Banach Journal of Mathematical Analysis

Perturbation analysis of the Moore–Penrose metric generalized inverse with applications

Jianbing Cao and Yifeng Xue

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Abstract

In this article, based on some geometric properties of Banach spaces and one feature of the metric projection, we introduce a new class of bounded linear operators satisfying the so-called (α,β)-USU (uniformly strong uniqueness) property. This new convenient property allows us to take the study of the stability problem of the Moore–Penrose metric generalized inverse a step further. As a result, we obtain various perturbation bounds of the Moore–Penrose metric generalized inverse of the perturbed operator. They offer the advantage that we do not need the quasiadditivity assumption, and the results obtained appear to be the most general case found to date. Closely connected to the main perturbation results, one application, the error estimate for projecting a point onto a linear manifold problem, is also investigated.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 3 (2018), 709-729.

Dates
Received: 25 August 2017
Accepted: 15 December 2017
First available in Project Euclid: 16 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1529114494

Digital Object Identifier
doi:10.1215/17358787-2017-0064

Mathematical Reviews number (MathSciNet)
MR3824748

Zentralblatt MATH identifier
06946078

Subjects
Primary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
Secondary: 46B20: Geometry and structure of normed linear spaces

Keywords
metric generalized inverse perturbation $(\alpha,\beta)$-USU operator best approximate solution metric projection

Citation

Cao, Jianbing; Xue, Yifeng. Perturbation analysis of the Moore–Penrose metric generalized inverse with applications. Banach J. Math. Anal. 12 (2018), no. 3, 709--729. doi:10.1215/17358787-2017-0064. https://projecteuclid.org/euclid.bjma/1529114494


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References

  • [1] A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht, 2004.
  • [2] J. Cao and Y. Xue, On the simplest expression of the perturbed Moore-Penrose metric generalized inverse, Ann. Univ. Buchar. Math. Ser. 4(LXII) (2013), no. 2, 433–446.
  • [3] J. Cao and Y. Xue, Perturbation analysis of bounded homogeneous operator generalized inverses in Banach spaces, Acta Math. Univ. Comenian. (N.S.) 83 (2014), no. 2, 181–194.
  • [4] J. Cao and W. Zhang, Perturbation of the Moore-Penrose metric generalized inverse in reflexive strictly convex Banach spaces, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 6, 725–735.
  • [5] J. Cao and W. Zhang, Perturbation bounds for the Moore–Penrose metric generalized inverse in some Banach spaces, Ann. Funct. Anal. 9 (2018), no. 1, 17–29.
  • [6] G. Chen and Y. Wei, Perturbation analysis for the projection of a point onto an affine set in a Hilbert space (in Chinese), Chinese Ann. Math. Ser. A 19 (1998), no. 4, 405–410; English translation in Chinese J. Contemp. Math. 19 (1998), no. 3, 245–252.
  • [7] 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.
  • [8] C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Math. 1965, Springer, London, 2009.
  • [9] J. Ding, Minimal distance upper bounds for the perturbation of least squares problems in Hilbert spaces, Appl. Math. Lett. 15 (2002), no. 3, 361–365.
  • [10] J. Ding, On the existence of solutions to equality constrained least-squares problems in infinite-dimensional Hilbert spaces, Appl. Math. Comput. 131 (2002), no. 2-3, 573–581.
  • [11] F. Du, Perturbation analysis for the Moore-Penrose metric generalized inverse of bounded linear operators, Banach J. Math. Anal. 9 (2015), no. 4, 100–114.
  • [12] F. Du and J. Chen, Perturbation analysis for the Moore-Penrose metric generalized inverse of closed linear operators in Banach spaces, Ann. Funct. Anal. 7 (2016), no. 2, 240–253.
  • [13] C. K. Giri and D. Mishra, Comparison results for proper multisplittings of rectangular matrices, Adv. Oper. Theory. 2 (2017), no. 3, 334–352.
  • [14] N. Karmarkar, A new polynomial–time algorithm for linear programming, Combinatorica 4 (1984), no. 4, 373–395.
  • [15] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer, New York, 1966.
  • [16] E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, New York, 1978.
  • [17] A. Kroó and A. Pinkus, On stability of the metric projection operator, SIAM J. Math. Anal. 45 (2013), no. 2, 639–661.
  • [18] P. Liu and Y. Wang, The best generalised inverse of the linear operator in normed linear space, Linear Algebra Appl. 420 (2007), no. 1, 9–19.
  • [19] H. Ma, H. Hudzik, and Y. Wang, Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 1, 45–56.
  • [20] H. Ma, Sh. Sun, Y. Wang, and W. Zheng, Perturbations of Moore–Penrose metric generalized inverses of linear operators in Banach spaces, Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 7, 1109–1124.
  • [21] M. Z. Nashed and G. F. Votruba, “A unified operator theory of generalized inverses” in Generalized Inverses and Applications (Madison, 1973), Publ. Math. Res. Center 32, Academic Press, New York, 1976, 1–109.
  • [22] R. Ni, Moore-Penrose metric generalized inverses of linear operators in arbitrary Banach spaces, Acta Math. Sinica (Chin. Ser.) 49 (2006), no. 6, 1247–1252.
  • [23] S. Shang and Y. Cui, Approximative compactness and continuity of the set-valued metric generalized inverse in Banach spaces, J. Math. Anal. Appl. 422 (2015), no. 2, 1363–1375.
  • [24] I. Singer, The Theory of Best Approximation and Functional Analysis, CBMS Reg. Conf. Ser. Math. 13, SIAM, Philadelphia, 1974.
  • [25] H. Wang and Y. Wang, Metric generalized inverse of linear operator in Banach space, Chinese Ann. Math. Ser. B 24 (2003), no. 4, 509–520.
  • [26] Y. Wang, Generalized Inverse of Operator in Banach Spaces and Applications, Science Press, Beijing, 2005.
  • [27] M. Wei, On the error estimate for the projection of a point onto a linear manifold, Linear Algebra Appl. 133 (1990), 53–75.
  • [28] Y. Xue, Stable Perturbations of Operators and Related Topics, World Scientific, Hackensack, N.J., 2012.
  • [29] Y. Xue and G. Chen, Some equivalent conditions of stable perturbation of operators in Hilbert spaces, Appl. Math. Comput. 147 (2004), no. 3, 765–772.