Banach代数元的广义逆与度量广义逆的研究
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摘要
论文主要内容可分为两大部分：第一部分主要研究Banach代数中广义逆的若干问题,重点文研究(p,q)型-广义逆,内容包括第二章和第三章；第二部分主要研究Banach空间中非线性算子广义逆的若干问题,重点研究有界齐性广义逆和Moore-Penrose度量广义逆的理论及应用,这部分内容包含第四章和第五章.具体内容可概括如下：
设A为一个具有单位元的Banach代数.设元素a∈A而p, q∈A为幂等元.在第二章中,我们对元素a定义了一种新的具有指定幂等元的广义逆ap,q(2,l),并深入研究了几种(p,q)型-广义逆的存在性条件和表示；设δa∈A而p',q'∈A为幂等元,令α=α+δα∈A.在第三章中,我们首先研究了代数A中几种(p,q)型-广义逆的稳定扰动及表示,然后借助于gap函数和幂等元的性质,得到了当元素a,p,q都有小的范数扰动时,范数和的上界估计
设X,Y为Banach空间,设T,δT：X→Y为有界线性算子.令T=T+δT.在第四章中，我们首先刻画了算子T的有界齐性广义逆Th的性质,然后研究了有界齐性广义逆Th和拟线性投影广义逆TH的稳定扰动及表示,同时利用Th的性质给出了度量广义逆的一个表示TM=(IX-πN(T))ThπR(T);在第五章中,利用Banach空间光滑性条件,我们首先给出了扰动算子T的Moore-Penrose度量广义逆TM具有最简表示形式TM(IY+δTTM)-1的等价条件.其次借助于子空间的gap函数,我们在Lp(Ω,μ)空间中给出了‖TM-TM‖的上界估计,同时也在一般的自反严格凸Banach空间中研究了TM的扰动界和TM的存在性条件.设b,b=b+δb∈Y,利用Moore-Penrose度景广义逆,我们也研究了算子方程Tx=b最佳逼近解的稳定性,即给出了的上界估计式.
The researches of this thesis mainly divide into two parts: The first part is con-cerned with some problems about the generalized inverses of Banach algebraic elements, we mainly study some kinds of (p,q)-generalized inverses, these results are included in Chapter2and Chapter3; The second one is devoted to the nonlinear operator generalized inverses in Banach spaces, we mainly investigate the homogeneous operator generalized inverse, the Moore-Penrose metric generalize inverse and its applications, these contain Chapter4and Chapter5. Our main results are as follows:
Let A be a complex Banach algebra with the unit1. Let α∈A. Let p, q∈A be idempotent elements. In Chapter2, we first define a new generalized inverse αp,q(2,l) with prescribed idempotents p, q for the element a. Then, some new characterizations and explicit representations for these generalized inverses, such as αp,q,(2) αp,q(1,2) and αp,q(2,l) are presented; Let δα∈A, let p', q'∈A be idempotent elements. Set α=α+δα∈A. In Chapter3, we first study the stable perturbations and expressions for the generalized inverses of perturbed element α, then, by using the gap function and idempotent elements, we investigate the error estimate for and when the elements p, q and α all have some small perturbations.
Let X, Y be Banach spaces. Let T,δT: X→Y be bounded linear operators from X to Y. Set T=T+δT. In Chapter4, we first characterize the existence of a homogeneous generalized inverse Th of the bounded linear operator T. Then, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse Th and quasi-linear projector generalized inverse TH of T. By using the property of Th. we also give a representation: TM=(Ix-πN(T))ThπR(T);In Chapter5, by means of smooth geometric assumptions of Bauach spaces, we first give some equivalent conditions for the Moore-Penrose metric generalized inverse of perturbed operator TM to have the simplest expression TM(IY+δTTM)-1. Then, by means of gap function, we obtain some error estimations of‖TN-TM‖in the Banach space Lp(Ω,μ). Simple results related to the existence and perturbed bound of TM are also given in a general reflexive and strictly convex Banach space. Finally, by using the Moore-Penrose metric generalized inverse, we give a further study on the stability of the best approximate solutions of the operator equation Tx=b. and also get the perturbed bound of where b＝b+δb∈Y.
