In this article, we study the complex symmetry of composition operators \(C_{\phi }f=f\circ \phi \) induced on the weighted Bergman spaces \(A^2_{\beta }(\mathbb {D}),\) by analytic self-maps of the unit disk. One of our main results shows that if \(C_\phi \) is complex symmetric then \(\phi \) must fix a point in \(\mathbb {D}\). From this, we prove that if \(\phi \) is neither constant nor an elliptic automorphism of \(\mathbb {D}\) and \(C_{\phi }\) is complex symmetric then \(C_{\phi }\) and \(C_{\phi }^*\) are cyclic operators. Moreover, by assuming \(\phi \) is an elliptic automorphism of \(\mathbb {D}\) which not a rotation and \(\beta \in \mathbb {N},\) we show that \(C_{\phi }\) is not complex symmetric whenever \(\phi \) has order greater than \(2(3+\beta ).\)

中文翻译：

---

加权Bergman空间上可逆复合算子的复对称性。

在本文中，我们研究在加权Bergman空间\（A ^ 2 _ {\ beta}（\ mathbb {D}）上诱导的合成算子\（C _ {\ phi} f = f \ circ \ phi \）的复杂对称性，\）通过分析单位磁盘的自映射。我们的主要结果之一表明，如果\（C_ \ phi \）是复对称的，那么\（\ phi \）必须在\（\ mathbb {D} \）中固定一个点。由此证明，如果\（\ phi \）既不是常数也不是\（\ mathbb {D} \）和\（C _ {\ phi} \）的椭圆自同构是对称的，则\（C _ {\ phi } \）和\（C _ {\ phi} ^ * \）是循环运算符。此外，通过假设\（\ phi \）是\（\ mathbb {D} \）的椭圆自同构而不是旋转，而\（\ beta \ in \ mathbb {N}，\）是椭圆自同构，我们证明\（C _ {\ phi} \）并非复杂对称\（\ phi \）的阶数大于\（2（3+ \ beta）。\）