A class of linear operators on Bergman spaces

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