International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2004 / Article

Open Access

Volume 2004 |Article ID 321430 | https://doi.org/10.1155/S0161171204310057

Yusuf Abu Muhanna, El-Bachir Yallaoui, "Absolutely continuous measures and compact composition operator on spaces of Cauchy transforms", International Journal of Mathematics and Mathematical Sciences, vol. 2004, Article ID 321430, 9 pages, 2004. https://doi.org/10.1155/S0161171204310057

Absolutely continuous measures and compact composition operator on spaces of Cauchy transforms

Received05 Oct 2003

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).

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.


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