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Operator Drazin Inverse

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Generalized Inverses: Theory and Computations

Part of the book series: Developments in Mathematics ((DEVM,volume 53))

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Abstract

Let X be a Hilbert space and L(X) be the vector space of the linear operators from X into X. We denote the set of bounded linear operators from X into X by B(X). In this chapter, we will investigate the definition, basic properties, representation theorem and computational methods for the Drazin inverse of an operator \(T \in B(X)\), \(\mathcal {R}(T^k)\) is closed, where \(k=\mathrm {Ind}(T)\) is the index of T.

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References

  1. A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, 2nd edn. (Wiley, New York-Chichester-Brisbane, 1980)

    Google Scholar 

  2. S. Qiao, The Drazin inverse of a linear operator on Banach spaces. J. Shanghai Normal Univ. 10, 11–18 (1981). in Chinese

    MATH  Google Scholar 

  3. J. Kuang, The representation and approximation for Drazin inverses of linear operators. Numer. Math., J. Chinese Univ., 4, 97–106 (1982). in Chinese

    Google Scholar 

  4. Y. Wei, S. Qiao, The representation and approximation of the Drazin inverse of a linear operator in Hilbert space. Appl. Math. Comput. 138, 77–89 (2003)

    Article  MathSciNet  Google Scholar 

  5. S.L. Campbell (ed.), Recent Application of Generalized Inverses (Pitman, London, 1982)

    Google Scholar 

  6. Y. Wei, A characterization and representation of the Drazin inverse. SIAM J. Matrix Anal. Appl. 17, 744–747 (1996)

    Google Scholar 

  7. Y. Wei, Representation and perturbation of the Drazin inverse in Banach space. Chinese J. Contemp. Math. 21, 39–46 (2000)

    MathSciNet  Google Scholar 

  8. Y. Chen, Representation and approximation for the Drazin inverse \(A^{(d)}\). Appl. Math. Comput. 119, 147–160 (2001)

    MathSciNet  MATH  Google Scholar 

  9. B.A. Barnes, Common operator properties of the linear operators \(RS\) and \(SR\). Proc. Amer. Math. Soc. 126, 1055–1061 (1998)

    Article  MathSciNet  Google Scholar 

  10. R. Walter, Functional Analysis, 2nd edn. (McGraw-Hill, New York, 1991)

    MATH  Google Scholar 

  11. F. Chatelin, Spectral Approximation of Linear Operators (Academic Press, New York, 1983)

    MATH  Google Scholar 

  12. C.D. Meyer, Limits and the index of a square matrix. SIAM J. Appl. Math. 26, 469–478 (1974)

    Article  MathSciNet  Google Scholar 

  13. Y. Wei, On the perturbation of the group inverse and oblique projection. Appl. Math. Comput. 98, 29–42 (1999)

    Article  MathSciNet  Google Scholar 

  14. Y. Wei, G. Wang, The perturbation theory for the Drazin inverse and its applications. Linear Algebra Appl. 258, 179–186 (1997)

    Google Scholar 

  15. J.J. Koliha, A generalized Drazin inverse. Glasgow Math. J. 38, 367–381 (1996)

    Article  MathSciNet  Google Scholar 

  16. V. Rakočević, Y. Wei, The perturbation theory for the Drazin inverse and its applications II. J. Austral. Math. Soc. 70, 189–197 (2001)

    Article  MathSciNet  Google Scholar 

  17. R.E. Cline, T.N.E. Greville, A Drazin inverse for rectangular matrices. Linear Algebra Appl. 29, 54–62 (1980)

    Article  MathSciNet  Google Scholar 

  18. S. Qiao, The weighted Drazin inverse of a linear operator on Banach spaces and its approximation. Numer. Math., J. Chinese Univ., 3, 1–8 (1981). in Chinese

    Google Scholar 

  19. V. Rakočević, Y. Wei, The representation and approximation of the W-weighted Drazin inverse of linear operators in Hilbert spaces. Appl. Math. Comput. 141, 455–470 (2003)

    Google Scholar 

  20. A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, 2nd edn. (Springer, New York, 2003)

    Google Scholar 

  21. V. Rakočević, Y. Wei, A weighted Drazin inverse and applications. Linear Algebra Appl. 350, 25–39 (2002)

    Article  MathSciNet  Google Scholar 

  22. G. Wang, Iterative methods for computing the Drazin inverse and the W-weighted Drazin inverse of linear operators based on functional interpolation. Numer. Math., J. Chinese Univ., 11, 269–280 (1989)

