Let φ and ψ be holomorphic maps on the open unit disk ? such that φ (?) ⊂ ? and H (?) be the space of holomorphic functions on ?. For a non-negative integer n, define a linear operator Dφ,ψn$D_{varphi ,psi }^n $ as Dφ,ψnf=ψ⋅(f(n)οφ)$D_{varphi ,psi }^n f = psi cdot left( {f^{(n)} circ varphi } right)$, f ∈ H (?). In this paper, we characterize boundedness and compactness of Dφ,ψn$D_{varphi ,psi }^n $ on the Bergman space ? 2 . We also compute the upper and lower bounds of essential norm of this operator on the Bergman space.
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