文档介绍：南京航空航天大学硕士学位论文
摘要
本文把 Hilbert C*-模上的离散框架和 Hilbert 空间中的广义框架两个概
念结合起来推广到了 Hilbert W*-模上去,得到了 Hilbert W*-模上广义框
架、广义紧框架、广义正规紧框架、广义对偶框架等概念,并着重研究了标
准的 Hilbert W*-模框架,且给出了广义框架的解析算子,框架算子,对偶
框架等概念,得出了类似于 Hilbert 空间和 Hilbert C*-模中的所谓的重构
公式. 同时给出了广义框架的强不相交,不相交,弱不相交的定义,及有关
它们的等价条件,巧妙的用算子理论的方法给出了这些条件的证明,并在强
不相交,不相交,弱不相交的条件下得到了一些重要的结果. 如在强不相交,
不相交,弱不相交的条件下,两个或更多的广义框架的和或直和何时仍是一
个广义框架等等. 本文还给出了广义框架摄动的许多与 Hilbert 空间中离散
框架摄动相类似的条件,研究了一个广义框架在什么样的微小波动下仍是一
个广义框架,并给出了相应的框架界的变化及它们的严格证明过程,还得到
了在摄动条件下广义框架的一些其它结果. 文章的最后主要讨论了有关
Hilbert W*-模上的广义框架的一些等式和不等式,尤其是对广义 Parseval
框架的等式进行了详细的研究.
关键字: Hilbert W*-模上的广义框架;对偶框架;框架算子;分析算子;不
相交性;摄动
I
Hilbert W*-模上的广义框架
Abstract
This thesis is devoted to generalizing the notion of frame to the Hilbert W*-
module setting bining the definitions of discrete frame in Hilbert C*-modules
and generalized frame in Hilbert spaces, and obtain the notions of generalized frame,
generalized tight frame, generalized normalized tight frame and generalized dual
frame in Hilbert W*-modules. In this thesis, the standard generalized frames in
Hilbert W*-modules are primarily studied, also some definitions such as analysis
operator, frame operator and dual frame for a given generalized frame are introduced,
and the reconstruction formula which is very important in Hilbert C*-modules and
Hilbert spaces is available. The definitions of strong disjointness, disjointness and
weak disjointness and their equivalent conditions which are similar to the case in
Hilbert spaces are given in chapter 2, while the proofs which we mainly use
operator-theoretic-methods are plicated. Some important results are
showed, such as under some other assumptions, the sum or the direct sum of two or
more generalized frames also form a generalized frame under the assumptions of
strong disjointness, disjointness or weak disjointness, respectively. It is known that
the most important problem of frames is perturbation, in this thesis, some
perturbation conditions ab