Banach空间中线性算子Moore-Penrose度量广义逆的扰动分析
|
摘要
Banach空间有界线性算子广义逆的扰动分析在算子理论的实际应用领域起到非常重要的作用,并且已经广泛应用于统计,优化,控制等学科.但由于度量广义逆一般为有界齐性的非线性算子,所以其扰动定理的证明与线性广义逆的扰动定理完全不同.
在本文中,我们更深入的研究了Banach空间中有界线性算子的Moore-Penrose(单值)度量广义逆的扰动分析.我们给出了在特定条件下的(单值)度量广义逆的形式表示,及在这种条件下的范数估计和误差界估计.首先应用度量稳定扰动的定义及广义正交分解定理,给出在特定条件下有界线性算子的Moore-Penrose单值度量广义逆的误差界估计,并推导出其度量广义逆扰动的范数估计.接下来我们在上述定理的基础上,应用度量投影算子的连续性,以及度量广义逆算子的拟可加性,给出了Moore-Penrose单值度量广义逆的形式表示,及其稳定扰动的范数估计和误差估计.
本文仅讨论了Banach有界线性算子的单值度量广义逆的扰动的范数估计及误差估计界.对于单值度量广义逆的稳定扰动的等价条件,集值度量广义逆的扰动,还有待进一步探究.
Perturbation analysis of Moore-Penrose metric generalizedinverse of linear op-erators in Banach space play an very important role in the actual application do-main of operator theory, and have already been applied to Statistics, Optimizes and Control. Because the Moore-Penrose metric generalized inverse is a bounded homogeneous nonlinear operator in general, the proof of its perturbation theorem is completely different from that of linear generalized inverse perturbation theorem.
In this paper, we further study the perturbation analysis for Moore-Penrose (single-valued) metric generalized inverse of bounded linear operators in Banach space. We give the complete description of (single-valued) metric generalized inverse under norm, and the norm estimate and the error bound estimate are also given.
Firstly, we apply the definition of stable perturbation and generalized orthog-onal decomposition theorem to give the Moore-Penrose metric generalized inverse error bound estimate under norm, and derive the norm estimate of the perturbation of Moore-Penrose metric generalized inverse. Next, by means of the above theorem, the continuity of the metric projection operator and the pseudo-additivity of metric generalized inverse, we give a complete description of Moore-Penrose single-valued metric generalized inverse, and the norm estimation and error estimation is also given in this paper.
In this paper, we only discuss the norm estimation and error estimation of perturbation for Moore-Penrose single-valued metric generalized inverse of bounded linear operators in Banach space. But the equivalent condition of the stable pertur-bation of single-valued metric generalized inverse, and the perturbation of set-valued metric generalized inverse still need further studies.
在本文中,我们更深入的研究了Banach空间中有界线性算子的Moore-Penrose(单值)度量广义逆的扰动分析.我们给出了在特定条件下的(单值)度量广义逆的形式表示,及在这种条件下的范数估计和误差界估计.首先应用度量稳定扰动的定义及广义正交分解定理,给出在特定条件下有界线性算子的Moore-Penrose单值度量广义逆的误差界估计,并推导出其度量广义逆扰动的范数估计.接下来我们在上述定理的基础上,应用度量投影算子的连续性,以及度量广义逆算子的拟可加性,给出了Moore-Penrose单值度量广义逆的形式表示,及其稳定扰动的范数估计和误差估计.
本文仅讨论了Banach有界线性算子的单值度量广义逆的扰动的范数估计及误差估计界.对于单值度量广义逆的稳定扰动的等价条件,集值度量广义逆的扰动,还有待进一步探究.
Perturbation analysis of Moore-Penrose metric generalizedinverse of linear op-erators in Banach space play an very important role in the actual application do-main of operator theory, and have already been applied to Statistics, Optimizes and Control. Because the Moore-Penrose metric generalized inverse is a bounded homogeneous nonlinear operator in general, the proof of its perturbation theorem is completely different from that of linear generalized inverse perturbation theorem.
In this paper, we further study the perturbation analysis for Moore-Penrose (single-valued) metric generalized inverse of bounded linear operators in Banach space. We give the complete description of (single-valued) metric generalized inverse under norm, and the norm estimate and the error bound estimate are also given.
