Banach Journal of Mathematical Analysis

Generalization of sharp and core partial order using annihilators

Dragan S. Rakic

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Abstract

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

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

Dates
First available in Project Euclid: 19 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1419001714

Digital Object Identifier
doi:10.15352/bjma/09-3-16

Mathematical Reviews number (MathSciNet)
MR3296136

Zentralblatt MATH identifier
1322.47037

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

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

Citation

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. https://projecteuclid.org/euclid.bjma/1419001714


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