设A为一个具有单位元的Banach代数.设元素a∈A而p, q∈A为幂等元.在第二章中,我们对元素a定义了一种新的具有指定幂等元的广义逆ap,q(2,l),并深入研究了几种(p,q)型-广义逆的存在性条件和表示；设δa∈A而p',q'∈A为幂等元,令α=α+δα∈A.在第三章中,我们首先研究了代数A中几种(p,q)型-广义逆的稳定扰动及表示,然后借助于gap函数和幂等元的性质,得到了当元素a,p,q都有小的范数扰动时,范数和的上界估计
设X,Y为Banach空间,设T,δT：X→Y为有界线性算子.令T=T+δT.在第四章中，我们首先刻画了算子T的有界齐性广义逆Th的性质,然后研究了有界齐性广义逆Th和拟线性投影广义逆TH的稳定扰动及表示,同时利用Th的性质给出了度量广义逆的一个表示TM=(IX-πN(T))ThπR(T);在第五章中,利用Banach空间光滑性条件,我们首先给出了扰动算子T的Moore-Penrose度量广义逆TM具有最简表示形式TM(IY+δTTM)-1的等价条件.其次借助于子空间的gap函数,我们在Lp(Ω,μ)空间中给出了‖TM-TM‖的上界估计,同时也在一般的自反严格凸Banach空间中研究了TM的扰动界和TM的存在性条件.设b,b=b+δb∈Y,利用Moore-Penrose度景广义逆,我们也研究了算子方程Tx=b最佳逼近解的稳定性,即给出了的上界估计式.
The researches of this thesis mainly divide into two parts: The first part is con-cerned with some problems about the generalized inverses of Banach algebraic elements, we mainly study some kinds of (p,q)-generalized inverses, these results are included in Chapter2and Chapter3; The second one is devoted to the nonlinear operator generalized inverses in Banach spaces, we mainly investigate the homogeneous operator generalized inverse, the Moore-Penrose metric generalize inverse and its applications, these contain Chapter4and Chapter5. Our main results are as follows:
Let A be a complex Banach algebra with the unit1. Let α∈A. Let p, q∈A be idempotent elements. In Chapter2, we first define a new generalized inverse αp,q(2,l) with prescribed idempotents p, q for the element a. Then, some new characterizations and explicit representations for these generalized inverses, such as αp,q,(2) αp,q(1,2) and αp,q(2,l) are presented; Let δα∈A, let p', q'∈A be idempotent elements. Set α=α+δα∈A. In Chapter3, we first study the stable perturbations and expressions for the generalized inverses of perturbed element α, then, by using the gap function and idempotent elements, we investigate the error estimate for and when the elements p, q and α all have some small perturbations.
Let X, Y be Banach spaces. Let T,δT: X→Y be bounded linear operators from X to Y. Set T=T+δT. In Chapter4, we first characterize the existence of a homogeneous generalized inverse Th of the bounded linear operator T. Then, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse Th and quasi-linear projector generalized inverse TH of T. By using the property of Th. we also give a representation: TM=(Ix-πN(T))ThπR(T);In Chapter5, by means of smooth geometric assumptions of Bauach spaces, we first give some equivalent conditions for the Moore-Penrose metric generalized inverse of perturbed operator TM to have the simplest expression TM(IY+δTTM)-1. Then, by means of gap function, we obtain some error estimations of‖TN-TM‖in the Banach space Lp(Ω,μ). Simple results related to the existence and perturbed bound of TM are also given in a general reflexive and strictly convex Banach space. Finally, by using the Moore-Penrose metric generalized inverse, we give a further study on the stability of the best approximate solutions of the operator equation Tx=b. and also get the perturbed bound of where b＝b+δb∈Y.
引文
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[2]V. Barbu and Th. Precpuanu, The Convexity and Optimization in Banach Spaces. Acad. Rep. Soc., Romania, Bucuresti, 1978.
[3]B. Beauzamy, Introduction to Banach Spaces and Their Geometry. North—Holland Mathematics Studies, 68, North-Holland Publishing Company, 1982.
[4]A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications. 2ed., Springer-Verlag, New York, 2003.
[5]N. Castro-Gonzalez and J. Koliha, Perturbation of Drazin inverse for closed linear operators, Integral Equations and Operator Theory. 36 (2000), 92-106.
[6]N. Castro-Gonzalez, J. Koliha and V. R.akocevie, Continuity and general pertur-bation of Drazin inverse for closed linear operators, Abstr. Appl. Anal., 7 (2002), 355-347.
[7]N. Castro-Gonzalez, J. Koliha and Y. Wei, On integral representation of the Drazin inverse in Banach algebra, Proc. Edinb. Math. Soc, 45 (2) (2002), 327-331.
[8]N. Castro-Gonzalez, J. Koliha and Y. Wei, Error bounds for perturbation of the Drazin inverse of closed operators with equal spectral idempotents, Appl. Anal., 81 (2002), 915-928.
[9]P. G. Cazassa and O. Christenseu, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl., 3 (5) (1997), 543-557.
