Annals of Functional Analysis

Perturbation bounds for the Moore–Penrose metric generalized inverse in some Banach spaces

Jianbing Cao and Wanqin Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let X,Y be Banach spaces, and let T, δT:XY be bounded linear operators. Put T¯=T+δT. In this article, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we first present some error estimates of the upper bound of T¯MTM in Lp (1<p<+) spaces. Then, by using the concept of strong uniqueness and modulus of convexity, we further investigate the corresponding perturbation bound T¯MTM in uniformly convex Banach spaces.

Article information

Source
Ann. Funct. Anal., Volume 9, Number 1 (2018), 17-29.

Dates
Received: 19 September 2016
Accepted: 30 January 2017
First available in Project Euclid: 12 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.afa/1499824816

Digital Object Identifier
doi:10.1215/20088752-2017-0020

Mathematical Reviews number (MathSciNet)
MR3758740

Zentralblatt MATH identifier
06841338

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 metric projection

Citation

Cao, Jianbing; Zhang, Wanqin. Perturbation bounds for the Moore–Penrose metric generalized inverse in some Banach spaces. Ann. Funct. Anal. 9 (2018), no. 1, 17--29. doi:10.1215/20088752-2017-0020. https://projecteuclid.org/euclid.afa/1499824816


Export citation

References

  • [1] X. Bai, Y. Wang, G. Liu, and J. Xia, Definition and criterion for a homogeneous generalized inverse, Acta Math. Sinica (Chin. Ser.) 52 (2009), no. 2, 353–360.
  • [2] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Editura Academiei, Bucharest, 1978.
  • [3] J. Cao and Y. Xue, Perturbation analysis of bounded homogeneous generalized inverses on 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] 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.
  • [6] 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.
  • [7] 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.
  • [8] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer, New York, 1966.
  • [9] A. Kroó and A. Pinkus, On stability of the metric projection operator, SIAM J. Math. Anal. 45 (2013), no. 2, 639–661.
  • [10] P. K. Lin, Strongly unique best approximation in uniformly convex Banach spaces, J. Approx. Theory 56 (1989), no. 1, 101–107.
  • [11] H. Ma, S. 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.
  • [12] M. Z. Nashed and G. F. Votruba, A unified approach to generalized inverses of linear operators, II: Extremal and proximal properties, Bull. Amer. Math. Soc. (N.S.) 80 (1974), 831–835.
  • [13] 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.
  • [14] I. Singer, The Theory of Best Approximation and Functional Analysis, CBMS-NSF Regional Conf. Ser. in Appl. Math. 13, Springer, New York, 1974.
  • [15] H. Wang and Y. Wang, Metric generalized inverse of linear operator in Banach space, Chin. Ann. Math. Ser. B 24 (2003), no. 4, 509–520.
  • [16] Y. Wang, Generalized Inverse of Operator in Banach Spaces and Applications, Science Press, Beijing, 2005, available online at http://www.cspm.net.cn/m_single.php?id=8841 (accessed 30 June 2017).
  • [17] Y. Xue, Stable Perturbations of Operators and Related Topics, World Scientific, Hackensack, 2012.
  • [18] 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.