Toeplitz Operators On Weighted Bergman Spaces Of The Unit Ball

In this paper, we investigate the boundedness and compactness of Toeplitz op-erators on weighted Bergman space of the unit ball. We construct different weightedBergman spaces by adding different weight, then we obtain some equivalent descrip-tions of the compactness of Toeplitz operators with different symbols on differentweighted Bergman spaces of the unit ball. In this paper, we characterize the compact-ness of Toeplitz operators by studying Berezin transform of the Toeplitz operators onboundary of the unit ball, then we get the conclusion that the Toeplitz operator S iscompact if and only if the Berezin transform of S vanishes at the boundary of the unitball.In the second chapter, we study the compactness of Toeplitz operators with sym-bol L~∞(B_n) on weighted Bergman space A~p(B_n,dV_φ~p)(1 < p <∞), and we showsome equivalent descriptions of the compactness of a finite sum of finite products ofToeplitz operators with symbol L~∞(B_n).In the third chapter, we study the boundedness and compactness of Toeplitz op-erators with symbol L~1(B_n) on weighted Bergman space A_α~2(B_n,dV_α), and we obtainsome equivalent descriptions of the boundedness and compactness of Toeplitz opera-tor with symbol L~1(B_n).In the fourth chapter, some estimates about the norm and essential norm ofToeplitz operators on A~p(B_n,dV )(1 < p <∞) with symbol in BT are obtained.