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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 55, Pages 2937-2945
http://dx.doi.org/10.1155/S0161171204310057

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 Hindawi Publishing Corporation. 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

The analytic self-map of the unit disk D, φ is said to induce a composition operator Cφ from the Banach space X to the Banach space Y if Cφ(f)=fφY for all fX. For zD and α>0, the families of weighted Cauchy transforms Fα are defined by f(z)=TKxα(z)dμ(x), where μ(x) is complex Borel measure, x belongs to the unit circle T, and the kernel Kx(z)=(1x¯z)1. In this paper, we will explore the relationship between the compactness of the composition operator Cφ acting on Fα and the complex Borel measures μ(x).