A class of linear operators on Bergman spaces

We study the boundedness of the linear operator $S$ on $L^{p}_{a}(dA_{\alpha})$ $(0<p<\infty)$. In particular, we obtain a sufficient and necessary condition for the compactness of the linear operator $S$ on $L^{p}_{a}(dA_{\alpha})$ $(1<p<\infty)$. Our results weaken the assumptions of earlier results of J. Miao and D. Zheng in a certain sense.