Skip to main content
SpringerLink
Log in

New results on common properties of bounded linear operators RS and SR

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let X, Y be Banach spaces, R: X → Y and S: Y → X be bounded linear operators. When λ ≠ 0, we investigate common properties of λI − SR and λI − RS. This work should be viewed as a continuation of researches of Barnes and Lin et al.. We also apply these results obtained to B-Fredholm theory, extensions and Aluthge transforms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aiena, P., González, M.: On the Dunford property (C) for bounded linear operators RS and SR. Integral Equations Operator Theory, 70, 561–568 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Barnes, B. A.: The spectral and Fredholm theory of extensions of bounded linear operators. Proc. Amer. Math. Soc., 105, 941–949 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benhida, C., Zerouali, E. H.: Local spectral theory of linear operators RS and SR. Integral Equations Operator Theory, 54, 1–8 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berkani, M.: Restriction of an operator to the range of its powers. Studia Math., 140, 163–175 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Berkani, M.: An Atkinson-type theorem for B-Fredholm operators. Studia Math., 148, 251–257 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berkani, M., Sarih, M.: On semi B-Fredholm operators. Glasg. Math. J., 43, 457–465 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berkani, M., Koliha, J. J.: Weyl type theorems for bounded linear operators. Acta Sci. Math. (Szeged), 69, 379–391 (2003)

    MathSciNet  Google Scholar 

  9. Grabiner, S.: Uniform ascent and descent of bounded operators. J. Math. Soc. Japan, 34, 317–337 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grabiner, S., Zemánek, J.: Ascent, descent, and ergodic properties of linear operators. J. Operator Theory, 48, 69–82 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Kaashoek, M. A.: Ascent, descent, nullity and defect, a note on a paper by A.E. Taylor. Math. Ann. (2), 172, 105–115 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kordula, V., Müller, V.: On the axiomatic theory of spectrum. Studia Math., 119, 109–128 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Labrousse, J. P.: Les operateurs quasi Fredholm: Une generalisation des operateurs semi Fredholm. Rend. Circ. Mat. Palermo (2), 29, 161–258 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin, C., Yan, Z., Ruan, Y.: Common properties of operators RS and SR and p-hyponormal operators. Integral Equations Operator Theory, 43, 313–325 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin, C., Yan, Z., Ruan, Y.: p-Hyponormal operators are subscalar. Proc. Amer. Math. Soc., 131, 2753–2759 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, C., Yan, Z., Ruan, Y.: w-Hyponormal operators are subscalar. Integral Equations Operator Theory, 50, 165–168 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lindeboom, L., Raubenheimer, H.: On regularities and Fredholm theory. Czechoslovak Math. J., 52, 565–574 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lubansky, R. A.: Koliha-Drazin invertibles form a regularity. Math. Proc. R. Ir. Acad., 107A, 137–141 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Martínez, Meléndez, A.: Topological algebras and α-spectrum. Contemp. Math., 341, 97–104 (2004)

    Article  Google Scholar 

  20. Mbekhta, M., Müller, V.: On the axiomatic theory of spectrum II. Studia Math., 119, 129–147 (1996)

    MathSciNet  MATH  Google Scholar 

  21. Müller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Second edition, Birkhäuser, Basel-Boston-Berlin, 2007

    MATH  Google Scholar 

  22. Orozco, J. S. C., Wawrzyńczyk, A.: Regularities and subspectra for commutative Banach algebras. Int. J. Math. Math. Sci., 2005, 2399–2407 (2005)

    Article  MATH  Google Scholar 

  23. Rakočević, V.: Apostol spectrum and generlizations: A brief survey. Facta Univ. Ser. Math. Inform., 14, 79–108 (1999)

    MATH  Google Scholar 

  24. Ruan, Y., Yan, Z.: Spectral structure and subdecomposability of p-Hyponormal operators. Proc. Amer. Math. Soc., 128, 2069–2074 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Živković-Zlatanović, S. Č., Djordjević, D. S., Harte, R. E.: Left-right Browder and left-right Fredholm operators. Integral Equations Operator Theory, 69, 347–363 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, P. R. China

    Qing Ping Zeng & Huai Jie Zhong

Authors
  1. Qing Ping Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huai Jie Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qing Ping Zeng.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 11171066), Special Funds of National Natural Science Foundation of China (Grant No. 11226113), Specialized Research Fund for the Doctoral Program of Higher Education (Grant Nos. 2010350311001 and 20113503120003), Natural Science Foundation of Fujian Province (Grant Nos. 2011J05002 and 2012J05003) and Foundation of the Education Department of Fujian Province (Grant No. JB10042)

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zeng, Q.P., Zhong, H.J. New results on common properties of bounded linear operators RS and SR . Acta. Math. Sin.-English Ser. 29, 1871–1884 (2013). https://doi.org/10.1007/s10114-013-1758-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-013-1758-3

Keywords

MR(2010) Subject Classification

Search

Navigation