Hilbert C*-模的模框架与框架的对称逼近
Modular frames for Hilbert C*-modules and symmetric approximation of frames
AI总结：
本文系统研究了Hilbert C*-模中的模框架构造及相关算子，通过几何膨胀方法建立与经典框架理论的联系，并利用框架算子范数距离刻画框架接近程度，特别探讨了最接近紧框架的存在唯一性及其与Löwdin正交化的关系。
摘要：
本文对Hilbert C*-模中一般的模框架构造及其上的相关模算子进行了全面介绍。Hilbert空间情形作为特例出现。所报告的研究依赖于几何膨胀到标准Hilbert C*-模的思想，这些标准模定义在具有单位元的C*-代数上，并且允许正交规范Riesz基。文中指出了与经典线性框架理论的相互联系及其应用。作为一个应用，我们描述了算子族{S_i}的性质，使得SUM_i S*_iS_i=id_H，其中H是一个Hilbert空间。借助Hilbert空间中的框架，我们讨论了一些度量框架对彼此接近程度的量。大多数度量用不同类型的框架算子的范数距离来表示。特别地，研究了与给定框架最接近的（规范化）紧框架的存在性和唯一性。对于具有某些限制的Riesz基，最接近紧框架的集合通常包含其对称正交化（即Löwdin正交化）的一个倍数。
英文摘要：
We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related modular operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modulesover unital C*-algebras that admit an orthonormal Riesz basis. Interrelations and applications to classical linear frame theory are indicated. As an application we describe the nature of families of operators {S_i} such that SUM_i S*_iS_i=id_H, where H is a Hilbert space. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. Most of the measures are expressed in terms of norm-distances of different kinds of frame operators. In particular, the existence and uniqueness of the closest (normalized) tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closetst tight frames often contains a multiple of its symmetric orthogonalization (i.e. Löwdin orthogonalization).
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