SpringerLink
Log in

On some classes of Saphar type operators

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper we define and study operators of Saphar type in a Banach space, their subclasses, essentially left (right) Drazin invertible operators and left (right) Weyl–Drazin invertible operators, by means of kernels and ranges of powers of an operator. A bounded linear operator T on a Banach space X is said to be of Saphar type if T is a direct sum of a Saphar operator and a nilpotent one. We prove that T is essentially left (right) Drazin invertible if and only if T is a direct sum of a left (right) Fredholm operator and a nilpotent one, as well as that T is left (right) Weyl–Drazin invertible if and only if T is a direct sum of a left (right) Weyl operator and a nilpotent operator. We also consider the corresponding spectra.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Spain)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aiena, P.: Semi-Fredholm Operators. Perturbation Theory and Localized SVEP, Mérida (2007)

    MATH  Google Scholar 

  2. Aiena, P.: Fredholm and local spectral theory II, with application to Weyl-type theorems, Lecture Notes in Mathematics, 2235, Springer (2018)

  3. Aiena, P., Rosas, E.: Single-valued extension property at the points of the approximate point spectrum. J. Math. Anal. Appl. 279, 180–188 (2003)

    Article  MathSciNet  Google Scholar 

  4. Berkani, M.: On a class of quasi-Fredholm operators. Integr. Equ. Oper. Theory 34, 244–249 (1999)

    Article  MathSciNet  Google Scholar 

  5. Berkani, M.: Restriction of an operator to the range of its powers. Studia Math. 140(2), 163–174 (2000)

    Article  MathSciNet  Google Scholar 

  6. Berkani, M.: Index of B-Fredholm operators and generalization of a Weyl’s theorem. Proc. Am. Math. Soc. 130, 1717–1723 (2002)

    Article  MathSciNet  Google Scholar 

  7. Berkani, M., Castro, N., Djordjević, S.V.: Single valued extension property and generalized Weyl’s theorem. Math. Bohem. 131(1), 29–38 (2006)

    Article  MathSciNet  Google Scholar 

  8. Boasso, E.: The Drazin spectrum in Banach algebras. An operator theory summer: Timisoara June 29-July 4 2010. In: International Book Series of Mathematical Texts, Theta Foundation, Bucharest, 21–28 (2012)

  9. Burlando, L.: Continuity of spectrum and spectral radius in algebras of operators, Annales Faculté des Sciences de Toulouse, Vol. IX (1) (1988)

  10. Caradus, S.R.: Generalized Inverses and Operator Theory, Queen’s Papers in Pure and Applied Mathematics, No. 38. Queen’s University, Kingston (1978)

  11. Cvetković, M.D., Živković-Zlatanović, S.Č: A note on Koliha–Drazin invertible operators and a-Browder’s theorem. Complex Anal. Oper. Theory 15(5), 81 (2021)

    Article  MathSciNet  Google Scholar 

  12. Ghorbel, A., Mnif, M.: On a generalization of left and right invertible operators on Banach spaces. Oper. Matrices 16(2), 401–413 (2022)

    Article  MathSciNet  Google Scholar 

  13. Grabiner, S.: Uniform ascent and descent of bounded operators. J. Math. Soc. Jpn. 34(2), 317–337 (1982)

    Article  MathSciNet  Google Scholar 

  14. Harte, R.E.: Invertibility and Singularity for Bounded Linear Operators. Marcel Dekker Inc, New York, Basel (1988)

    MATH  Google Scholar 

  15. Jiang, Q., Zhong, H.: Generalized Kato decomposition, single-valued extension property and approximate point spectrum. J. Math. Anal. Appl. 356, 322–327 (2009)

    Article  MathSciNet  Google Scholar 

  16. Jiang, Q., Zhong, H., Zeng, Q.: Topological uniform descent and Localized SVEP. J. Math. Anal. Appl. 390, 355–361 (2012)

    Article  MathSciNet  Google Scholar 

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

  18. King, C.: A note on Drazin inverses. Pacific J. Math. 70, 383–390 (1977)

    Article  MathSciNet  Google Scholar 

  19. Lay, D.C.: Spectral analysis using ascent, descent, nullity and defect. Math. Ann. 184, 197–214 (1970)

    Article  MathSciNet  Google Scholar 

  20. Mbekhta, M., Müller, V.: On the axiomatic theory of spectrum II. Studia Math. 119(2), 129–147 (1996)

    Article  MathSciNet  Google Scholar 

  21. Miller, T.L., Miller, V.G., Smith, R.C.: Bishop’s property (\(\beta \)) and \(Ces\acute{a}ro operator\). J. Lond. Math. Soc. (2) 58, 197–207 (1998)

    Article  Google Scholar 

  22. Müller, V.: On the Kato decomposition of quasi-Fredholm operators and B - Fredholm operators. In: Proc. Workshop Geometry in Functional Analysis, Erwin Schrodinger Institute, Wien (2000)

  23. Müller, V.: Spectral theory of linear operators and spectral systems in Banach algebras, Birkhäuser (2007)

  24. Pietch, A.: Zur Theorie der -Transformationen in lokalkonvexen Vektorräumen. Math. Nachr. 21, 347–369 (1960)

    Article  MathSciNet  Google Scholar 

  25. Svensson, L.: Sums of complemented subspaces in locally convex spaces. Ark. Mat. 25(1), 147–153 (1987)

    Article  MathSciNet  Google Scholar 

  26. Živković-Zlatanović, S.Č, Berkani, M.: Topological uniform descent, quasi-Fredholmness and operators originated from semi-B-Fredholm theory. Complex Anal. Oper. Theory 13, 3595–3622 (2019)

    Article  MathSciNet  Google Scholar 

  27. Živković-Zlatanović, S.Č, Cvetković, M.D.: Generalized Kato–Riesz decomposition and generalized Drazin–Riesz invertible operators. Linear Multilinear Algebra 65(6), 1171–1193 (2017)

    Article  MathSciNet  Google Scholar 

  28. Živković-Zlatanović, S.Č, Harte, R.E.: Polynomially Riesz perturbations II. Filomat 29(9), 2125–2136 (2015)

    Article  MathSciNet  Google Scholar 

  29. Živković-Zlatanović, S.Č, Djordjević, D.S., Harte, R.E.: On left and right Browder operators. J. Korean Math. Soc. 48, 1053–1063 (2011)

    Article  MathSciNet  Google Scholar 

  30. Živković-Zlatanović, S. C., Djordjević, D. S., Harte, R. E.: Left–right browder and left–right Fredholm operators. Integral Equ. Oper. Theory, 1-18 (2011)

  31. Živković-Zlatanović, S.Č, Djordjević, D.S., Harte, R.E.: Polynomially Riesz perturbations. J. Math. Anal. Appl. 408, 442–451 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, 18000, Niš, Serbia

    Snežana Č. Živković-Zlatanović

  2. Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias F’indsico Matemáticas, R’indo Verde y Av. San Claudio, San Manuel, Puebla, Pue, 72570, Mexico

    Slaviša V. Djordjević

Authors
  1. Snežana Č. Živković-Zlatanović
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Slaviša V. Djordjević
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Snežana Č. Živković-Zlatanović.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author is supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant no. 451-03-68/2020-14/200124 .

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Živković-Zlatanović, S.Č., Djordjević, S.V. On some classes of Saphar type operators. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 170 (2022). https://doi.org/10.1007/s13398-022-01314-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-022-01314-5

Keywords

Mathematics Subject Classification

Search

Navigation