We obtain a new boundedness criterion for the difference of two composition operators from a weighted Bergman space \(A^p_{\alpha }\) into a Lebesgue space \(L^q(\mu )\), where \(0< q < p\) and \(\alpha > -1\). As a consequence, we provide a direct proof that such a bounded difference operator is necessarily compact. We also characterize compact differences of composition operators from \(A^p_{\alpha }\) into \(A^q_{\beta }\) explicitly for \(0 < p \le q\) and \(\alpha , \beta > -1\).

中文翻译：

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加权Bergman空间上复合算子的差异。

我们为两个合成算子的差值从加权Bergman空间\（A ^ p _ {\ alpha} \）到Lebesgue空间\（L ^ q（\ mu）\）中获得了一个新的有界性准则，其中\（0 < q <p \）和\（\ alpha> -1 \）。结果，我们提供了直接证明，该有界差分算子必定是紧凑的。我们还针对\（0 <p \ le q \）和\（\ alpha，显式地描述了组成运算符从\（A ^ p _ {\ alpha} \）到\（A ^ q _ {\ beta} \）的紧凑差异。\ beta> -1 \）。