关于线性算子广义Bott-Duffin逆的一点注记

数学物理学报 ›› 2011, Vol. 31 ›› Issue (4): 1071-1076.

关于线性算子广义Bott-Duffin逆的一点注记

杜鸿科^1,2|邵春芳^1,2*|许俊莲¹|姬淑凤¹

1.陕西师范大学 |数学与信息科学学院西安 710062|
2.安康学院 |数学系陕西安康 725000

收稿日期:2008-08-17 修回日期:2010-07-28 出版日期:2011-08-25 发布日期:2011-08-25
通讯作者: 邵春芳 E-mail:scfwzy@yahoo.com.cn
基金资助:
国家自然科学基金(10871224)资助

A Note on the Generalized Bott-Duffin Inverse of Linear Operators

DU Hong-Ke^1,2, SHAO Chun-Fang^1,2*, XU Jun-Lian¹, JI Shu-Feng¹

1.College of Mathematics and Information Science, Shaanxi Normal University, |Xi'an 710062;
2.Department of Mathematics, Ankang University, Shaanxi Ankang 725000

Received:2008-08-17 Revised:2010-07-28 Online:2011-08-25 Published:2011-08-25
Contact: SHAO Chun-Fang E-mail:scfwzy@yahoo.com.cn
Supported by:
国家自然科学基金(10871224)资助

摘要/Abstract

摘要：

该文推广了邓彬和陈果良在文献[7]中所得的关于矩阵的广义Bott-Duffin逆的若干结果. 基于算子的空间分解理论,证明了这些结果对无穷维希尔伯特空间上的有界线性算子也是成立的. 值得指出的是该文所用的数学思想和方法与文献[7]中所用的方法是完全不同的.

关键词: 广义逆, Bott-Duffin 逆, 算子矩阵, 幂等算子

Abstract:

In this note, two theorems involving generalized Bott-Duffin inverses of bounded linear operators on a Hilbert space are proved, which are generalizations of that obtained in a finite dimensional space by Deng and Chen^[7]. It is worthwhile to point out that the idea and the method used in this note are different from those used by Deng and Chen.

Key words: Generalized inverse, Bott-Duffin inverse, Operator matrix, Idempotent

中图分类号:

47A05

杜鸿科, 邵春芳, 许俊莲, 姬淑凤. 关于线性算子广义Bott-Duffin逆的一点注记[J]. 数学物理学报, 2011, 31(4): 1071-1076.

DU Hong-Ke, SHAO Chun-Fang, XU Jun-Lian, JI Shu-Feng. A Note on the Generalized Bott-Duffin Inverse of Linear Operators[J]. Acta mathematica scientia,Series A, 2011, 31(4): 1071-1076.

参考文献

[1] Ben Israel A, Greville T N E. Generalized Inverse: Theory and Applications. New York: Springer-Verlag, 2003

[2] Bott R, Duffin R J. On the algebra of network. Trans Amer Math Soc, 1953, 74: 99--109

[3] Chen G, Liu G, Xue Y. Perturbation analysis of the generalized Bott-Duffin inverse of L-zero matrices. Linear Multilinear Algebra, 2003, 51: 11--20

[4] Chen G, Liu G, Xue Y. Perturbation theory for the generalized Bott-Duffin inverse and its applications. Appl Math Comput, 2002, 129: 145--155

[5] Chen G, Xue Y. The expression of the generalized inverse of the perturbed operator under type I perturbation in Hilbert spaces. Linear Algebra Appl, 1998, 285: 1--6

[6] Chen Y. The generalized Bott-Duffin inverse and its applications. Linear Algebra Appl, 1990, 134: 71--91

[7] Deng B, Chen G. A note on the generalized Bott-Duffin inverse. Appl Math Lett, 2007, 20: 746--750

[8] Du H K, Deng C Y. A new characterization of gaps between two subspaces. Proc Amer Math Soc, 2005, 133: 3065--3070

[9] Du H K, Yao X Y, Deng C Y. Invertibility of linear combinations of two idempotents. Proc Amer Math Soc, 2005, 134: 1451--1457

[10] Deng C Y. On the minimum gap and the angle between two subspaces. J Comput Appl Math, 2007, 206: 625--630

[11] Groetsch G W. Generalized Inverse of Linear Operators. New York: Marcel Dekker, 1977

[12] Horn R A, Johnson C R. Matrix Analysis. Cambridge: Cambridge University Press, 1985

[13] Halmos P R. Two subspaces. Trans Amer Math Soc, 1969, 114: 381--389

[14] Shao C F, Tian L C, Du H K. The representations of the generalized Bott-Duffin inverse of linear operators. Basic Sciences Journal of Textile Universities (Chinese Series), 2008, 21(1): 48--51

[15] Wang G, Wei Y, Qiao S. Generalized Inverse: Theory and Computations. Beijing, New York: Science press, 2004

[16] Wei Y, Xu W. Condition number of Bott-Duffin inverse and their condition numbers. Appl Math Comput, 2003, 142: 79--97

[17] Xue Y, Chen G. The expression of the generalized Bott-Duffin inverse and its perturbation theory. Appl Math Comput, 2002, 132: 437--444