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Essential norms of composition operators between Bloch type spaces. (English) Zbl 1190.47028

Summary: For \( \alpha>0\), the \( \alpha\)-Bloch space is the space of all analytic functions \( f\) on the unit disk \( D\) satisfying \[ \| f\| _{B^{\alpha}}=\sup_{z\in D}| f^{\prime}(z)|(1-| z|^2)^{\alpha}<\infty. \] Let \( \varphi\) be an analytic self-map of \( D\). We show that for \( 0<\alpha,\beta<\infty\), the essential norm of the composition operator \( C_{\varphi}\) mapping from \( B^{\alpha}\) to \( B^{\beta}\) can be given by the following formula: \[ \| C_{\varphi}\| _e=\left(\frac{e}{2\alpha}\right)^{\alpha}\limsup_{n\to\infty} n^{\alpha-1}\|\varphi^n\| _{B^{\beta}}. \]

MSC:

47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
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[1] M. D. Contreras and A. G. Hernandez-Diaz, Weighted composition operators in weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 69 (2000), no. 1, 41 – 60. · Zbl 0990.47018
[2] Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017
[3] Barbara D. MacCluer and Ruhan Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), no. 4, 1437 – 1458. · Zbl 1061.30023 · doi:10.1216/rmjm/1181075473
[4] Kevin M. Madigan, Composition operators on analytic Lipschitz spaces, Proc. Amer. Math. Soc. 119 (1993), no. 2, 465 – 473. · Zbl 0793.47037
[5] Kevin Madigan and Alec Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2679 – 2687. · Zbl 0826.47023
[6] Alfonso Montes-Rodríguez, The essential norm of a composition operator on Bloch spaces, Pacific J. Math. 188 (1999), no. 2, 339 – 351. · Zbl 0932.30034 · doi:10.2140/pjm.1999.188.339
[7] Alfonso Montes-Rodríguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. London Math. Soc. (2) 61 (2000), no. 3, 872 – 884. · Zbl 0959.47016 · doi:10.1112/S0024610700008875
[8] Raymond C. Roan, Composition operators on a space of Lipschitz functions, Rocky Mountain J. Math. 10 (1980), no. 2, 371 – 379. · Zbl 0433.46023 · doi:10.1216/RMJ-1980-10-2-371
[9] Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. · Zbl 0791.30033
[10] Hasi Wulan, Dechao Zheng, and Kehe Zhu, Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3861 – 3868. · Zbl 1194.47038
[11] J. Xiao, Composition operators associated with Bloch-type spaces, Complex Variables Theory Appl. 46 (2001), no. 2, 109 – 121. · Zbl 1044.47020
[12] Ke He Zhu, Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 139, Marcel Dekker, Inc., New York, 1990.
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