解析Hilbert模的Lorentz群及一种相关的子模分类方式

In recent years, research in the Hardy space over the bidisk is one of the most active fronts in multivariable operator theory, an intriguing question is whether there is a meaningful classification of its submodules. Unitary equivalence is a well-known equivalence relation in set of submodules, however the rigidity phenomenon suggests that it lacks the flexibility. Congruence equivalence relation works well when classifying the submodules with a finite rank core operator, but it is still far from a complete classification. This project intends to consider.the group (called Lorentz group) of isometric self-maps of a submodule with respect to a indefinite metric, and its subgroup little Lorentz group , to characterize the structure of submodules and classify them in new perspective. On the other hand, the project intends to generalize this method into Bergman space via root operator, and explore invariant subspaces of Bergman shift operator. Furthermore, we will give a new description of the famous invariant subspace problem and try to get essential breakthrough.

近年来双圆盘Hardy空间的相关研究一直处在多变量算子理论的前沿，一个有趣的问题即是对它的子模进行分类。酉等价是子模分类中常用的一种等价关系，然而刚性现象的发现表明其缺乏一定的灵活性。而相合等价虽可以划分具有有限核算子秩的子模，但是离子模的完全分类仍有一定的距离。本项目拟考虑不定度量下子模上的等距自映射群-Lorentz群和它的特殊子群Little Lorentz群，从这一新角度对双圆盘Hardy空间的子模结构进行刻画和分类。另一方面，本项目拟通过根算子将此手法推广至Bergman空间，对Bergman移位不变子空间进行探索，并进一步对经典不变子空间问题进行新的描述寻求本质突破。

关于双圆盘 Hardy 空间中子模结构的研究，核算子是一种较为有效的工具。因此利用核算子诱导的两类可交换乘法群，Lorentz 群及 Little Lorentz 群，去刻画子模的特征并对其进行有效分类是值得尝试的。另外，将此研究方法推广运用至Bergman空间中有助于进一步探索移位不变子空间的结构。本项目按照研究计划主要取得了以下研究成果：（1）得到了核算子秩有限时, Lorentz群同构的充要条件；（2）刻画了内函数序列生成子模的特征及分类方式；（3）从群的定义出发，对度量进行了改良并应用。