2003), 105–124.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    A.E. Richman, Composition operators with complex symbol having subnormal adjoint, Houston J. Math. 29 (2003), 371–384.zbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Christopher Hammond
    • 1
  • Linda J. Patton
    • 2
  1. 1.Department of MathematicsConnecticut CollegeNew LondonUSA
  2. 2.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA

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