  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.
