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http://dx.doi.org/10.4134/JKMS.2010.47.4.831

PERTURBATION ANALYSIS OF THE MOORE-PENROSE INVERSE FOR A CLASS OF BOUNDED OPERATORS IN HILBERT SPACES  

Deng, Chunyuan (COLLEGE OF MATHEMATICS SCIENCE SOUTH CHINA NORMAL UNIVERSITY)
Wei, Yimin (SCHOOL OF MATHEMATICAL SCIENCES FUDAN UNIVERSITY)
Publication Information
Journal of the Korean Mathematical Society / v.47, no.4, 2010 , pp. 831-843 More about this Journal
Abstract
Let $\cal{H}$ and $\cal{K}$ be Hilbert spaces and let T, $\tilde{T}$ = T + ${\delta}T$ be bounded operators from $\cal{H}$ into $\cal{K}$. In this article, two facts related to the perturbation bounds are studied. The first one is to find the upper bound of $\parallel\tilde{T}^+\;-\;T^+\parallel$ which extends the results obtained by the second author and enriches the perturbation theory for the Moore-Penrose inverse. The other one is to develop explicit representations of projectors $\parallel\tilde{T}\tilde{T}^+\;-\;TT^+\parallel$ and $\parallel\tilde{T}^+\tilde{T}\;-\;T^+T\parallel$. In addition, some spectral cases related to these results are analyzed.
Keywords
generalized inverse; Moore-Penrose inverse; perturbation; block operator matrix;
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1 Y. Wei and G. Chen, Perturbation of least squares problem in Hilbert spaces, Appl. Math. Comput. 121 (2001), no. 2-3, 177-183.   DOI   ScienceOn
2 Y. Wei and G. Chen, Some equivalent conditions of stable perturbation of operators in Hilbert spaces, Appl. Math. Comput. 147 (2004), no. 3, 765-772.   DOI   ScienceOn
3 Y.Wei and J. Ding, Representations for Moore-Penrose inverses in Hilbert spaces, Appl. Math. Lett. 14 (2001), no. 5, 599-604.   DOI   ScienceOn
4 J. Zhou and G. Wang, Block idempotent matrices and generalized Schur complement, Appl. Math. Comput. 188 (2007), no. 1, 246-256.   DOI   ScienceOn
5 C. Zhu, J. Cai, and G. Chen, Perturbation analysis for the reduced minimum modulus of bounded linear operator in Banach spaces, Appl. Math. Comput. 145 (2003), no. 1, 13-21.   DOI   ScienceOn
6 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), no. 1-3, 1-6.   DOI   ScienceOn
7 J. Ding, New perturbation results on pseudo-inverses of linear operators in Banach spaces, Linear Algebra Appl. 362 (2003), 229-235.   DOI   ScienceOn
8 J. Ding, On the expression of generalized inverses of perturbed bounded linear operators in Banach spaces, Missouri J. Math. Sci. 15 (2003), no. 1, 40-47.
9 J. Ding and L. J. Huang, Perturbation of generalized inverses of linear operators in Hilbert spaces, J. Math. Anal. Appl. 198 (1996), no. 2, 506-515.   DOI   ScienceOn
10 J. Ding and L. J. Huang, On the continuity of generalized inverses of linear operators in Hilbert spaces, Linear Algebra Appl. 262 (1997), 229-242.   DOI   ScienceOn
11 J. Ding and L. J. Huang, On the perturbation of the least squares solutions in Hilbert spaces, Linear Algebra Appl. 212/213 (1994), 487-500.   DOI   ScienceOn
12 J. Ding and Y.Wei, Relative errors versus residuals of approximate solutions of weighted least squares problems in Hilbert space, Comput. Math. Appl. 44 (2002), no. 3-4, 407-411.   DOI   ScienceOn
13 C. W. Groetsch, Generalized Inverses of Linear Operators: representation and approximation, Monographs and Textbooks in Pure and Applied Mathematics, No. 37. Marcel Dekker, Inc., New York-Basel, 1977.
14 M. Z. Nashed, Generalized Inverses and Applications, Academic Press, New York, 1976.
15 G. W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev. 19 (1977), no. 4, 634-662.   DOI   ScienceOn
16 Y. Wei, The representation and approximation for the weighted Moore-Penrose inverse in Hilbert space, Appl. Math. Comput. 136 (2003), no. 2-3, 475-486.   DOI   ScienceOn
17 G. Chen, Y. Wei, and Y. Xue, The generalized condition numbers of bounded linear operators in Banach spaces, J. Aust. Math. Soc. 76 (2004), no. 2, 281-290.   DOI
18 R. H. Bouldin, Generalized inverses and factorizations, Recent applications of generalized inverses, pp. 233–249, Res. Notes in Math., 66, Pitman, Boston, Mass.-London, 1982.
19 S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Surveys and Reference Works in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.
20 G. Chen, Y. Wei, and Y. Xue, Perturbation analysis of the least squares solution in Hilbert spaces, Linear Algebra Appl. 244 (1996), 69-80.   DOI   ScienceOn
21 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.   DOI   ScienceOn