On the compactness of Bergman-type integral operators

2020/04/28 by Lijia Ding, Ding, Lijia, Junmei Fan +1
Mathematics · #32A25 #42B20 #47B05 #47B10 #Advanced Harmonic Analysis Research #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2004.13635

openalex publication_date 2020/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Bergman-type integral operators are classical operators in complex analysis and operator theory. Recently, the first author and his collaborator \citeDiW completely characterized the L^p-L^q boundedness of Bergman-type integral operators K_α,K_α⁺ and the L^p-L^q compactness of K_α on the unit ball. In this paper, we will use a substantially new method to completely characterize the L^p-L^q compactness of K_α⁺, but also prove that the L^p-L^q compactness of operators K_α,K_α⁺ is in fact equivalent. Moreover, we completely characterize Schatten class and Macaev class Bergman-type integral operator K_α on L² space and Bergman space via inequalities related to the dimension of the unit ball, and we also give an intrinsic characterization by introducing the concept of Hausdorff dimension of compact operators. The Dixmier trace of K_α are also calculated in this paper.

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