Skip to main content
SpringerLink
Log in

Core-Nilpotent Endomorphisms of Infinite-Dimensional Vector Spaces

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

The aim of this work is to develop a general theory of core-nilpotent endomorphisms of arbitrary vector spaces, such that endomorphisms of finite-dimensional vector spaces and finite potent endomorphisms of infinite-dimensional vector spaces are particular cases of the CN-endomorphisms studied in this theory. For these CN-endomorphisms, we introduce an index that generalizes the index of a finite square matrix and we prove the existence of the Drazin inverse and reflexive generalized inverses. In particular, we characterize all endomorphisms that have Drazin inverse on arbitrary vector spaces. Moreover, we offer a method to study infinite linear systems associated with CN-endomorphisms.

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. Argerami, M., Szechtman, F., Tifenbach, R.: On Tate’s trace. Linear Multilinear Algebra 55(6), 515–520 (2007)

    Article  MathSciNet  Google Scholar 

  2. Cabezas Sánchez, V., Pablos Romo, F.: Explicit solutions of infinite systems of linear equations from reflexive generalized inverses of finite potent endomorphisms. Linear Algebra Appl. 559, 125–144 (2018)

    Article  MathSciNet  Google Scholar 

  3. Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65(7), 506–514 (1958)

    Article  MathSciNet  Google Scholar 

  4. Mitra, S.K., Bhimasankaram, P., Malik, S.B.: Matrix partial orders, shorted operators and applications, series in algebra, vol. 10. World Scientific Publishing Company, Singapore (2010). https://doi.org/10.1142/7170

  5. Pablos Romo, F.: Classification of finite potent endomorphisms. Linear Algebra Appl. 440, 266–277 (2014)

    Article  MathSciNet  Google Scholar 

  6. Pablos Romo, F.: Core-Nilpotent decomposition and new generalized inverses of finite potent endomorphisms. Linear Multilinear Algebra 68(11), 2254–2275 (2020)

    Article  MathSciNet  Google Scholar 

  7. Pablos Romo, F.: On the classification of endomorphisms on infinite-dimensional vector spaces. Georg. Math. J. (2020). https://doi.org/10.1515/gmj-2020-2054

  8. Pablos Romo, F.: On the Drazin inverse of finite potent endomorphisms. Linear Multilinear Algebra 67(10), 2135–2146 (2019)

    Article  MathSciNet  Google Scholar 

  9. Robert, P.: On the group-inverse of a linear transformation. J. Math. Anal. Appl. 22, 658–669 (1968)

    Article  MathSciNet  Google Scholar 

  10. Tate, J.: Residues of differentials on curves. Ann. Sci. Éc. Norm. Sup., 4a série 1, 149–159 (1968)

Download references

Acknowledgements

The author would like to thank the anonymous reviewer for his/her valuable comments to improve the quality of the paper.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas and Instituto Universitario de Física Fundamental y Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008, Salamanca, Spain

    Fernando Pablos Romo

Authors
  1. Fernando Pablos Romo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fernando Pablos Romo.

Additional information

To my friends of the group EMMM for their support during the preparation of this work.

Publisher's Note

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

This work is partially supported by the Spanish Government research projects nos. MTM2015-66760-P and PGC2018-099599-B-I00 and the Regional Government of Castile and Leon research project no. J416/463AC03.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pablos Romo, F. Core-Nilpotent Endomorphisms of Infinite-Dimensional Vector Spaces. Mediterr. J. Math. 18, 84 (2021). https://doi.org/10.1007/s00009-021-01732-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-021-01732-6

Keywords

Mathematics Subject Classification

Search

Navigation