Following the idea of the paper by Contreras and Hernández-Díaz, we first give the characterization of the boundedness of the weighted composition operator \(W_{\varphi ,\phi }\) on \(B^p\). Then, we investigate the boundedness and the compactness of composition operators \(C_{\varphi}\) on \(B^p\) and study the spectrum of the multiplication operator \(M_{\phi}\) on \(B^p\), as well. Finally, motivated by the paper by Čučković and Paudyal, we describe the relationships between the invariant subspaces of \(M_{z}\) on \(B^p_{0}\) and T on \(A^p\), where T is the sum of multiplication operator and Volterra operator. Moreover, we provide some Beurling-type invariant subspaces of \(M_{z}\) on \(B^p\) and \(B^p_0\), respectively.

遵循Contreras和Hernández-Díaz提出的论文的思想，我们首先给出\（B ^ p \）上加权合成算子\（W _ {\ varphi，\ phi} \）的有界性的刻画。然后，我们研究了有界和组成运营商的紧凑\（C _ {\ varphi} \）上\（B ^ P \）和研究乘法运算符的频谱\（M _ {\披} \）上\（B ^ p \）。最后，根据Čučković和Paudyal的论文，我们描述了\（B ^ p_ {0} \）上\（M_ {z } \）和\（A ^ p \）上T的不变子空间之间的关系，其中Ť是乘法运算符和Volterra运算符的总和。此外，我们分别在\（B ^ p \）和\（B ^ p_0 \）上提供了\（M_ {z} \）的一些Beurling型不变子空间。