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

Yusuf Abu Muhanna and El-Bachir Yallaoui

International Journal of Mathematics and Mathematical Sciences, 2004, vol. 2004, 1-9

Abstract:

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

Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:321430

DOI: 10.1155/S0161171204310057

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