Perturbation analysis of bounded homogeneous generalized inverses on Banach spaces
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Abstract
Let $X, Y$ be Banach spaces and $T: X \to Y$ be a bounded linear operator. In this paper, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse $T^h$ and quasi-linear projector generalized inverse $T^H$ of $T$. Some applications to the representations and perturbations of the Moore-Penrose metric generalized inverse $T^M$ of $T$ are also given. The obtained results in this paper extend some well-known results for linear operator generalized inverses in this field.
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How to Cite
Xue, Y., & Cao, J.
(2014).
Perturbation analysis of bounded homogeneous generalized inverses on Banach spaces.
Acta Mathematica Universitatis Comenianae, 83(2), 181-194.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/94/84
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References
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[12] 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., (Accepted).
[13] Q. Huang, On perturbations for oblique projection generalized inverses of closed linear operators in Banach spaces, Linear Algebra Appl., 434 (2011), no. 12, 2468–2474.
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[19] 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.
[20] Y. Wang, Generalized Inverse of Operator in Banach Spaces and Applications, Science Press, Beijing, 2005.
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[22] Y. Wang and S. Li, Homogeneous generalized inverses of linear operators in Banach spaces, Acta Math. Sinica, 48 (2) (2005), 253–258.
[23] Y. Wang and Sh. Pan, An approximation problem of the finite rank operator in Banach spaces, Sci. Chin. A, 46 (2) (2003) 245–250.
[24] Y. Wei and J. Ding, Representations for Moore–Penrose inverses in Hilbert spaces, Appl. Math. Lett., 14 (2001) 599–604.
[25] Y. Xue, Stable Perturbations of Operators and Related Topics, World Scientific, 2012.
[26] Y. Xue and G. Chen, Some equivalent conditions of stable perturbation of operators in Hilbert spaces, Applied Math. Comput., 147 (2004), 765–772.
[27] Y. Xue and G. Chen, Perturbation analysis for the Drazin inverse under stable perturbation in Banach space, Missouri J. Math. Sci., 19 (2007), 106–120.
[28] Y. Xue, Stable perturbation in Banach algebras, J. Aust. Math. Soc., 83 (2007), 1–14.
[2] P. Cazassa and O. Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl., 3 (5) (1997) 543–557.
[3] N. Castro–González and J. Koliha, Perturbation of Drazin inverse for closed linear operators, Integral Equations and Operator Theory, 36 (2000), 92–106.
[4] N. Castro–González, J. Koliha and V. Rakočevic, Continuity and general perturbation of Drazin inverse for closed linear operators, Abstr. Appl. Anal., 7 (2002), 355–347.
[5] N. Castro–González, 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.
[6] G. Chen, M.Wei and Y. Xue, Perturbation analysis of the least square solution in Hilbert spaces, Linear Algebra Appl., 244 (1996), 69–80.
[7] 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.
[8] 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.
[9] G. Chen and Y. Xue, The expression of generalized inverse of the perturbed operators under type I perturbation in Hilbert spaces, Linear Algebra Appl., 285 (1998), 1–6.
[10] J. Ding, New perturbation results on pseudo-inverses of linear operators in Banach spaces, Linear Algebra Appl., 362 (2003), no. 1, 229–235.
[11] J. Ding, On the expression of generalized inverses of perturbed bounded linear operators, Missouri J. Math. Sci., 15 (2003), no. 1, 40–47.
[12] 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., (Accepted).
[13] Q. Huang, On perturbations for oblique projection generalized inverses of closed linear operators in Banach spaces, Linear Algebra Appl., 434 (2011), no. 12, 2468–2474.
[14] J. Koliha, Error bounds for a general perturbation of Drazin inverse, Appl. Math. Comput., 126 (2002), 181–185.
[15] I. Singer, The Theory of Best Approximation and Functional Analysis, Springer-Verlag, New York, 1970.
[16] T. Kato. Perturbation Theory for Linear Operators. Springer-Verlag, New York, 1984.
[17] P. Liu and Y. Wang, The best generalized inverse of the linear operator in normed linear space, Linear Algebra Appl., 420 (2007) 9–19.
[18] M. Z. Nashed (Ed.), Generalized inverse and Applications ,Academic Press, New York, 1976.
[19] 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.
[20] Y. Wang, Generalized Inverse of Operator in Banach Spaces and Applications, Science Press, Beijing, 2005.
[21] H. Wang and Y. Wang, Metric generalized inverse of linear operator in Banach space, Chin. Ann. Math. B, 24 (4) (2003) 509–520.
[22] Y. Wang and S. Li, Homogeneous generalized inverses of linear operators in Banach spaces, Acta Math. Sinica, 48 (2) (2005), 253–258.
[23] Y. Wang and Sh. Pan, An approximation problem of the finite rank operator in Banach spaces, Sci. Chin. A, 46 (2) (2003) 245–250.
[24] Y. Wei and J. Ding, Representations for Moore–Penrose inverses in Hilbert spaces, Appl. Math. Lett., 14 (2001) 599–604.
[25] Y. Xue, Stable Perturbations of Operators and Related Topics, World Scientific, 2012.
[26] Y. Xue and G. Chen, Some equivalent conditions of stable perturbation of operators in Hilbert spaces, Applied Math. Comput., 147 (2004), 765–772.
[27] Y. Xue and G. Chen, Perturbation analysis for the Drazin inverse under stable perturbation in Banach space, Missouri J. Math. Sci., 19 (2007), 106–120.
[28] Y. Xue, Stable perturbation in Banach algebras, J. Aust. Math. Soc., 83 (2007), 1–14.