    Google Scholar 

  23. Y. Wei, A characterization and representation of the generalized inverse \(A_{T, S}^{(2)}\) and its applications. Linear Algebra Appl. 280, 87–96 (1998)

    Article  MathSciNet  Google Scholar 

  24. Y. Wei, Index splitting for Drazin inverse and the singular linear system. Appl. Math. Comput. 95, 115–124 (1998)

    Google Scholar 

  25. D. Cai, The Drazin generalized inverse of linear operators. J. Math. (Wuhan) 1, 81–88 (1985). in Chinese

    MathSciNet  MATH  Google Scholar 

  26. H. Du, C. Deng, The representation and characterization of Drazin inverses of operators on a Hilbert space. Linear Algebra Appl. 407, 117–124 (2005)

    Article  MathSciNet  Google Scholar 

  27. P.S. Stanimirović, V.N. Katsikis, H. Ma, Representations and properties of the W-weighted Drazin inverse. Linear Multilinear Algebra 65(6), 1080–1096 (2017)

    Article  MathSciNet  Google Scholar 

  28. C. Deng, H. Du, The reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces. Proc. Amer. Math. Soc. 134, 3309–3317 (2006)

    Article  MathSciNet  Google Scholar 

  29. S.R. Caradus, Generalized Inverses and Operator Theory, vol. 50, Queen’s Papers in Pure and Applied Mathematics (Queen’s University, Kingston, Ontario, 1978)

    Google Scholar 

  30. S.L. Campbell, The Drazin inverse of an operator, in Recent Applications of Generalized Inverses, ed. by S.L. Cambpell (Pitman, 1982)

    Google Scholar 

  31. D. Huang, Group inverses and Drazin inverses over Banach algebras. Integral Equ. Oper. Theory 17, 54–67 (1993)

    Article  MathSciNet  Google Scholar 

  32. N.C. González, J.J. Koliha, Y. Wei, On integral representations of the Drazin inverse in Banach algebras. Proc. Edinburgh Math. Soc. 45, 327–331 (2002)

    Article  MathSciNet  Google Scholar 

  33. D.S. Djordjević, Y. Wei, Additive results for the generalized Drazin inverse. J. Austral. Math. Soc. 73, 115–125 (2002)

    Article  MathSciNet  Google Scholar 

  34. J.J. Koliha, V. Rakčević, Differentiability of the g-Drazin inverse. Studia Math. 168(3), 193–201 (2005)

    Article  MathSciNet  Google Scholar 

  35. C. Deng, Y. Wei, New additive results for the generalized Drazin inverse. J. Math. Anal. Appl. 370, 313–321 (2010)

    Article  MathSciNet  Google Scholar 

  36. A. Dajić, J.J. Koliha, The weighted g-Drazin inverse for operators. J. Aust. Math. Soc. 82(2), 163–181 (2007)

    Article  MathSciNet  Google Scholar 

  37. G. Wang, Y. Wei, Iterative methods for computing the W-weighted Drazin inverses of bounded linear operators in Banach spaces. J. Shanghai Normal Univ. 28, 1–7 (1999). in Chinese

    Google Scholar 

  38. G. Wang, Several approximate methods for the W-weighted Drazin inverse of bounded linear operator in Banach spaces. Numer. Math., J. Chinese Univ., 10, 76–81 (1988). in Chinese

    Google Scholar 

  39. N. Castro González, J.J. Koliha, Y. Wei, Error bounds for perturbation of the Drazin inverse of closed operators with equal spectral projections. Appl. Anal. 81(4), 915–928 (2002)

    Article  MathSciNet  Google Scholar 

  40. X. Wang, H. Ma, R. Nikolov, A note on the perturbation bounds of W-weighted Drazin inverse. Linear Multilinear Algebra 64(10), 1960–1971 (2016)

    Article  MathSciNet  Google Scholar 

  41. X. Wang, H. Ma, M. Cvetković-Ilić, A note on the perturbation bounds of W-weighted Drazin inverse of linear operator in Banach space. Filomat 31(2), 505–511 (2017)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, China

    Guorong Wang

  2. Department of Mathematics, Fudan University, Shanghai, China

    Yimin Wei

  3. Department of Computing and Software, McMaster University, Hamilton, ON, Canada

    Sanzheng Qiao

Authors
  1. Guorong Wang
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  2. Yimin Wei
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  3. Sanzheng Qiao
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Correspondence to Guorong Wang .

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Wang, G., Wei, Y., Qiao, S. (2018). Operator Drazin Inverse. In: Generalized Inverses: Theory and Computations. Developments in Mathematics, vol 53. Springer, Singapore. https://doi.org/10.1007/978-981-13-0146-9_12

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