Banach Journal of Mathematical Analysis

Some new perturbation results for generalized inverses of closed linear operators in Banach spaces

Qianglian Huang, Jiena Yu, and Lanping Zhu

Full-text: Open access

Abstract

We consider the perturbation and expression for the generalized inverse and Moore--Penrose inverse of closed linear operator under a weaker perturbation condition. As a application, we also investigate the perturbation for the Moore--Penrose inverse of closed $EP$ operator. Some new and interesting perturbation results and examples are obtained in this paper.

Article information

Source
Banach J. Math. Anal., Volume 6, Number 2 (2012), 58-68.

Dates
First available in Project Euclid: 13 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1342210160

Digital Object Identifier
doi:10.15352/bjma/1342210160

Mathematical Reviews number (MathSciNet)
MR2945988

Zentralblatt MATH identifier
1257.47004

Subjects
Primary: 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
Secondary: 46A32: Spaces of linear operators; topological tensor products; approximation properties [See also 46B28, 46M05, 47L05, 47L20] 47A58: Operator approximation theory

Keywords
Moore--Penrose inverse, , , , generalized inverse closed linear operator EP operator T-boundedness

Citation

Huang, Qianglian; Zhu, Lanping; Yu, Jiena. Some new perturbation results for generalized inverses of closed linear operators in Banach spaces. Banach J. Math. Anal. 6 (2012), no. 2, 58--68. doi:10.15352/bjma/1342210160. https://projecteuclid.org/euclid.bjma/1342210160


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References

  • A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Springer-Verlag, New York, 2003.
  • E. Boasso, On the Moore–Penrose inverse, EP Banach space operators and EP Banach algebra elements, J. Math. Anal. Appl. 339 (2008), no. 2, 1003–1014.
  • S.L. Campbell and C.D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979. Dover, New York, 1991.
  • P.G. Cazassa and O. Christensen,Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3 (1997), no. 5, 543–557.
  • 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.
  • 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.
  • J. Ding, On the expression of generalized inverses of perturbed bounded linear operators, Missouri J. Math. Sci. 15 (2003), no. 1, 40–47.
  • J. Ding, New perturbation results on pseudo-inverses of linear operators in Banach spaces, Linear Algebra Appl. 362 (2003), no. 1, 229–235.
  • J. Ding and L. Huang, On the continuity of generalized inverses of linear operators in Hilbert spaces, Linear Algebra Appl. 262 (1997), no. 1, 229–242.
  • 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.
  • Q. Huang and J. Ma, Continuity of generalized inverses of linear operators in Banach spaces and its applications, Appl. Math. Mech. 26 (2005), no. 12, 1657–1663.
  • Q. Huang and W. Zhai, Perturbations and expressions for generalized inverses in Banach spaces and Moore–Penrose inverses in Hilbert spaces of closed linear operators, Linear Algebra Appl. 435 (2011), no. 1, 117–127.
  • T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1984.
  • J. Ma, Complete rank theorem of advanced calculus and singularities of bounded linear operators, Front. Math. China. 3 (2008), no. 2, 305–316.
  • D. Mosic and D.S. Djordjevic, EP elements in Banach algebras, Banach J. Math. Anal. 5 (2011), no. 2, 25–32.
  • M.S. Moslehian and Gh. Sadeghi, Perturbation of closed range operators, Turkish J. Math. 33 (2009), no. 2, 143–149.
  • M.Z. Nashed (Ed.), Generalized Inverses and Applications, Academic Press, New York, 1976.
  • M. Reed and B. Simon, Methods of modern mathematical physics $(I, II)$, Academic Press, New York, 1980.
  • G. Wang, Y. Wei and S. Qiao, Generalized Inverses: Theory and Computations, Science Press, Beijing, New York, 2004.
  • Y. Wang and H. Zhang, Perturbation analysis for oblique projection generalized inverses of closed linear operators in Banach spaces,Linear Algebra Appl. 426 (2007), no. 1, 1–11.
  • Y. Wei and G. Chen, Perturbation of least squares problem in Hilbert spaces, Appl. Math. Comp. 121 (2001), no. 2-3, 177–183.
  • Y. Wei and J. Ding, Representations for Moore–Penrose inverses in Hilbert spaces, Appl. Math. Letter. 14 (2001), no. 5, 599–604.
  • J. Weidmann, Linear operators in Hilbert spaces, Springer-Verlag, New York, 1980.
  • Y. Xue and G. Chen, Some equivalent conditions stable perturbation of operators in Hilbert spaces, Appl. Math. Comput. 147 (2004), no. 3, 765–772.
  • X. Yang and Y. Wang, Some new perturbation theorems for generalized inverses of linear operators in Banach spaces, Linear Algebra Appl. 433 (2010), no. 11-12, 1939–1949.