International Journal of Pure and Applied Mathematics Volume 78 No. 2 2012, 161-183 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu P A ijpam.eu WEIGHTED COMPOSITION OPERATORS ON WEIGHTED BERGMAN SPACES OF THE UNIT BALL Waleed Al-Rawashdeh 1 § , Sivaram K. Narayan 2 1 Department of Mathematical Sciences Montana Tech of The University of Montana 1300, West Park Street, Butte, Montana, 59701, USA 2 Department of Mathematics Central Michigan University Washington Street, Mount Pleasant, Michigan, 48859, USA Abstract: Suppose ϕ is an analytic self map of B n and ψ is analytic on B n . Then a weighted composition operator induced by ϕ with weight ψ is given by (W ψ,ϕ f )(z)= ψ(z)f (ϕ(z)) for z in B n and f analytic on B n . Given W ψ,ϕ : A p α (B n ) → A q β (B n ) we characterize boundedness and compactness of W ψ,ϕ , where 0 <p,q< ∞ and −1 < α,β < ∞. We also characterize the Schatten p-class weighted composition operators S p ( A 2 α (B n ) ) for 0 <p< ∞ and −1 <α< ∞. AMS Subject Classification: 47B38, 32A35, 32A36 Key Words: weighted composition operators, Bergman spaces, Toeplitz op- erators, the Berezin transform, Schatten p-class, unit ball of C n 1. Introduction Let B n denote the open unit ball in the complex n-dimensional Euclidean space C n . Let H (B n ) denote the space of all analytic functions in B n . For α> −1, Received: December 27, 2011 c  2012 Academic Publications, Ltd. url: www.acadpubl.eu § Correspondence author