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A class of Hausdorff–Berezin operators on the unit ball

Part of: Special classes of linear operators Linear function spaces and their duals Integral, integro-differential, and pseudodifferential operators

Published online by Cambridge University Press:  23 October 2024

Alexey Karapetyants Open the ORCID record for Alexey Karapetyants [Opens in a new window]  and
Adolf Mirotin
Alexey Karapetyants*
Affiliation:
Institute of Mathematics, Mechanics and Computer Sciences & Regional Mathematical Center, Southern Federal University, Rostov-on-Don, 344090, Russia
Adolf Mirotin
Affiliation:
Department of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Gomel, 246019, Belarus and Regional Mathematical Center, Southern Federal University, Rostov-on-Don, 344090, Russia e-mail: amirotin@yandex.ru
*
e-mail: karapetyants@gmail.com
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Abstract

The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

Keywords

Hausdorff operatorsBerezin transformHaar measureMöbius group

MSC classification

Primary: 47G10: Integral operators 47B38: Operators on function spaces (general) 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 4 , December 2024 , pp. 1069 - 1080
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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