This paper finds some lower and upper bounds for the essential norm of the weighted composition operator from
$\alpha $-Bloch spaces to the weighted-type space ${H}_{\mu }^{\infty }$ on the unit ball for the case $\alpha \geq 1$.
References
A. L. Shields and D. L. Williams, ``Bonded projections, duality, and multipliers in spaces of analytic functions,'' Transactions of the American Mathematical Society, vol. 162, pp. 287--302, 1971.
S. Stević, ``Weighted composition operators between mixed norm spaces and $H_\alpha ^\infty $ spaces in the unit ball,'' Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007.
X. Tang, ``Weighted composition operators between Bers-type spaces and Bergman spaces,'' Applied Mathematics. A Journal of Chinese Universities B, vol. 22, no. 1, pp. 61--68, 2007.
D. D. Clahane and S. Stević, ``Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,'' Journal of Inequalities and Applications, vol. 2006, Article ID 61018, 11 pages, 2006.
C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.
K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005.
X. Fu and X. Zhu, ``Weighted composition operators on some weighted spaces in the unit ball,'' Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008.
P. Gorkin and B. D. MacCluer, ``Essential norms of composition operators,'' Integral Equations and Operator Theory, vol. 48, no. 1, pp. 27--40, 2004.
S. Li and S. Stević, ``Composition followed by differentiation between Bloch type spaces,'' Journal of Computational Analysis and Applications, vol. 9, no. 2, pp. 195--205, 2007.
S. Li and S. Stević, ``Weighted composition operators from Bergman-type spaces into Bloch spaces,'' Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 117, no. 3, pp. 371--385, 2007.
S. Li and S. Stević, ``Weighted composition operators from $\alpha $-Bloch space to $H^\infty $ on the polydisc,'' Numerical Functional Analysis and Optimization, vol. 28, no. 7-8, pp. 911--925, 2007.
S. Li and S. Stević, ``Weighted composition operators from $H^\infty $ to the Bloch space on the polydisc,'' Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007.
S. Li and S. Stević, ``Generalized composition operators on Zygmund spaces and Bloch type spaces,'' Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1282--1295, 2008.
S. Li and S. Stević, ``Weighted composition operators between $H^\infty $ and $\alpha $-Bloch space čommentComment on ref. [11?]: Please update the information of this reference, if possible. in the unit ball,'' to appear in Taiwanese Journal of Mathematics.
L. Luo and S.-I. Ueki, ``Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of $\mathbbC^n$,'' Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 88--100, 2007.
B. D. MacCluer and R. Zhao, ``Essential norms of weighted composition operators between Bloch-type spaces,'' The Rocky Mountain Journal of Mathematics, vol. 33, no. 4, pp. 1437--1458, 2003.
A. Montes-Rodríguez, ``Weighted composition operators on weighted Banach spaces of analytic functions,'' Journal of the London Mathematical Society, vol. 61, no. 3, pp. 872--884, 2000.
S. Ohno, ``Weighted composition operators between $H^\infty $ and the Bloch space,'' Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 555--563, 2001.
S. Ohno, K. Stroethoff, and R. Zhao, ``Weighted composition operators between Bloch-type spaces,'' The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 191--215, 2003.
J. Shi and L. Luo, ``Composition operators on the Bloch space of several complex variables,'' Acta Mathematica Sinica, vol. 16, no. 1, pp. 85--98, 2000.
S. Stević, ``Composition operators between $H^\infty $ and $\alpha $-Bloch spaces on the polydisc,'' Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 457--466, 2006.
S. Stević, ``Norm of weighted composition operators from Bloch space to $H_\mu ^\infty $ on the unit ball,'' Ars Combinatoria, vol. 88, pp. 125--127, 2008.
S.-I. Ueki and L. Luo, ``Compact weighted composition operators and multiplication operators between Hardy spaces,'' Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008.
X. Zhu, ``Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces,'' Indian Journal of Mathematics, vol. 49, no. 2, pp. 139--150, 2007.
X. Zhu, ``Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces,'' Integral Transforms and Special Functions, vol. 18, no. 3, pp. 223--231, 2007.
S. Stević, ``On an integral operator on the unit ball in $\mathbbC^n$,'' Journal of Inequalities and Applications, vol. 2005, no. 1, pp. 81--88, 2005.
S. Li and S. Stević, ``Riemann-Stieltjes-type integral operators on the unit ball in $\mathbbC^n$,'' Complex Variables and Elliptic Equations, vol. 52, no. 6, pp. 495--517, 2007.
S. Stević, ``Boundedness and compactness of an integral operator on a weighted space on the polydisc,'' Indian Journal of Pure and Applied Mathematics, vol. 37, no. 6, pp. 343--355, 2006.
S. Stević, ``Boundedness and compactness of an integral operator in a mixed norm space on the polydisk,'' Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 48, no. 3, pp. 694--706, 2007. \endthebibliography