On the essential norm and the compactness of the weighted composition operator on Bergman spaces

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In this research, we considered the Bergman space which consists of holomorphic functions on the unit ball and the weighted composition operator on these spaces. Our purposes are to estimate the essential norm for this type of operator and to characterize the compactness of it. In 2008 we had estimated the essential norm of the weighted composition operator on the Bergman space or the Hardy space. In 2009 we had estimate the essential norm of the composition operator on the Bergman space in terms of the condition on the angular derivative of φ at the boundary point. Also we characterized the compactness of the weighted composition operator from the Bergman space into the Hardy space and estimated the essential norm of it.