Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T19:59:20.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear Combinations of Composition Operators on the Bloch Spaces

Published online by Cambridge University Press:  20 November 2018

Takuya Hosokawa ,
Pekka J. Nieminen  and
Shûichi Ohno
Takuya Hosokawa
Affiliation:
Faculty of Engineering, Ibaraki University, Hitachi, Ibaraki 316-8511, Japan email: [email protected]
Pekka J. Nieminen
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland email: [email protected]
Shûichi Ohno
Affiliation:
Nippon Institute of Technology, Miyashiro, Minami-Saitama 345-8501, Japan email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterize the compactness of linear combinations of analytic composition operators on the Bloch space. We also study their boundedness and compactness on the little Bloch space.

Keywords

47B3330D4547B07composition operatorcompactnessBloch space
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 4 , 01 August 2011 , pp. 862 - 877
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Anderson, J. M., Clunie, J., and Pommerenke, Ch., On Bloch functions and normal functions. J. Reine Angew. Math. 270(1974), 12–37.Google Scholar
[2] Cowen, C. C. and MacCluer, B. D., Composition operators on spaces of analytic functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.Google Scholar
[3] Garnett, J. B., Bounded analytic functions. Graduate Texts in Mathematics, 236, Springer, New York, 2007.Google Scholar
[4] Ghatage, P. and Zheng, D., Hyperbolic derivatives and generalized Schwarz-Pick estimates. Proc. Amer. Math. Soc. 132(2004), no. 11, 3309–3318. doi:10.1090/S0002-9939-04-07479-9Google Scholar
[5] Gorkin, P. and Mortini, R., Norms and essential norms of linear combinations of endomorphisms. Trans. Amer. Math. Soc. 358(2006), no. 2, 553–571. doi:10.1090/S0002-9947-04-03633-5Google Scholar
[6] Hedenmalm, H., Korenblum, B., and Zhu, K., Theory of Bergman spaces. Graduate Texts in Mathematics, 199, Springer-Verlag, New York, 2000.Google Scholar
[7] Hosokawa, T., Differences of weighted composition operators on the Bloch spaces. Complex Anal. Oper. Theory 3(2009), no. 4, 847–866. doi:10.1007/s11785-008-0062-1Google Scholar
[8] Hosokawa, T. and Ohno, S., Topological structure of the sets of composition operators on the Bloch spaces. J. Math. Anal. Appl. 314(2006), no. 2, 736–748. doi:10.1016/j.jmaa.2005.04.080Google Scholar
[9] Hosokawa, T. and Ohno, S., Differences of composition operators on the Bloch spaces. J. Operator Theory 57(2007), no. 2, 229–242.Google Scholar
[10] Izuchi, K. J. and Ohno, S., Linear combinations of composition operators on H1. J. Math. Anal. Appl. 338(2008), no. 2, 820–839. doi:10.1016/j.jmaa.2007.05.061Google Scholar
[11] Kriete, T. and Moorhouse, J., Linear relations in the Calkin algebra for composition operators. Trans. Amer. Math. Soc. 359(2007), no. 6, 2915–2944. doi:10.1090/S0002-9947-07-04166-9Google Scholar
[12] Madigan, K. and Matheson, A., Compact composition operators on the Bloch space. Trans. Amer. Math. Soc. 347(1995), no. 7, 2679–2687. doi:10.2307/2154848Google Scholar
[13] Nieminen, P. J., Compact differences of composition operators on Bloch and Lipschitz spaces. Comput. Methods Funct. Theory 7(2007), no. 3, 325–344.Google Scholar
[14] Zhu, K., Operator theory in function spaces. Second ed., Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007.Google Scholar