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Complex Analysis and Operator Theory

, Volume 12, Issue 7, pp 1767–1778 | Cite as

Quasinormal and Hyponormal Weighted Composition Operators on \(H^2\) and \(A^2_{\alpha }\) with Linear Fractional Compositional Symbol

  • Mahsa Fatehi
  • Mahmood Haji Shaabani
  • Derek Thompson
Article

Abstract

In this paper, we study quasinormal and hyponormal composition operators \(W_{\psi ,\varphi }\)  with linear fractional compositional symbol \(\varphi \) on the Hardy and weighted Bergman spaces. We characterize the quasinormal composition operators induced on \(H^{2}\) and \(A_{\alpha }^{2}\) by these maps and many such weighted composition operators, showing that they are necessarily normal in all known cases. We eliminate several possibilities for hyponormal weighted composition operators but also give new examples of hyponormal weighted composition operators on \(H^2\) which are not quasinormal.

Keywords

Weighted composition operator Composition operator Hyponormal Quasinormal 

Mathematics Subject Classification

47B33 

References

  1. 1.
    Bourdon, P.S.: Invertible weighted composition operators. Proc. Am. Math. Soc. 142, 289–299 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bourdon, P.S., Narayan, S.K.: Normal weighted composition operators on the Hardy space \(H^{2}(\mathbb{D})\). J. Math. Anal. Appl. 367, 278–286 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Conway, J.B.: Functions of One Complex Variable, 2nd edn. Springer, New York (1978)CrossRefGoogle Scholar
  4. 4.
    Conway, J.B.: The Theory of Subnormal Operators. American Mathematical Society, Providence (1991)CrossRefGoogle Scholar
  5. 5.
    Cowen, C.C.: Linear fractional composition operators on \(H^{2}\). Integral Equ. Oper. Theory 11, 151–160 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cowen, C.C., Jung, S., Ko, E.: Normal and cohyponormal weighted composition operators on \(H^{2}\). Oper. Theory Adv. Appl. 240, 69–85 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cowen, C.C., Ko, E., Thompson, D., Tian, F.: Spectra of some weighted composition operators on \(H^{2}\), Acta Sci. Math. (Szeged) 82, 221–234 (2016)Google Scholar
  8. 8.
    Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  9. 9.
    Douglas, R.: On majorization, factorization, and range inclusion of operators on Hilbert Space. Proc. Am. Math. Soc. 17, 413–416 (1966)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fatehi, M., Haji Shaabani, M.: Which weighted composition operators are hyponormal on the Hardy and weighted Bergman spaces? https://arxiv.org/abs/1505.00684
  11. 11.
    Fatehi, M., Haji Shaabani, M.: Normal, cohyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. J. Korean Math. Soc. 54(2), 599–612 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fatehi, M., Haji Shaabani, M.: Some essentially normal weighted composition operators on the weighted Bergman spaces. Complex Var. Elliptic Equ. 60, 1205–1216 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fatehi, M., Robati, B.K.: Essential normality for certain weighted composition operators on the Hardy space \(H^{2}\). Turk. J. Math. 36, 583–595 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gunatillake, G.: Invertible weighted composition operators. J. Funct. Anal. 261, 831–860 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hyvärinen, O., Lindström, M., Nieminen, I., Saukko, E.: Spectra of weighted composition operators with automorphic symbols. J. Funct. Anal. 265, 1749–1777 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hurst, P.: Relating composition operators on different weighted Hardy spaces. Arch. Math. (Basel) 68, 503–513 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jung, S., Kim, Y., Ko, E.: Characterizations of binormal composition operators with linear fractional symbols on \(H^2\). Appl. Math. Comput. 261, 252–263 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kriete, T.L., MacCluer, B.D., Moorhouse, J.L.: Toeplitz-composition \(C^{\ast }\)-algebras. J. Oper. Theory 58, 135–156 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kriete, T.L., Moorhouse, J.L.: Linear relations in the Calkin algebra for composition operators. Trans. Am. Math. Soc. 359, 2915–2944 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Le, T.: Self-adjoint, unitary, and normal weighted composition operators in several variables. J. Math. Anal. Appl. 395, 596–607 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    MacCluer, B.D., Narayan, S.K., Weir, R.J.: Commutators of composition operators with adjoints of composition operators on weighted Bergman spaces. Complex Var. Elliptic Equ. 58, 35–54 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sadraoui, H.: Hyponormality of Toeplitz and Composition Operators. Purdue University, Thesis (1992)Google Scholar
  23. 23.
    Stampfli, J.G.: Hyponormal operators. Pac. J. Math. 12(4), 1453–1458 (1962)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zorboska, N.: Hyponormal composition operators on the weighted Hardy spaces. Acta Sci. Math. (Szeged) 55, 399–402 (1991)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Mahsa Fatehi
    • 1
  • Mahmood Haji Shaabani
    • 2
  • Derek Thompson
    • 3
  1. 1.Department of Mathematics, Shiraz BranchIslamic Azad UniversityShirazIran
  2. 2.Department of MathematicsShiraz University of TechnologyShirazIran
  3. 3.Department of MathematicsTaylor UniversityUplandUSA

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