Open Access

The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications

Journal of Inequalities and Applications20062006:90742

DOI: 10.1155/JIA/2006/90742

Received: 27 December 2005

Accepted: 22 July 2006

Published: 29 October 2006

Abstract

Let be the unit polydisc of and a holomorphic self-map of . , and denote the -Bloch space, little -Bloch space, and little star -Bloch space in the unit polydisc , respectively, where . This paper gives the estimates of the essential norms of bounded composition operators induced by between ( or ) and ( or ). As their applications, some necessary and sufficient conditions for the (bounded) composition operators to be compact from ( or ) into ( or ) are obtained.

[12345678910111213]

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin University

References

  1. Madigan K, Matheson A: Compact composition operators on the Bloch space. Transactions of the American Mathematical Society 1995,347(7):2679–2687. 10.2307/2154848MathSciNetView ArticleMATHGoogle Scholar
  2. Montes-Rodríguez A: The essential norm of a composition operator on Bloch spaces. Pacific Journal of Mathematics 1999,188(2):339–351. 10.2140/pjm.1999.188.339MathSciNetView ArticleMATHGoogle Scholar
  3. Shapiro JH: The essential norm of a composition operator. Annals of Mathematics 1987,125(2):375–404. 10.2307/1971314MathSciNetView ArticleMATHGoogle Scholar
  4. Shi JH, Luo L: Composition operators on the Bloch space of several complex variables. Acta Mathematica Sinica. English Series 2000,16(1):85–98. 10.1007/s101149900028MathSciNetView ArticleMATHGoogle Scholar
  5. Timoney RM: Bloch functions in several complex variables. I. The Bulletin of the London Mathematical Society 1980,12(4):241–267. 10.1112/blms/12.4.241MathSciNetView ArticleMATHGoogle Scholar
  6. Timoney RM: Bloch functions in several complex variables. II. Journal für die reine und angewandte Mathematik 1980, 319: 1–22.MathSciNetMATHGoogle Scholar
  7. Zhou Z: Composition operators on the Lipschitz space in polydiscs. Science in China. Series A 2003,46(1):33–38.MathSciNetView ArticleMATHGoogle Scholar
  8. Zhou Z, Shi JH: Compact composition operators on the Bloch space in polydiscs. Science in China. Series A 2001,44(3):286–291. 10.1007/BF02878708MathSciNetView ArticleMATHGoogle Scholar
  9. Zhou Z, Shi JH: Composition operators on the Bloch space in polydiscs. Complex Variables 2001,46(1):73–88.MathSciNetView ArticleMATHGoogle Scholar
  10. Zhou Z, Shi JH: Compactness of composition operators on the Bloch space in classical bounded symmetric domains. The Michigan Mathematical Journal 2002,50(2):381–405. 10.1307/mmj/1028575740MathSciNetView ArticleMATHGoogle Scholar
  11. Zhou Z, Shi JH: The essential norm of a composition operator on the Bloch space in polydiscs. Chinese Annals of Mathematics. Series A 2003,24(2):199–208. Chinese Journal of Contemporary Mathematics 24 (2003), no. 2, 175–186 Chinese Journal of Contemporary Mathematics 24 (2003), no. 2, 175–186MathSciNetMATHGoogle Scholar
  12. Zhou Z, Zeng HG: Composition operators between-Bloch space and-Bloch space in the unit ball. Progress in Natural Science 2003,13(3):233–236.MathSciNetMATHGoogle Scholar
  13. Zhu K: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Volume 226. Springer, New York; 2005:x+271.Google Scholar

Copyright

© Z. Zhou and Y. Liu. 2006

This article is published under license to BioMed Central Ltd. 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.