Compact Bergman Type Operators

We characterize the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}–Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^q$$\end{document} compactness of Bergman type operators, which are singular integral operators induced by the modified Bergman kernel on the complex unit ball. Moreover, we characterize Schatten class and Macaev class Bergman type integral operators on the Lebesgue space and the Bergman space by the methods of spectral estimates and operator inequalities; we also give a relatively intrinsic characterization by introducing a concept of dimension of a compact operator. The Dixmier trace of Bergman type operators is also calculated.