AI 中文总结
本文证明内函数模自同构后复合与加权Bergman空间循环子空间之间的对应为双射，核心方法是通过Liouville映射建立互逆关系。
AI 中文摘要
我们研究了文献[critical-structures]中提出的内函数模单位圆盘自同构后复合与加权Bergman空间$A^2_1$的循环子空间之间的对应关系。该对应将内函数$I$映至不变子空间$[I']$，反之，对非零函数$H \in A^2_1$，赋予与Gauss曲率方程$\Delta u = |H|^2 e^{2u}$的典范解相关联的Liouville映射$I_H$。我们证明了$I'_H$生成与$H$相同的循环子空间。结合文献[critical-structures]中的结果，这表明这两个映射互为逆映射，因此对应$I \to [I']$是一个双射。
英文摘要
We study the correspondence proposed in \cite{critical-structures} between inner functions modulo post-compositions with automorphisms of the unit disk and cyclic subspaces of the weighted Bergman space $A^2_1$. The correspondence sends an inner function $I$ to the invariant subspace $[I']$, and in the opposite direction, assigns to a non-zero function $H \in A^2_1$ the Liouville map $I_H$ associated to the canonical solution of the Gauss curvature equation $Δu = |H|^2 e^{2u}$. We prove that $I'_H$ generates the same cyclic subspace as $H$. Combined with the results in \cite{critical-structures}, this shows that these two mappings are inverses of one another, and hence the correspondence $I \to [I']$ is a bijection.
Comments11 pages
