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The Invertibility of U -Fusion Cross Gram Matrices of Operators
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-07-15 , DOI: 10.1007/s00009-020-01536-0 Mitra Shamsabadi , Ali Akbar Arefijamaal , Peter Balazs
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-07-15 , DOI: 10.1007/s00009-020-01536-0 Mitra Shamsabadi , Ali Akbar Arefijamaal , Peter Balazs
Finding matrix representations 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. Recently, frames have been used here. In this paper, we apply fusion frames for this task, a generalization motivated by a block representation, respectively, a domain decomposition. We interpret the operator representation using fusion frames as a generalization of fusion Gram matrices. We present the basic definition of U-fusion cross Gram matrices of operators for a bounded operator U. We give necessary and sufficient conditions for their (pseudo-)invertibility and present explicit formulas for the (pseudo-)inverse. More precisely, our attention is on how to represent the inverse and pseudo-inverse of such matrices as U-fusion cross Gram matrices. In particular, we characterize fusion Riesz bases and fusion orthonormal bases by such matrices. Finally, we look at which perturbations of fusion Bessel sequences preserve the invertibility of the fusion Gram matrix of operators.
中文翻译:
算符的U-融合交叉革矩阵的可逆性
查找矩阵表示形式是算子理论的重要组成部分。对于算子方程的数值解,计算这样的离散化方案同样重要。传统上在两个领域中,这都是使用基数完成的。最近,这里已经使用了框架。在本文中,我们将融合帧用于此任务,该融合帧分别由块表示和域分解所激发。我们将使用融合框架的算子表示形式解释为融合Gram矩阵的概括。我们给出了有界算子U的U-融合克跨矩阵的基本定义。我们为它们的(伪)可逆性提供了必要和充分的条件,并给出了(伪)逆的显式。更精确地,我们的注意力在于如何表示诸如U- fusion cross Gram矩阵之类的矩阵的逆和伪逆。特别是,我们通过此类矩阵来表征融合Riesz碱基和融合正交正态碱基。最后,我们看一下融合贝塞尔序列的哪些扰动保留了算符的融合Gram矩阵的可逆性。
更新日期:2020-07-15
中文翻译:
算符的U-融合交叉革矩阵的可逆性
查找矩阵表示形式是算子理论的重要组成部分。对于算子方程的数值解,计算这样的离散化方案同样重要。传统上在两个领域中,这都是使用基数完成的。最近,这里已经使用了框架。在本文中,我们将融合帧用于此任务,该融合帧分别由块表示和域分解所激发。我们将使用融合框架的算子表示形式解释为融合Gram矩阵的概括。我们给出了有界算子U的U-融合克跨矩阵的基本定义。我们为它们的(伪)可逆性提供了必要和充分的条件,并给出了(伪)逆的显式。更精确地,我们的注意力在于如何表示诸如U- fusion cross Gram矩阵之类的矩阵的逆和伪逆。特别是,我们通过此类矩阵来表征融合Riesz碱基和融合正交正态碱基。最后,我们看一下融合贝塞尔序列的哪些扰动保留了算符的融合Gram矩阵的可逆性。