In this article we study the boundedness and the compactness for an operator related to composition operators, differentiation operators and multiplication operators from Bergman type spaces to Bers type spaces.

The boundedness and compactness of the generalized weighted composition operator on weighted Bergman spaces are discussed.

Let ????_n be the unit ball of ℂⁿ, H(????_n) the space of all holomorphic functions in ????_n. Let u????H(????_n) and α be a holomorphic self‐map of ????_n. For f????H(????_n), the weigthed composition operator uC_α is defined by (uC_αf)(z) = u(z)f(α(z)), z????????_n. The boundedness and compactness of the weighted composition operator on some weighted spaces on the unit ball are studied in this paper.

New criteria for the boundedness and the compactness of the generalized weighted composition operators from Bloch spaces into Bers-type spaces are given in this paper.

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