[10]G. Chen and Y. Xue, Perturbation analysis for the operator equation Tx = b in Banach spaces. J. Math. Anal. Appl.. 212 (1997), no. 1,107-125.
[11]G. Chen and Y. Xue. The expression of the generalized inverse of the perturbed operator under Type Ⅰ perturbation in Hilbert spaces, Linear Algebra. Appl., 285 (1998), 16.
[12]G. Chen, Y. Wei and Y. Xue. The generalized condition numbers of bounded linear operators in Banach spaces. J. Aust. Math. Soc, 76 (2004). 281-290.
[13]G. Chen. Y. Wei and Y. Xue. Perturbation analysis of the least square solution in Hilbert spaces. Linear Algebra Appl., 2-14 (1996), 69-80.
[14]T. Chen, H. Hudzik, W. Kowalewski, et al., Approximative compactness and con-tinuity of metric projector in Banach spaces and applications. Sci. China, Ser. A, 51(2), (2008),293-303.
[15]J. B. Conway, A Course in Functional Analysis,2nd Edition, Springer-Verlag,1990.
[16]D. Cvetkovic-Ilic, X. Liu and J. Zhong, On the (p,q)-outer generalized inverse in Banach algebras, App. Math. Comp.,209 (2) (2009),191-196.
[17]J. Ding, New perturbation results on pseudo-inverses of linear operators in Banach spaces, Linear Algebra Appl.,362 (2003), no.1,229-235.
[18]J. Ding, On the expression of generalized inverses of perturbed bounded linear op-erators, Missouri J. Math. Sci.,15 (2003), no.1,40-47.
[19]J. Ding, Lower and upper bounds in the perturbation of general linear algebraic equations, Appl. Math. Lett.,17 (2004),55-58.
[20]J. Ding and L. Huang, On the perturbation of the least squares solutions in Hilbert spaces, Linear Alg. Appl.,212 (1994),487-500.
[21]J. Ding and L. Huang, A generalization of a classic theorem in the perturbation of linear operators, J. Math. Anal. Appl.,239 (1999),118-123.
[22]J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Mathe-matics, Springer-Verlag, New York,1975.
[23]R. G. Douglas, Banach Algebra Techniques in Operator Theory. Academic Pres New York,1972.
[24]M. Drazin, Pseudoinverse in associative rings and semigroups, Amer. Math. Mon 65 (1958),506-514.
[25]Dragan S. Djordjevic and Y. Wei, Outer generalized inverses in rings, Commun. Algebra,33 (2005),3051-3060.
[26]Dragan S. Djordjevic and Predrag S. Stanimirovic, Splittings of operators and gen-eralized inverses, Publ. Math. Debrecen,59 (1-2) (2001),147-159.
[27]Dragan S. Djordjevic and Predrag S. Stanimirovic, On the generalized Drazin inverse and generalized resolvent, J. Czechoslovak Math.,51 (126) (2001),617-634.
[28]F. Du and Y. Xue, Perturbation analysis of AT,S[2] on Banach spaces, Electronic J. Linear Algebra,23 (2012),586-598.
29] F. Du and Y. Xue, The perturbation of the group inverse under the stable pertur-bation in a unital ring, Filomat,27 (2013),61-70.
30] F. Du and Y. Xue, The characterizations of the stable perturbation of a closed operator by a linear operator in Banach spaces, Linear Algebra Appl..438 (2013 2046-2053.
[311 R. Harte and M. Mbekhta, On generalized inverses in C*-algebras, Stuclia Math., 103 (1992),71-77.
[321 H. Hudzik, Y. Wang and W. Zheng, Criteria for the metric generalized inverse and its selection in Banach spaces, Set-Valued Analysis,16, (2008) 51-61
[33]Q. Huang. L. Zhu, W. Gerig and J. Yu, Perturbation and expression for inner inverses in Banach spaces and its applications, Linear Algebra Appl., 436 (2012), 3721-373.
[34]Q. Huang', L. Zhu and Y. Jian, On stable perturbations for outer inverses of linear operators in Banach spaces, Linear Algebra Appl., 437 (2012). 1942-1954.
[35]Q. Huang and J. Ma. Perturbation analysis of generalized inverses of linear operators in Banach spaces, Linear Algebra Appl., 389 (2004), 355-364.
[36]Q. Huang, On perturbations for oblique projection generalized inverses of closed lin-ear operators in Banach spaces, Linear Algebra Appl., 434 (2011), no. 12, 2468-2474.
[37]T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag. New York, 1984.
[38]J. Koliha, A generalized Drazin inverse, Glasgow Math. J., 38 (1996), 367-381.
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