Hilbert-Schmidt框架的广义交叉Gram矩阵  

Generalized Cross Gram Matrices with Respect to Hilbert-Schmidt Frames

作　　者：张伟 李云章 Wei ZHANG;Yun Zhang LI(School of Mathematics and Information Sciences,Henan University of Economics and Law,Zhengzhou 450046,P.R.China;School of Mathematics,Statistics and Mechanics,Beijing University of Technology,Beijing 100124,P.R.China)

机构地区：[1]河南财经政法大学数学与信息科学学院,郑州450046 [2]北京工业大学数学统计学与力学学院,北京100124

出　　处：《数学学报（中文版）》2024年第6期1077-1090,共14页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金(12371091,12461016);河南省科技攻关项目(242102210049)。

摘　　要：寻找算子的矩阵表示是算子理论的一个重要问题,计算这种离散形式对于算子方程的数值解也同样重要.传统上,二者都是通过基来完成,而本文通过HS-框架来完成.首先引入HS-框架广义交叉Gram矩阵的概念,讨论若干基本性质;接下来给出其可逆的充分必要条件及逆矩阵精确公式;特别地,例子展示矩阵是不可逆的若构成矩阵的序列是HS-框架而不是HS-Riesz基.最后得到若干稳定性结果,准确地说,证明了在小扰动下广义交叉Gram矩阵的可逆性是保持的.Finding matrix representations of operators is an important part of operator theory.Calculating such a discretization scheme is equally important for the numerical solution of operator equations.Traditionally in both fields,this was done using bases,Hilbert-Schmidt frames have been used here.Firstly,we introduce the concept of generalized cross gram matrix with respect to HS-frame,discuss some basic properties.Then,we give necessary and sufficient conditions for their invertibility and present explicit formulas for the inverse.In particular,the example shows that invertibility of generalized cross Gram matrix is not possible when the associated sequences are HS-frames rather than HS-Riesz bases.Finally,we obtain some stability results.More precisely,it is shown that the invertibility of generalized cross Gram matrices is preserved under small perturbations.

关 键 词：HS-框架 广义交叉Gram矩阵 稳定性 

分 类 号：O174.2[理学—数学]