Abstract
First of all, we obtain a necessary and sufficient condition for a certain operator to be compact on . Next, we give a short proof for Proposition 2.5 which was proved by MacCluer, Narayan and Weir. Then, we characterize the essentially normal weighted composition operators on the weighted Bergman spaces , when is not an automorphism and is continuous at a point which has a finite angular derivative. After that we find some non-trivially essentially normal weighted composition operators, when is not an automorphism. In the last section, for and , we characterize the essentially normal weighted composition operators on and investigate some essentially normal weighted composition operators on and . Finally, we find some non-trivially essentially normal weighted composition operators on and , when and .
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