算子广义逆的若干理论及其应用

The generalized inverse, which is always a noteworthy object, has various applications in numerical algebra, differential equation, control theory, vibration mechanics and so on. Much progress has been made in the study of representations, perturbations and applications of the generalized inverse. Yet, little has been done in such areas as multiplicative perturbations of the M-P inverse, representations for the Drazin inverse of a block operator matrix with small Drazin index, and approximations of an infinite–dimensional M-P inverse by certain finite-dimensional M-P inverses. . In this project, we will study some aspects of the generalized inverse systematically by using the theory of functional analysis, the techniques of matrix decompositions, together with some examples as well as associated numerical computations. More precisely, we will focus on such topics as norm estimations for the rank-preserving (stable) perturbations of matrices (operators), representations and norm estimations for the M-P inverse associated to multiplicative perturbations of operators, norm estimations for the solutions of certain (weighted) least squares problems, approximations of an infinite–dimensional M-P inverse by certain finite-dimensional M-P inverses, representations for the Drazin inverse of a blocked operator matrix with small Drazin index, the comparison of lattice orders associated to different orders and so on. . The research of this project will expand and deepen the known results on the generalized inverse, which is helpful to promote the development of the associated disciplines.

广义逆在数值代数、微分方程、控制论、振动力学等学科领域有广泛的应用，始终是一个值得关注的研究对象。表示、扰动和应用是广义逆研究的三大主题，相关的研究工作丰富多样。尽管如此，至今国内外在诸如M-P逆的乘法扰动、低指标分块算子矩阵的Drazin 逆的表示、无限秩M-P逆的有限逼近等方面的研究还很少。. 本项目将综合运用泛函分析和矩阵分解等理论与技巧，结合应用实例及数值计算，围绕广义逆研究方面的若干核心课题，对算子广义逆进行系统的研究。具体地，我们将研究矩阵保秩扰动和算子稳定扰动的M-P逆的误差估计、算子乘法扰动的M-P逆的表示及误差估计、（加权）最小二乘问题解的误差估计、无限秩M-P逆的有限逼近、低指标分块算子矩阵的Drazin逆的表示及应用、不同序关系下序结构性质及其相互关系等。. 本项目的研究将拓展和深化国内外在广义逆方面的现有成果，促进相关学科领域的发展。

受国家自然科学基金委的资助(2017.1-2020.12)，四年来我们按计划主要对算子广义逆的若干理论及其应用进行了系统而又深入的研究，得到了一系列富有特色的研究成果。具体地， 我们研究了矩阵保秩扰动和算子稳定扰动的M-P逆的误差估计、算子乘法扰动的M-P逆的表示及误差估计、（加权）最小二乘问题解的误差估计、无限秩M-P逆的有限逼近、低指标分块算子矩阵的Drazin逆的表示及应用、不同序关系下序结构性质及其相互关系这些既定的课题。同时，我们还对算子与矩阵的极分解及其应用、推广的Douglas定理及其应用、 Hilbert C*-模上两个投影的Halmos分解及其应用、Hilbert C*-模上可共轭算子的并联和、不定内积空间的加权M-P逆等新的课题进行了研究，也得到了一系列较为深入的研究成果。本项目的部分研究成果已以论文的方式加以了呈现。截止2020年12月底，我们正式发表论文27篇，录用且有校样的文章11篇。发表和录用的38篇文章中，有36篇发表或将发表于SCI/SCIE期刊上。