The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ A α p with $$p \in (0, \infty )$$ p ∈ ( 0 , ∞ ) and $$ \alpha \in (-1, \infty )$$ α ∈ ( - 1 , ∞ ) . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

中文翻译：

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伯格曼空间上（加权）复合算子空间的路径分量

已经在单位圆盘上的各种函数空间（例如哈代空间、有界全纯函数空间、具有超范数的全纯函数的加权巴拿赫空间、希尔伯特伯格曼空间）上研究了（加权）复合算子集的拓扑结构。在本文中，我们考虑所有 Bergman 空间的这个问题 $$A_{\alpha }^p$$ A α p with $$p \in (0, \infty )$$ p ∈ ( 0 , ∞ ) and $$ \ alpha \in (-1, \infty )$$ α ∈ ( - 1 , ∞ ) 。在这个设置中，我们建立了两个组合算子在组合算子空间中线性连接的标准；此外，对于加权组合算子的空间，我们证明了紧凑加权组合算子的集合是路径连通的，但它不是一个组件。