Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and hardy spaces of character-automorphic functions
Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and hardy spaces of character-automorphic functions
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具有齐次谱、无限维雅可比反演和字符自守函数 Hardy 空间的几乎周期性雅可比矩阵
DOI：
10.1007/bf02921627
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发表时间：
1997
期刊：
影响因子：
--
通讯作者：
P. Yuditskii
中科院分区：
文献类型：
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作者：
M. Sodin;P. Yuditskii
设JE是一类作用于l ~ 2（n）的Jacobi矩阵，具有齐次谱E（见定义3.2），其预解式R（m，m; z）的对角元具有纯虚边值a.e.一个.对于这类，我们扩展了有关有限带的基本结果（即，代数几何）算子。特别地，我们证明了类JE的矩阵是概周期的。我们的主要工具是关于Fuchsian群的特征自守函数理论，它统一了预解式域。对于Widom型群，我们找到了傅里叶基的自然模拟，对于Widom-Carleson型群，我们刻画了来自哈代空间H2的特征自守函数的正交补。这种技术使我们能够研究无限维Abel映射并找到无限维的真实的版本的Jacobi逆，这在我们研究类矩阵中发挥着主要作用JE。
All three subjects reflected in the title are closely intertwined in the paper.LetJE be a class of Jacobi matrices acting inl2(ℤ) with a homogeneous spectrumE (see Definition 3.2) and with diagonal elements of the resolventR(m, m; z) having pure imaginary boundary values a.e. onE. For this class, we extend fundamental results pertaining to the finite-band (i.e., algebraic-geometrical) operators. In particular, we prove that matrices of the classJE are almost periodic.Our main tool is a theory of character-automorphic functions with respect to the Fuchsian group uniformizing the resolvent domain. For Widom type groups we find a natural analog of the Fourier basis and for Widom-Carleson type groups we characterize the orthogonal complement to character-automorphic functions from the Hardy spaceH2.This technique allows us to study the infinite dimensional Abel map and to find an infinite dimensional real version of the Jacobi inversion, which play a principal role in our investigation of matrices of the classJE.