International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 55, Pages 2937-2945
Absolutely continuous measures and compact composition operator on spaces of Cauchy transforms
1Department of Mathematics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates
2College of Arts and Sciences, Abu Dhabi University, P.O. Box 1790, Al Ain, United Arab Emirates
Received 5 October 2003
Copyright © 2004 Yusuf Abu Muhanna and El-Bachir Yallaoui. 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.
The analytic self-map of the unit disk , is said to induce a composition operator from the Banach space to the Banach space if for all . For and , the families of weighted Cauchy transforms are defined by , where is complex Borel measure, belongs to the unit circle , and the kernel . In this paper, we will explore the relationship between the compactness of the composition operator acting on and the complex Borel measures .