On the compactness of Bergman-type integral operators

TL;DR

This paper provides a comprehensive characterization of the compactness, Schatten class, and Macaev class properties of Bergman-type integral operators on the unit ball, introducing new methods and concepts like Hausdorff dimension.

Contribution

It introduces a new method to characterize the $L^p$-$L^q$ compactness of $K_eta^+$ and establishes the equivalence of compactness for $K_eta$ and $K_eta^+$, along with intrinsic operator characterizations.

Findings

Complete characterization of $L^p$-$L^q$ compactness of $K_eta^+$.

Equivalence of $L^p$-$L^q$ compactness for $K_eta$ and $K_eta^+$.

Calculation of Dixmier trace for $K_eta$.

Abstract

Bergman-type integral operators are classical operators in complex analysis and operator theory. Recently, the first author and his collaborator \cite{DiW} completely characterized the $L^p$-$L^q$ boundedness of Bergman-type integral operators $K_\alpha,K_\alpha^+$ and the $L^p$-$L^q$ compactness of $K_\alpha$ on the unit ball. In this paper, we will use a substantially new method to completely characterize the $L^p$-$L^q$ compactness of $K_\alpha^+,$ but also prove that the $L^p$-$L^q$ compactness of operators $K_\alpha,K_\alpha^+$ is in fact equivalent. Moreover, we completely characterize Schatten class and Macaev class Bergman-type integral operator $K_\alpha$ on $L^2$ 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 $K_\alpha$ are also calculated in this paper.