The reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces
Authors:
ChunYuan Deng and HongKe Du
Journal:
Proc. Amer. Math. Soc. 134 (2006), 33093317
MSC (2000):
Primary 47A05, 46C07, 15A09
Published electronically:
May 12, 2006
MathSciNet review:
2231916
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: In this article, we study the reduced minimum modulus of the Drazin inverse of an operator on a Hilbert space and give lower and upper bounds of the reduced minimum modulus of an operator and its Drazin inverse, respectively. Using these results, we obtain a characterization of the continuity of Drazin inverses of operators on a Hilbert space.
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S. L. Campbell, Recent applications of generalized inverses, Pitman, London, 1982. MR 0666720 (83h:65010)
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J. Ding, L. J. Huang, Perturbation of Generalized Inverses of Linear Operators in Hilbert Spaces, Journal of Mathematical Analysis and Applications, 198(1996), 506515. MR 1376277 (97i:47017)
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D. S. Djordjevic, Y. Wei, Additive results for the generalized Drazin inverse, J. Austral Math. Soc., 73(2002), 115125. MR 1916312 (2003e:47002)
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H. Du, C. Deng, A new characterization of gaps between two subspaces, Proceedings of the American Mathematical Society, 133(2005), 30653070. MR 2159786
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H. Du, C. Deng, The representation and characterization of Drazin inverses of operators on a Hilbert space, Linear Algebra and its Applications, 407(2005), 117124. MR 2161918 (2006d:47001)
 7.
R. E. Hartwig, Gworong Wang, Yimin Wei, Some additive results on Drazin inverses, Linear Algebra and its Applications, 322(2001), 207217. MR 1804521 (2001k:15004)
 8.
D. T. Kato, Perturbation theory for linear operators, Springer, New York, 1966. MR 0203473 (34:3324)
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A. E. Taylar, D. C. Lay, Introduction to functional analysis 2nd Edition, John Wiley Sons, New York, Chichester, Brisbane, Toronto, 1980. MR 0564653 (81b:46001)
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G. Wang, Y. Wei, S. Qiao, Generalized inverses: Theory and Computions, Science Press, Beijing, New York, 2004.
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Yimin Wei, Guoliang Chen, Perturbation of least squares problem in Hilbert spaces, Applied Mathematics and Computation, 121(2001), 177183. MR 1830868 (2002b:47022)
 12.
Yimin Wei, Sanzheng Qiao, The representation and approximation of the Drazin inverse of a linear operator in Hilbert space, Applied Mathematics and Computation, 138(2003), 7789. MR 1950346 (2003j:47001)
 13.
Yimin Wei, A characterization and representation of the generalized inverse and its applications, Linear Algebra and its Applications, 280(1998), 8796. MR 1645022 (99g:15006)
 14.
Yimin Wei, A characterization and representation of the Drazin inverse, SIAM J. Matrix Anal. Appl., 17(1996), 744747. MR 1410699 (97k:15008)
 15.
Liping Zhang, A characterization of the Drazin inverse, Linear Algebra and its Applications, 335(2001), 183188. MR 1850823 (2002f:15007)
 16.
Chao Zhu, Jing Cai, Guoliang Chen, Perturbation analysis for the reduced minimum modulus of bounded linear operator in Banach spaces, Applied Mathematics and Computation, 145(2003), 1321. MR 2005984 (2004g:47005)
 1.
 A. BenIsrael, T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd Edition, SpringerVerlag, New York, 2003. MR 1987382 (2004b:15008)
 2.
 S. L. Campbell, Recent applications of generalized inverses, Pitman, London, 1982. MR 0666720 (83h:65010)
 3.
 J. Ding, L. J. Huang, Perturbation of Generalized Inverses of Linear Operators in Hilbert Spaces, Journal of Mathematical Analysis and Applications, 198(1996), 506515. MR 1376277 (97i:47017)
 4.
 D. S. Djordjevic, Y. Wei, Additive results for the generalized Drazin inverse, J. Austral Math. Soc., 73(2002), 115125. MR 1916312 (2003e:47002)
 5.
 H. Du, C. Deng, A new characterization of gaps between two subspaces, Proceedings of the American Mathematical Society, 133(2005), 30653070. MR 2159786
 6.
 H. Du, C. Deng, The representation and characterization of Drazin inverses of operators on a Hilbert space, Linear Algebra and its Applications, 407(2005), 117124. MR 2161918 (2006d:47001)
 7.
 R. E. Hartwig, Gworong Wang, Yimin Wei, Some additive results on Drazin inverses, Linear Algebra and its Applications, 322(2001), 207217. MR 1804521 (2001k:15004)
 8.
 D. T. Kato, Perturbation theory for linear operators, Springer, New York, 1966. MR 0203473 (34:3324)
 9.
 A. E. Taylar, D. C. Lay, Introduction to functional analysis 2nd Edition, John Wiley Sons, New York, Chichester, Brisbane, Toronto, 1980. MR 0564653 (81b:46001)
 10.
 G. Wang, Y. Wei, S. Qiao, Generalized inverses: Theory and Computions, Science Press, Beijing, New York, 2004.
 11.
 Yimin Wei, Guoliang Chen, Perturbation of least squares problem in Hilbert spaces, Applied Mathematics and Computation, 121(2001), 177183. MR 1830868 (2002b:47022)
 12.
 Yimin Wei, Sanzheng Qiao, The representation and approximation of the Drazin inverse of a linear operator in Hilbert space, Applied Mathematics and Computation, 138(2003), 7789. MR 1950346 (2003j:47001)
 13.
 Yimin Wei, A characterization and representation of the generalized inverse and its applications, Linear Algebra and its Applications, 280(1998), 8796. MR 1645022 (99g:15006)
 14.
 Yimin Wei, A characterization and representation of the Drazin inverse, SIAM J. Matrix Anal. Appl., 17(1996), 744747. MR 1410699 (97k:15008)
 15.
 Liping Zhang, A characterization of the Drazin inverse, Linear Algebra and its Applications, 335(2001), 183188. MR 1850823 (2002f:15007)
 16.
 Chao Zhu, Jing Cai, Guoliang Chen, Perturbation analysis for the reduced minimum modulus of bounded linear operator in Banach spaces, Applied Mathematics and Computation, 145(2003), 1321. MR 2005984 (2004g:47005)
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Additional Information
ChunYuan Deng
Affiliation:
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
Email:
cydeng@263.net
HongKe Du
Affiliation:
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
Email:
hkdu@snnu.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002993906083778
PII:
S 00029939(06)083778
Keywords:
Drazin inverse,
reduced minimum modulus,
gap between two subspaces
Received by editor(s):
May 11, 2005
Received by editor(s) in revised form:
May 31, 2005
Published electronically:
May 12, 2006
Additional Notes:
This research was partially supported by the National Natural Science Foundation of China (10571113).
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