Firstly, we apply the definition of stable perturbation and generalized orthog-onal decomposition theorem to give the Moore-Penrose metric generalized inverse error bound estimate under norm, and derive the norm estimate of the perturbation of Moore-Penrose metric generalized inverse. Next, by means of the above theorem, the continuity of the metric projection operator and the pseudo-additivity of metric generalized inverse, we give a complete description of Moore-Penrose single-valued metric generalized inverse, and the norm estimation and error estimation is also given in this paper.
In this paper, we only discuss the norm estimation and error estimation of perturbation for Moore-Penrose single-valued metric generalized inverse of bounded linear operators in Banach space. But the equivalent condition of the stable pertur-bation of single-valued metric generalized inverse, and the perturbation of set-valued metric generalized inverse still need further studies.
引文
1 E. H. Moore. On the reciprocal of the general algebra matrix (Abstract). Bull. Amer. Math. Soc.,1920,26:394-395
2 R. Penrose. A generalized inverse for matrices. Proc, Cambridge. Philos, Soc., 1995,51:406-413
3 M. P. Drazin. Pesudoinverse in associative rings and semigroups. Amer. Math. Monthly,1958,65:506-514
4 M. Z. Nashed. Ed. Generalized Inverse and Applications. New York/London: Academic Press,1976
5 Y. Y. Tseng. The charateristic value problem of Hermitam functional operators in a non-Hilbert spaces. Doctoral Dissertation in Math. University of Chicago, 1933
6 Y. Y. Tseng. Generalized inverses of unbounded operators between two unitary spaces. Dokl. Akad. Nauk. SSSR(N.S.),1941,67:431-434
7 Y. Y. Tseng. Properties and classification of generalized inverses of closed oper-ators. Dokl. Akad. Nauk. SSSR(N.S.),1949,67:607-610
8 Y. Y. Tseng. Sur les solutions des equations operatrices functionnelless entre les espaces. Unitaires, C. R. Acad. Sci. Paris,1949.228:640-641
9 Y. Y. Tseng. Virtual solutions and general inversions. Uspehi. Mat.Nauk. (N.S.),1956,11:213-215
10马吉溥.关于R(Ax)闭的连续算子族Ax的Moore-Pense广义逆A+x连续的充要条件.中国科学,A(1990)：561～568
11李绍宽Hilbert空间的算子广义逆.数学年刊,4(1990)
12 M. Z. Nashed, G. F. Votruba. A Unified Operator Theory of Generalized In-verse, Nashed Ed. Academic Press, New York,1976:101-109
13魏益民,匡蛟勋Banach空间中计算线性算子Drazin逆的迭代法.复旦学报,1996,35：407～413
14 M. Z. Nashed. Inner, outer and generalized inverse in Banach and Hilbert spaces. Numer. Funct. Anal. Optim.,1987,9:261-325
15 M. Z. Nashed. A new approach to classification and regularzation of ill-posed operator equations, Inverse and Ill-posed problems, (sankt wolfgarg,1986), 53-75, Notes Rep. Math. Sci. Engrg.4, Boston:Academic Press, MA,1987
16 J. P. Ma. (1,2)-Inverse of operators between Banach spaces and local conjugacy theorem. Chin.Ann.Math.,1999,20B(1):57-62
17 J. P. Ma. Rank theorems of operators between Banach Spaces, Science in China, 2000,43A(1):1-5
18蔡东汉.线性算子的Drazin广义逆.数学杂志,1985,5(1)：81～87
19乔三正Banach空间线性算子的Drazin逆.上海师范学院学报(自然科学版),1981.2：11～18
20乔三正Banach空间线性算子的带权Drazin逆和Drazin逆的逼近.高等学校计算数学学报,1981,4：296～305
21匡蛟勋.线性算子Drazin逆的表示与逼近.高等学校计算数学学报,1982,4(2)：97-106
22王国荣Banach空间中线性算子的带权Drazin逆的逼近方法.高校计算数学学报,1988,16：76～81
