Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T00:39:41.318Z Has data issue: false hasContentIssue false
  • English
  • Français

On Bounded and Compact Composition Operators in Polydiscs

Published online by Cambridge University Press:  20 November 2018

F. Jafari
F. Jafari*
Affiliation:
Bowdoin College, Department of Mathematics, Brunswick, Maine 04011, U. S. A.
Rights & Permissions [Opens in a new window]

Extract

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.

Recently MacCluer and Shapiro [6] have characterized the compact composition operators in Hardy and weighted Bergman spaces of the disc, and MacCluer [5] has made an extensive study of these opertors in the unit ball of Cn. Angular derivatives and Carleson measures have played an essential role in these studies. In this article we study composition operators in poly discs and characterize those operators which are bounded or compact in Hardy and weighted Bergman spaces. In addition to Carleson measure theorems resembling those of [5], [6], we give necessary and sufficient conditions satisfied by the maps inducing bounded or compact composition operators. We conclude by considering the role of angular derivatives on the compactness question explicitly.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 5 , 01 October 1990 , pp. 869 - 889
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Garnett, J., Bounded analytic functions, (Academic Press, New York, 1980).Google Scholar
2. Jafari, F., The extremal problem of Schwarz and the Julia-Caratheodory theorems in polydiscs, Preprint.Google Scholar
3. Jafari, F., Carleson measures in Hardy and weighted Bergman spaces of polydiscs , Proc. A. M. S. (1990), to appear.Google Scholar
4. Jafari, F., Composition operators in polydiscs, (Dissertation, University of Wisconsin, Madison, 1989).Google Scholar
5. MacCluer, B.D., Compact composition operators on fP(BN), Michigan Math. J., 32 (1985), 237–248.Google Scholar
6. MacCluer, B.D., Shapiro, J.H., Angular derivatives and compact composition operators on the Hardyand Bergman spaces, Can. J. Math., 38 (1986), 878–906.Google Scholar
7. Rudin, W., Function theory in polydiscs, (Benjamin, New York, 1969).Google Scholar
8. Ryff, J.V., Subordinate HP functions, Duke Math. J., 33 (1966), 347–354.Google Scholar
9. Sharma, S.D., Singh, R.K., Composition operators and several complex variables, Bull. Austral. Math. Soc., 23 (1981), 237–247.Google Scholar
10. Shapiro, J.H., Taylor, P.D., Compact, nuclear, and Hilbert-Schmidt composition operators, Indiana Univ. Math. J., 23 (1973), 471–496.Google Scholar