Abstract and Applied Analysis
Volume 2007 (2007), Article ID 48478, 13 pages
doi:10.1155/2007/48478
Weighted Composition Operators from H∞ to the Bloch Space on the Polydisc
Songxiao Li1
and Stevo Stević2
1Department of Mathematics, Shantou University, Shantou 515063, GuangDong, China
2Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia
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).