Czechoslovak Mathematical Journal

, Volume 62, Issue 4, pp 901–917 | Cite as

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

  • Mahsa Fatehi
  • Bahram Khani Robati


In 1999 Nina Zorboska and in 2003 P. S.Bourdon, D. Levi, S.K.Narayan and J.H. Shapiro investigated the essentially normal composition operator \({C_\varphi }\), when φ is a linear-fractional self-map of D. In this paper first, we investigate the essential normality problem for the operator T w \({C_\varphi }\) on the Hardy space H 2, where w is a bounded measurable function on ∂D which is continuous at each point of F(φ), φS(2), and T w is the Toeplitz operator with symbol w. Then we use these results and characterize the essentially normal finite linear combinations of certain linear-fractional composition operators on H 2.


Hardy spaces essentially normal composition operator linear-fractional transformation 

MSC 2010



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. B. Aleksandrov: Multiplicity of boundary values of inner functions. Izv. Akad. Nauk Arm. SSR, Ser. Mat. 22 (1987), 490–503.Google Scholar
  2. [2]
    P. S. Bourdon: Components of linear-fractional composition operators. J. Math. Anal. Appl. 279 (2003), 228–245.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    P. S. Bourdon, D. Levi, S.K. Narayan, J. H. Shapiro: Which linear-fractional composition operators are essentially normal? J. Math. Anal. Appl. 280 (2003), 30–53.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    G. A. Chacón, G. R. Chacón: Some properties of composition operators on the Dirichlet space. Acta Math. Univ. Comen., New Ser. 74 (2005), 259–272.MATHGoogle Scholar
  5. [5]
    D.N. Clark: One-dimensional perturbations of restricted shifts. J. Anal. Math. 25 (1972), 169–191.MATHCrossRefGoogle Scholar
  6. [6]
    C. C. Cowen: Linear fractional composition operators on H 2. Integral Equations Oper. Theory 11 (1988), 151–160.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    C. C. Cowen, B. D. MacCluer: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, 1995.MATHGoogle Scholar
  8. [8]
    P. L. Duren: Theory of H p Spaces. Academic Press, New York, 1970.MATHGoogle Scholar
  9. [9]
    K. Heller, B. D. MacCluer, R. J. Weir: Compact differences of composition operators in several variables. Integral Equations Oper. Theory 69 (2011), 247–268.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    T. L. Kriete, B. D. MacCluer, J. L. Moorhouse: Toeplitz-composition C*-algebras. J. Oper. Theory 58 (2007), 135–156.MathSciNetMATHGoogle Scholar
  11. [11]
    T. L. Kriete, J. L. Moorhouse: Linear relations in the Calkin algebra for composition operators. Trans. Am. Math. Soc. 359 (2007), 2915–2944.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    B. D. MacCluer, R. J. Weir: Essentially normal composition operators on Bergman spaces. Acta Sci. Math. 70 (2004), 799–817.MathSciNetMATHGoogle Scholar
  13. [13]
    B. D. MacCluer, R. J. Weir: Linear-fractional composition operators in several variables. Integral Equations Oper. Theory 53 (2005), 373–402.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    J. Moorhouse: Compact differences of composition operators. J. Funct. Anal. 219 (2005), 70–92.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    A. G. Poltoratski: The boundary behavior of pseudocontinuable functions. St. Petersb. Math. J. 5 (1994), 389–406; translation from, Algebra Anal. 5 (1993), 189–210.Google Scholar
  16. [16]
    J. V. Ryff: Subordinate H p functions. Duke Math. J. 33 (1966), 347–354.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    D. E. Sarason: Sub-Hardy Hilbert Spaces in the Unit Disk. John Wiley & Sons, New York, 1994.MATHGoogle Scholar
  18. [18]
    H. J. Schwartz: Composition operators on H p. Ph.D. Thesis. University of Toledo, 1969.Google Scholar
  19. [19]
    J. H. Shapiro: Composition Operators and Classical Function Theory. Springer, New York, 1993.MATHCrossRefGoogle Scholar
  20. [20]
    J. H. Shapiro, P.D. Taylor: Compact, nuclear, and Hilbert-Schmidt composition operators on H 2. Indiana Univ. Math. J. 23 (1973), 471–496.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    N. Zorboska: Closed range essentially normal composition operators are normal. Acta Sci. Math. 65 (1999), 287–292.MathSciNetMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesShiraz UniversityShirazIran

Personalised recommendations