K-框架的交织和框架算子  

Weaving and frame operator of K-frames

作　　者：肖祥春 陈淑铌 周国荣 王琛晖[1] XIAO Xiangchun;CHEN Shuni;ZHOU Guorong;WANG Chenhui(School of Mathematics and Statistics,Xiamen University of Technology,Xiamen 361024,China;School of Mathematical Sciences,Xiamen University,Xiamen 361005,China)

机构地区：[1]厦门理工学院数学与统计学院,福建厦门361024 [2]厦门大学数学科学学院,福建厦门361005

出　　处：《厦门大学学报（自然科学版）》2024年第6期1118-1123,共6页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(12101519);福建省自然科学基金(2020J01267,2021J011192);福建省高校数学学科联盟计划(2025SXLMMS06)。

摘　　要：[目的]K-框架是Gavruta L于2013年为研究Hilbert空间的原子分解而提出的一种与Hilbert空间有界线性算子K有关的一种更一般的框架.正是因为K-框架与一个有界线性算子K相结合,使得K-框架相比较于经典框架而具有很多独特的性质.经典框架的交错对偶涉及的两个序列是可交织的,而本文将探索K-框架的K-对偶涉及的两个序列是否可交织,同时将讨论K-框架的框架算子的特征值的若干刻画.[方法]借助Banach空间的的算子理论中的算子范数、序列求和、正交性以及代数中特征值等工具来研究K-框架的K-对偶序列的交织和K-框架的特征值刻画.[结果] Hilbert空间H的K-框架的K-对偶涉及的两个序列一般情况在整个空间K-中不可交织,接着在K-对偶涉及的两个序列的基础上构造了一组新的序列,使得它们在H中可K-交织.得到了K-框架和经典框架的特征值和对应特征向量的两个刻画.[结论]对比经典的框架而言,本文得到的K-框架的K-对偶所涉及的两个序列的K-交织以及K-框架的框架算子的特征值的刻画的结论都更具一般性.当K为Hilbert空间H的恒等算子时,本文所得K-框架结论即为经典框架对应的结论.[Objective] The K-frame is a more general kind of frame related to a bounded linear operator K in Hilbert spaces,it was first proposed by Gavruta L.in 2013 for the purpose of studying the atomic decompositions in Hilbert spaces.It is the combination of the K-frame with a bounded linear operator K that gives the K-frame many unique properties compared to the classical frames.It is well known that the pair of sequences of an alternate dual of classical frames are woven,in this paper we explore whether the two involved two sequences of a K-dual with K-frames are woven.At the same time,we discuss several characterizations of the eigenvalues of the frame operator of the K-frames.[Methods] We will study the weaving of the pair of sequences of a K-dual and the characterizations of the eigenvalues of the frame operator of the K-frames with the help of tools such as operator norm,sequence summation,orthogonality of the the operator theory of Banach spaces and eigenvalues in algebra.[Results] The two involved two sequences of a K-dual with K-frames are generally not woven in the whole space.Then we construct a new pair of sequences based on the two sequences involved in the K-dual,making them K-woven in H.Two characterizations of eigenvalues and corresponding eigenvectors of K-frame and classical frame are obtained.[Conclusions] Compared with classical frames,we get more general conclusions about the weaving of two sequences involved in the K-duality of K-frames and the characterizations of the eigenvalues of the frame operators of K-frames.When K is the identity operator of Hilbert space,the obtained K-frame conclusions in this paper are the corresponding results of classical frames.

关 键 词：K-框架 K-对偶 K-交织 框架算子 

分 类 号：O177.1[理学—数学]