Compact Operators On Weighted Bergman Spaces Of The Unit Ball

In this paper, we define an operator S: and get a sufficient and necessary condition for S to be compact on weighted Berglnan space A_φ^p(B)(p＞1). That isTheorem 3.1 Suppose 1＜p＜∞,α＞(p-1)b and S is a bounded operator on A_φ^p(B) such that for some k＞M(M can be saw in page 13), then S is compact on A_φ^p(B) if and only if S(z)→O as z→(?)B.When n is finite and positive, and S is a finite sum of operators of the form T_μ1T_μ2...T_μn, where eachμ_j∈L^∞(B) and T_μj is the Toeplitz operator with symbolμ_j, we get a sufficient and necessary condition for S to be compact. That isTheorem 3.2 Suppose 1＜p＜∞and S is a finite sum of operators of the form T_μ1T_μ2...T_μn, where eachμ_j∈L^∞(B). Then S is compact on A_φ^p(B)if and only if S(z)→O as z→(?)B.