23 Y. W. Wang. The generalized inverse operator in Banach spaces. Bull. Polish. Aca. Sci. Math.,1989,37(7-12):433-441
24王玉文,王辉,王润洁Banach空间中线性算子的集值度量右逆的表示及应用.应用泛函分析学报,1999,1(3)：255～260
25王玉文,李志伟Banach空间中Moore-Penrose广义逆与不适定的边值问题.系统科学与数学,20(2),1995
26 Y. W. Wang, R. J. Wang. Pseudo-inverse and Two Objective Optimal Control in Banach Spaces. Functions et Approx., XXI,1992, UAM,140-160
27李志伟,王玉文Banach空间中非光滑指标奇异最优控制.系统科学与数学,1995,15(4)：305～311
28王玉文,季大琴Banach空间中无限维矩阵的Tseng度量广义逆.哈师大自然科学学报,Vol 14,No.4,1998：11-14
29王玉文,季大琴Banach空间中线性算子的Tseng度量广义逆.系统科学与数学,2000,20(2)：203～209
30季大琴,全玉锦,林秀芹Banach空间中线性算子的Tseng度量广义逆与Moore-penrose度量广义逆.丹东纺专学报,2001,8(3)：46～48
31王玉文,潘少荣Banach空间线性算子(集值)度量广义逆及其齐性选择.数学学报,2003,46(3)：273～280
32 H. Wang, Y. W. Wang. Metric Generalized inverse of linear operator in Ba-nach spaces. Chin. Ann. Math.,2003.24B(4):509-520
33 Y. W. Wang, J. Liu.Metric generalized inverse of linear maniford and extremal solution of linear inclusion in Banach Spaces. J. Math. Anal. Apll.2005.302: 360-371
34倪仁兴.任意Banach空间中线性算子的Moore-Penrose度量广义逆.数学学报,2006,49(6)：1247～1252
35倪仁兴Banach空间中线性算子的度量右逆.系统科学与数学,2008,28(6)：35～56
36陈述涛,王玉文Banach空间的逼近紧性与度量投影算子的连续性及其应用.中国数学(A辑),2007,11：235～256
37孙继广.矩阵扰动分析.北京：科学出版社,2001
38 Y. W. Wang, Y. M. Wei, S. Zh. Qiao. Generalized Inverses:Theory and Com-putations. Beijing/New York:SCIENCE PRESS,2004
39 M. Wei. The perturbation of consistent least squares problems. Linear Algebra Appl.,1989,112:231-245
40 M. Wei. The perturbation of least squares problems. Linear Algebra Appl.,1986, 141:177-182
41 G. L. Chen, Y. F. Xue. Pertubation Analysis for the Operator Equation Tx=b in Banach Spaces. Journal of Mathematical Analysis and Application,1997,212: 107-125
42 G. Chen, M. Wei and Y. Xue. Perturbation analysis of the least squares solu-tion in Hilbert spaces. Linear Algebra Appl.,1996,244:69-80
43 J. Ding, L. J. Huang. On the Perturbation of the Least Solutions in Hilbert Spaces. Linear Algebra Appl.,1994,212/213:487-500
44 J. Ding. Minimal Distance Upper Bounds for the Perturbation of Least Squares Problems in Hilbert Spaces. Applied Mathematics Letters,2002,15:361-365
45 Y.M. Wei. Perturbation bound of the Drazin inverse,Applied Mathematics and Computation.Vol.125,(2002),231-244.
46 Y. M. Wei, G. R. Wang.The pertubation theory for the Drazin inverse and its applications. Linear Algebra Appl.,1997,258:178-186
47魏益民Banach空间中的Dazin逆的表示和扰动.数学年刊,2000,21A：910：33～38
48 J. Cai, G. L Chen. Pertubation Analysis for the Drazin Inverses of Bounded Lin-ear Operators on a Banach Space. Joural of Mathematical Research/Exposition, 2004, Vol.24, No.3:421-429
49陈果良,薛以锋,蔡静Hilbert空间中稳定扰动下广义逆的误差估计界.华东师范大学学报,2002,3：1～6
50 Stewart G W, Sun J G. Matrix Perturbation Theory[M]. New York:Academic Press,1976
51 G. L Chen,Y. F. Xue. The expression of the generalized inverse of the perturbed operator under Type Ⅰ perturbation in Hilbert spaces[J]. Linear Algebra Appl, 1998,285:1-6
52 Ding. J. Perturbation results for projecting a point onto a linear manifold[J]. SIAM Matrix Anal Appl,1998,19(3):696-700
53定光桂Banach空间引论.北京：科学出版社.1984
54王玉文.巴拿赫空间中算子广义逆理论及其应用.北京：科学出版社.2005.1
55王玉文,潘少荣Banach空间中有限秩算子的逼近问题.中国科学,2002,32A(9)：837～841
56马海风,李双臻,刘冠琦,王玉文Banach空间中有界线性算子的Moore-Pen-rose度量广义逆的扰动分析.哈尔滨师范大学自然科学学报,2008,24(6)：1-3.
57马海凤,王玉文.单值度量广义逆的扰动分析.哈尔滨师范大学自然科学学报,2006,22(2)：8-10.
