Weighted Composition Operators from <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:mrow></mml:math> to the Bloch Space on the Polydisc

Li, Songxiao; Stevic, Stevo

doi:https://doi.org/10.1155/2007/48478

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2007 | Article ID 048478 | https://doi.org/10.1155/2007/48478

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

Songxiao Li^1,2and Stevo Stevic³

Academic Editor: Ferhan Merdivenci Atici

Received09 Feb 2007

Revised16 Apr 2007

Accepted03 May 2007

Published19 Jun 2007

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).

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Copyright

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.

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