Annals of Functional Analysis

On the perturbation of outer inverses of linear operators in Banach spaces

Abstract

The main concern of this article is the perturbation problem for outer inverses of linear bounded operators in Banach spaces. We consider the following perturbed problem. Let $T\in B(X,Y)$ with an outer inverse $T^{\{2\}}\in B(Y,X)$ and $\delta T\in B(X,Y)$ with $\Vert \delta TT^{\{2\}}\Vert \lt 1$. What condition on the small perturbation $\delta T$ can guarantee that the simplest possible expression $B=T^{\{2\}}(I+\delta TT^{\{2\}})^{-1}$ is a generalized inverse, Moore–Penrose inverse, group inverse, or Drazin inverse of $T+\delta T$? In this article, we give a complete solution to this problem. Since the generalized inverse, Moore–Penrose inverse, group inverse, and Drazin inverse are outer inverses, our results extend and improve many previous results in this area.

Article information

Source
Ann. Funct. Anal. (2018), 10 pages.

Dates
Accepted: 10 July 2017
First available in Project Euclid: 11 December 2017

https://projecteuclid.org/euclid.afa/1512982817

Digital Object Identifier
doi:10.1215/20088752-2017-0041

Subjects
Secondary: 47A58: Operator approximation theory

Citation

Zhu, Lanping; Pan, Weiwei; Huang, Qianglian; Yang, Shi. On the perturbation of outer inverses of linear operators in Banach spaces. Ann. Funct. Anal., advance publication, 11 December 2017. doi:10.1215/20088752-2017-0041. https://projecteuclid.org/euclid.afa/1512982817

References

• [1] A. Ben-Israel and T. N. E. Greville,Generalized Inverses: Theory and Applications, 2nd ed., CMS Books Math./Ouvrages Math. SMC15, Springer, New York, 2003.
• [2] N. Castro-González and J. Y. Vélez-Cerrada,On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces, J. Math. Anal. Appl.341(2008), no. 2, 1213–1223.
• [3] 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.
• [4] J. Ding,On the expression of generalized inverses of perturbed bounded linear operators, Missouri J. Math. Sci.15(2003), no. 1, 40–47.
• [5] 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.
• [6] 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.
• [7] Q. Huang and M. S. Moslehian,Relationship between the Hyers-Ulam stability and the Moore-Penrose inverse, Electron. J. Linear Algebra.23(2012), no. 3, 891–905.
• [8] Q. Huang, L. Zhu, W. Geng, and J. Yu,Perturbation and expression for inner inverses in Banach spaces and its applications, Linear Algebra Appl.436(2012), no. 9, 3721–3735.
• [9] Q. Huang, L. Zhu, and Y. Jiang,On stable perturbations for outer inverses of linear operators in Banach spaces, Linear Algebra Appl.437(2012), no. 7, 1942–1954.
• [10] Q. Huang, L. Zhu, and J. Yu,Some new perturbation results for generalized inverses of closed linear operators in Banach spaces, Banach J. Math. Anal.6(2012), no. 2, 58–68.
• [11] X. Li and Y. Wei,An improvement on the perturbation of the group inverse and oblique projection, Linear Algebra Appl.338(2001), no. 1-3, 53–66.
• [12] J. Ma,Complete rank theorem of advanced calculus and singularities of bounded linear operators, Front. Math. China3(2008), no. 2, 305–316.
• [13] D. Mosić, H. Zou, and J. Chen,The generalized Drazin inverse of the sum in a Banach algebra, Ann. Funct. Anal.8(2017), no. 1, 90–105.
• [14] M. Z. Nashed, “Generalized inverses, normal solvability, and iteration for singular operator equations” inNonlinear Functional Analysis and Applications (Madison, WI, 1970), Academic Press, New York, 1971, 311–359.
• [15] M. Z. Nashed,Inner, outer, and generalized inverses in Banach and Hilbert spaces, Numer. Funct. Anal. Optim.9(1987), no. 3-4, 261–325.
• [16] M. Z. Nashed and X. Chen,Convergence of Newton-like methods for singular operator equations using outer inverses, Numer. Math.66(1993), no. 2, 235–257.
• [17] M. D. Petković,Generalized Schultz iterative methods for the computation of outer inverses, Comput. Math. Appl.67(2014) no. 10, 1837–1847.
• [18] Y. Wei, “Recent results on the generalized inverse $A_{T,S}^{(2)}$” inLinear Algebra Research Advances, Nova Science, New York, 2007, 231–250.
• [19] Q. Xu, C. Song, and Y. Wei,The stable perturbation of the Drazin inverse of the square matrices, SIAM J. Matrix Anal. Appl.31(2009), no. 3, 1507–1520.