58 Tosio Kato. Pertubation Theory for Linear Operators. Springer-Verlag. Berlin Heidel berg. New York.1980
59 R.B.Holmes. Geometric Functional Analysis and Its Application. New York, Heideberg, Berlin:Epring-Verlage,1975.
60张恭庆,林源渠.泛函分析讲义.北京：北京大学出版社.2003
2 R. Penrose. A generalized inverse for matrices. Proc, Cambridge. Philos, Soc., 1995,51:406-413
3 M. P. Drazin. Pesudoinverse in associative rings and semigroups. Amer. Math. Monthly,1958,65:506-514
4 M. Z. Nashed. Ed. Generalized Inverse and Applications. New York/London: Academic Press,1976
5 Y. Y. Tseng. The charateristic value problem of Hermitam functional operators in a non-Hilbert spaces. Doctoral Dissertation in Math. University of Chicago, 1933
6 Y. Y. Tseng. Generalized inverses of unbounded operators between two unitary spaces. Dokl. Akad. Nauk. SSSR(N.S.),1941,67:431-434
7 Y. Y. Tseng. Properties and classification of generalized inverses of closed oper-ators. Dokl. Akad. Nauk. SSSR(N.S.),1949,67:607-610
8 Y. Y. Tseng. Sur les solutions des equations operatrices functionnelless entre les espaces. Unitaires, C. R. Acad. Sci. Paris,1949.228:640-641
9 Y. Y. Tseng. Virtual solutions and general inversions. Uspehi. Mat.Nauk. (N.S.),1956,11:213-215
10马吉溥.关于R(Ax)闭的连续算子族Ax的Moore-Pense广义逆A+x连续的充要条件.中国科学,A(1990)：561～568
11李绍宽Hilbert空间的算子广义逆.数学年刊,4(1990)
12 M. Z. Nashed, G. F. Votruba. A Unified Operator Theory of Generalized In-verse, Nashed Ed. Academic Press, New York,1976:101-109
13魏益民,匡蛟勋Banach空间中计算线性算子Drazin逆的迭代法.复旦学报,1996,35：407～413
14 M. Z. Nashed. Inner, outer and generalized inverse in Banach and Hilbert spaces. Numer. Funct. Anal. Optim.,1987,9:261-325
15 M. Z. Nashed. A new approach to classification and regularzation of ill-posed operator equations, Inverse and Ill-posed problems, (sankt wolfgarg,1986), 53-75, Notes Rep. Math. Sci. Engrg.4, Boston:Academic Press, MA,1987
16 J. P. Ma. (1,2)-Inverse of operators between Banach spaces and local conjugacy theorem. Chin.Ann.Math.,1999,20B(1):57-62
17 J. P. Ma. Rank theorems of operators between Banach Spaces, Science in China, 2000,43A(1):1-5
18蔡东汉.线性算子的Drazin广义逆.数学杂志,1985,5(1)：81～87
19乔三正Banach空间线性算子的Drazin逆.上海师范学院学报(自然科学版),1981.2：11～18
20乔三正Banach空间线性算子的带权Drazin逆和Drazin逆的逼近.高等学校计算数学学报,1981,4：296～305
21匡蛟勋.线性算子Drazin逆的表示与逼近.高等学校计算数学学报,1982,4(2)：97-106
22王国荣Banach空间中线性算子的带权Drazin逆的逼近方法.高校计算数学学报,1988,16：76～81
23 Y. W. Wang. The generalized inverse operator in Banach spaces. Bull. Polish. Aca. Sci. Math.,1989,37(7-12):433-441
24王玉文,王辉,王润洁Banach空间中线性算子的集值度量右逆的表示及应用.应用泛函分析学报,1999,1(3)：255～260
25王玉文,李志伟Banach空间中Moore-Penrose广义逆与不适定的边值问题.系统科学与数学,20(2),1995
26 Y. W. Wang, R. J. Wang. Pseudo-inverse and Two Objective Optimal Control in Banach Spaces. Functions et Approx., XXI,1992, UAM,140-160
27李志伟,王玉文Banach空间中非光滑指标奇异最优控制.系统科学与数学,1995,15(4)：305～311
28王玉文,季大琴Banach空间中无限维矩阵的Tseng度量广义逆.哈师大自然科学学报,Vol 14,No.4,1998：11-14
29王玉文,季大琴Banach空间中线性算子的Tseng度量广义逆.系统科学与数学,2000,20(2)：203～209
