Weighted Composition Operators from α-Bloch Space to H^∞ on the Polydisc

Abstract

Let ⁿ be the unit polydisc of ℂⁿ, ϕ(z) = (ϕ₁(z),…, ϕ_n(z)) be a holomorphic self-map of ⁿ, and ψ(z) a holomorphic function on ⁿ. Let H^∞(ⁿ) be the space of all bounded holomorphic functions on ⁿ and ℬ^α(ⁿ) the α-Bloch space. Necessary and sufficient conditions for the weighted composition operator ψC_ϕ induced by ϕ(z) and ψ(z) to be bounded and compact from ℬ^α(ⁿ) to H^∞(ⁿ) are given in this paper. One of our results corrects and extends Theorem 1 in the paper by S. Ohno, Weighted composition operators between H^∞ and the Bloch space, Taiwanese J. Math. 2001; 5:555–563.

Keywords:

• Bloch space
• Boundedness
• Compactness
• Weighted composition operator

AMS Subject Classification:

• 47B33
• 47B38

ACKNOWLEDGMENTS

The first author is supported in part by the NNSF of China (no. 10671115), grants from Specialized Research Fund for the doctoral program of Higher Education (no. 20060560002), and NSF of Guangdong Province (no. 06105648).