×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[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
[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
[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
[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
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.