30季大琴,全玉锦,林秀芹Banach空间中线性算子的Tseng度量广义逆与Moore-penrose度量广义逆.丹东纺专学报,2001,8(3)：46～48
31王玉文,潘少荣Banach空间线性算子(集值)度量广义逆及其齐性选择.数学学报,2003,46(3)：273～280
32 H. Wang, Y. W. Wang. Metric Generalized inverse of linear operator in Ba-nach spaces. Chin. Ann. Math.,2003.24B(4):509-520
33 Y. W. Wang, J. Liu.Metric generalized inverse of linear maniford and extremal solution of linear inclusion in Banach Spaces. J. Math. Anal. Apll.2005.302: 360-371
34倪仁兴.任意Banach空间中线性算子的Moore-Penrose度量广义逆.数学学报,2006,49(6)：1247～1252
35倪仁兴Banach空间中线性算子的度量右逆.系统科学与数学,2008,28(6)：35～56
36陈述涛,王玉文Banach空间的逼近紧性与度量投影算子的连续性及其应用.中国数学(A辑),2007,11：235～256
37孙继广.矩阵扰动分析.北京：科学出版社,2001
38 Y. W. Wang, Y. M. Wei, S. Zh. Qiao. Generalized Inverses:Theory and Com-putations. Beijing/New York:SCIENCE PRESS,2004
39 M. Wei. The perturbation of consistent least squares problems. Linear Algebra Appl.,1989,112:231-245
40 M. Wei. The perturbation of least squares problems. Linear Algebra Appl.,1986, 141:177-182
41 G. L. Chen, Y. F. Xue. Pertubation Analysis for the Operator Equation Tx=b in Banach Spaces. Journal of Mathematical Analysis and Application,1997,212: 107-125
42 G. Chen, M. Wei and Y. Xue. Perturbation analysis of the least squares solu-tion in Hilbert spaces. Linear Algebra Appl.,1996,244:69-80
43 J. Ding, L. J. Huang. On the Perturbation of the Least Solutions in Hilbert Spaces. Linear Algebra Appl.,1994,212/213:487-500
44 J. Ding. Minimal Distance Upper Bounds for the Perturbation of Least Squares Problems in Hilbert Spaces. Applied Mathematics Letters,2002,15:361-365
45 Y.M. Wei. Perturbation bound of the Drazin inverse,Applied Mathematics and Computation.Vol.125,(2002),231-244.
46 Y. M. Wei, G. R. Wang.The pertubation theory for the Drazin inverse and its applications. Linear Algebra Appl.,1997,258:178-186
47魏益民Banach空间中的Dazin逆的表示和扰动.数学年刊,2000,21A：910：33～38
48 J. Cai, G. L Chen. Pertubation Analysis for the Drazin Inverses of Bounded Lin-ear Operators on a Banach Space. Joural of Mathematical Research/Exposition, 2004, Vol.24, No.3:421-429
49陈果良,薛以锋,蔡静Hilbert空间中稳定扰动下广义逆的误差估计界.华东师范大学学报,2002,3：1～6
50 Stewart G W, Sun J G. Matrix Perturbation Theory[M]. New York:Academic Press,1976
51 G. L Chen,Y. F. Xue. The expression of the generalized inverse of the perturbed operator under Type Ⅰ perturbation in Hilbert spaces[J]. Linear Algebra Appl, 1998,285:1-6
52 Ding. J. Perturbation results for projecting a point onto a linear manifold[J]. SIAM Matrix Anal Appl,1998,19(3):696-700
53定光桂Banach空间引论.北京：科学出版社.1984
54王玉文.巴拿赫空间中算子广义逆理论及其应用.北京：科学出版社.2005.1
55王玉文,潘少荣Banach空间中有限秩算子的逼近问题.中国科学,2002,32A(9)：837～841
56马海风,李双臻,刘冠琦,王玉文Banach空间中有界线性算子的Moore-Pen-rose度量广义逆的扰动分析.哈尔滨师范大学自然科学学报,2008,24(6)：1-3.
57马海凤,王玉文.单值度量广义逆的扰动分析.哈尔滨师范大学自然科学学报,2006,22(2)：8-10.
58 Tosio Kato. Pertubation Theory for Linear Operators. Springer-Verlag. Berlin Heidel berg. New York.1980
59 R.B.Holmes. Geometric Functional Analysis and Its Application. New York, Heideberg, Berlin:Epring-Verlage,1975.
60张恭庆,林源渠.泛函分析讲义.北京：北京大学出版社.2003