Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 90742, 22 pages
doi:10.1155/JIA/2006/90742
The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications
Department of Mathematics, Tianjin University, Tianjin 300072, China
Received 27 December 2005; Revised 26 June 2006; Accepted 22 July 2006
Copyright © 2006 Zehua Zhou and Yan Liu. 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 Un be the unit polydisc of ℂn and
φ=(φ1,…,φn) a holomorphic self-map of Un. ℬp(Un), ℬ0p(Un), and B0*p(Un) denote the p-Bloch space, little p-Bloch space,
and little star p-Bloch space in the unit polydisc Un, respectively, where p,q>0. This paper gives the estimates of the
essential norms of bounded composition operators Cφ induced
by φ between ℬp(Un) (ℬ0p(Un)
or B0*p(Un)) and ℬq(Un) (ℬ0q(Un)
or B0*p(Un)). As their applications, some necessary and
sufficient conditions for the (bounded) composition operators
Cφ to be compact from ℬp(Un) (ℬ0p(Un)
or ℬ0∗p(Un)) into ℬq(Un) (ℬ0q(Un)
or B0*p(Un)) are obtained.