Banach Journal of Mathematical Analysis

Generalization of sharp and core partial order using annihilators

Dragan S. Rakic

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The sharp order is a well known partial order defined on the set of complex matrices with index less or equal one. Following Šemrl's approach, Efimov extended this order to the set of those bounded Banach space operators $A$ for which the closure of the image and kernel are topologically complementary subspaces. In order to extend the sharp order to arbitrary ring $R$ (particulary to Rickart and Rickart $*$-rings) we use the notions of annihilators. The concept of the sharp order is extended to the set $\mathcal{I}_R$ of those elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent. It is proved that the sharp order is a partial order relation on $\mathcal{I}_R$. Following the idea we also extend and discuss the recently introduced concept of core partial order.

Article information

Banach J. Math. Anal., Volume 9, Number 3 (2015), 228-242.

First available in Project Euclid: 19 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47C10: Operators in $^*$-algebras
Secondary: 06A06: Partial order, general 15A09: Matrix inversion, generalized inverses 16U99: None of the above, but in this section

Sharp partial order core partial order linear bounded operator Rickart ring annihilator


Rakic, Dragan S. Generalization of sharp and core partial order using annihilators. Banach J. Math. Anal. 9 (2015), no. 3, 228--242. doi:10.15352/bjma/09-3-16.

Export citation


  • O.M. Baksalary and G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra 58 (2010), no. 6, 681–697.
  • A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, 2nd Edition, Springer, New York, 2003.
  • S.K. Berberian, Baer $^{*}$-rings, Springer-Verlag, New York, 1972.
  • D.S. Djordjević, D.S. Rakić and J. Marovt, Minus partial order in Rickart rings, preprint (available at
  • D.S. Djordjević and V. Rakočevi\' c, Lectures on generalized inverses, Faculty of Scince and Mathematics, University of Niš, 2008
  • M.P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978), 139–141.
  • M.A. Efimov, On the $\stackrel{\sharp}{\leq}$-order on the set of linear bounded operators in Banach Space, Math. Notes 93 (2013), no. 5, 784–788.
  • R.E. Hartwig, How to partialy order regular elements, Math. Japon. 25 (1980), 1–13.
  • N. Jacobson, Structure of rings, Amer. Math. Soc., 1968.
  • I. Kaplansky, Rings of operators, W. A. Benjamin, 1968.
  • S.B. Malik, L. Reuda and N. Thome, Further properties on the core partial order and other matrix partial orders, Linear Multilinear Algebra 62 (2014), no. 12, 1629–1648.
  • J. Marovt, On partial orders in Rickart rings, Linear Multilinear Algebra, DOI: 10.1080/03081087.2014.972314
  • J. Marovt, D.S. Raki\' c and D.S. Djordjević, Star, left-star, and right-star partial orders in Rickart $*$-rings, Linear Multilinear Algebra 63 (2015), no. 2, 343–365.
  • S.K. Mitra, On Group Inverses and the Sharp Order, Linear Algebra Appl. 92 (1987), 17–37.
  • S.K. Mitra, P. Bhimasankaram and S.B. Malik, Matrix partial orders, shorted operators and applications, World Scientific, 2010.
  • P. Šemrl, Automorphisms of $B(H)$ with respect to minus partial order, J. Math. Anal. Appl. 369 (2010), 205–213.