Compact operators on the weighted Bergman space A1(ψ)

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We show that a bounded linear operator $S$ on the weighted Bergman space $A^1(\psi )$ is compact and the predual space $A_0(\varphi )$ of $A^1(\psi )$ is invariant under $S^\ast $ if and only if $Sk_z \rightarrow 0$ as $z\rightarrow \partial D$, where $k_