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.
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24 June 2017
An erratum to this article has been published.
References
Bourdon, P.S.: Invertible weighted composition operators. Proc. Am. Math. Soc. 142, 289–299 (2014)
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)
Conway, J.B.: Functions of One Complex Variable, 2nd edn. Springer, New York (1978)
Conway, J.B.: The Theory of Subnormal Operators. American Mathematical Society, Providence (1991)
Cowen, C.C.: Linear fractional composition operators on \(H^{2}\). Integral Equ. Oper. Theory 11, 151–160 (1988)
Cowen, C.C., Jung, S., Ko, E.: Normal and cohyponormal weighted composition operators on \(H^{2}\). Oper. Theory Adv. Appl. 240, 69–85 (2014)
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)
Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)
Douglas, R.: On majorization, factorization, and range inclusion of operators on Hilbert Space. Proc. Am. Math. Soc. 17, 413–416 (1966)
Fatehi, M., Haji Shaabani, M.: Which weighted composition operators are hyponormal on the Hardy and weighted Bergman spaces? https://arxiv.org/abs/1505.00684
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)
Fatehi, M., Haji Shaabani, M.: Some essentially normal weighted composition operators on the weighted Bergman spaces. Complex Var. Elliptic Equ. 60, 1205–1216 (2015)
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)
Gunatillake, G.: Invertible weighted composition operators. J. Funct. Anal. 261, 831–860 (2011)
Hyvärinen, O., Lindström, M., Nieminen, I., Saukko, E.: Spectra of weighted composition operators with automorphic symbols. J. Funct. Anal. 265, 1749–1777 (2013)
Hurst, P.: Relating composition operators on different weighted Hardy spaces. Arch. Math. (Basel) 68, 503–513 (1997)
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)
Kriete, T.L., MacCluer, B.D., Moorhouse, J.L.: Toeplitz-composition \(C^{\ast }\)-algebras. J. Oper. Theory 58, 135–156 (2007)
Kriete, T.L., Moorhouse, J.L.: Linear relations in the Calkin algebra for composition operators. Trans. Am. Math. Soc. 359, 2915–2944 (2007)
Le, T.: Self-adjoint, unitary, and normal weighted composition operators in several variables. J. Math. Anal. Appl. 395, 596–607 (2012)
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)
Sadraoui, H.: Hyponormality of Toeplitz and Composition Operators. Purdue University, Thesis (1992)
Stampfli, J.G.: Hyponormal operators. Pac. J. Math. 12(4), 1453–1458 (1962)
Zorboska, N.: Hyponormal composition operators on the weighted Hardy spaces. Acta Sci. Math. (Szeged) 55, 399–402 (1991)
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Communicated by Isabelle Chalendar.
The original version of this article was revised: The title of section 3.4 was incorrect. Now, it has been corrected.
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Fatehi, M., Haji Shaabani, M. & Thompson, D. Quasinormal and Hyponormal Weighted Composition Operators on \(H^2\) and \(A^2_{\alpha }\) with Linear Fractional Compositional Symbol. Complex Anal. Oper. Theory 12, 1767–1778 (2018). https://doi.org/10.1007/s11785-017-0683-3
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DOI: https://doi.org/10.1007/s11785-017-0683-3