论文标题
分析自动形态组和分析功能的类似表示
Analytic automorphism group and similar representation of analytic functions
论文作者
论文摘要
在几何组理论中，里程碑之一是格罗莫夫M.的多项式生长定理：有限生成的组具有多项式生长，并且仅当它们实际上是nilpotent时。受M. Gromov的作品的启发，我们介绍了加权强度的增长类型。在本文中，我们专注于多项式生长的加权耐寒空间，这些空间涵盖了经典的耐寒空间，加权的伯格曼空间，加权的迪里奇（Dirichlet）空间，并且更广泛。我们的主要结果如下。 $（1）$我们获得了组成操作员的界限，其符号具有作用于多项式增长的加权强壮空间的单位开放式磁盘的符号，这意味着乘法$ m_z $类似于任何分析性自动形态$φ$ $m_zφ$。此外，我们获得了由分析函数在单位封闭磁盘上引起的组成算子的界限。 $（2）$对于任何Blaschke产品$ b $的订单$ m $，$ m_b $类似于$ \ bigoplus_ {1}^m m_z $，这是对2007年R. R. Douglas提出的一个问题的肯定答案，2007年。中间生长的强大空间可能是无限的，这表明需要设定多项式生长条件。然后，多项式增长的加权强度空间的收集几乎是最大的阶级，因此道格拉斯的问题具有肯定的答案。 $（4）$最后，我们将Jordan表示定理和单位闭合磁盘上的分析功能的相似性分类作为乘法运算符，以在多项式增长的加权耐寒空间上。
In geometry group theory, one of the milestones is M. Gromov's polynomial growth theorem: Finitely generated groups have polynomial growth if and only if they are virtually nilpotent. Inspired by M. Gromov's work, we introduce the growth types of weighted Hardy spaces. In this paper, we focus on the weighted Hardy spaces of polynomial growth, which cover the classical Hardy space, weighted Bergman spaces, weighted Dirichlet spaces and much broader. Our main results are as follows. $(1)$ We obtain the boundedness of the composition operators with symbols of analytic automorphisms of unit open disk acting on weighted Hardy spaces of polynomial growth, which implies the multiplication operator $M_z$ is similar to $M_φ$ for any analytic automorphism $φ$ on the unit open disk. Moreover, we obtain the boundedness of composition operators induced by analytic functions on the unit closed disk on weighted Hardy spaces of polynomial growth. $(2)$ For any Blaschke product $B$ of order $m$, $M_B$ is similar to $\bigoplus_{1}^m M_z$, which is an affirmative answer to a generalized version of a question proposed by R. Douglas in 2007. $(3)$ We also give counterexamples to show that the composition operators with symbols of analytic automorphisms of unit open disk acting on a weighted Hardy space of intermediate growth could be unbounded, which indicates the necessity of the setting of polynomial growth condition. Then, the collection of weighted Hardy spaces of polynomial growth is almost the largest class such that Douglas's question has an affirmative answer. $(4)$ Finally, we give the Jordan representation theorem and similarity classification for the analytic functions on the unit closed disk as multiplication operators on a weighted Hardy space of polynomial growth.
