Abstract and Applied Analysis
Volume 2007 (2007), Article ID 48478, 13 pages
doi:10.1155/2007/48478
Research Article

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

Songxiao Li1,2 and Stevo Stević3

1Department of Mathematics, Shantou University, Shantou 515063, GuangDong, China
2Department of Mathematics, JiaYing University, Meizhou 514015, GuangDong, China
3Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia

Received 9 February 2007; Revised 16 April 2007; Accepted 3 May 2007

Academic Editor: Ferhan Merdivenci Atici

Copyright © 2007 Songxiao Li and Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Dn be the unit polydisc of ℂn, ϕ(z)=(ϕ1(z),…,ϕn(z)) be a holomorphic self-map of Dn, and ψ(z) a holomorphic function on Dn. Let H(Dn) denote the space of all holomorphic functions with domain Dn, H∞(Dn) the space of all bounded holomorphic functions on Dn, and B(Dn) the Bloch space, that is, B(Dn)={f∈H(Dn)|‖f‖B=|f(0)|+supz∈Dn∑k=1n|(∂f/∂zk)(z)|(1−|zk|2)<+∞}. We give necessary and sufficient conditions for the weighted composition operator ψCϕ induced by ϕ(z) and ψ(z) to be bounded and compact from H∞(Dn) to the Bloch space B(Dn).