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2015 Generalization of sharp and core partial order using annihilators
Dragan S. Rakic
Banach J. Math. Anal. 9(3): 228-242 (2015). DOI: 10.15352/bjma/09-3-16


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.


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Dragan S. Rakic. "Generalization of sharp and core partial order using annihilators." Banach J. Math. Anal. 9 (3) 228 - 242, 2015.


Published: 2015
First available in Project Euclid: 19 December 2014

zbMATH: 1322.47037
MathSciNet: MR3296136
Digital Object Identifier: 10.15352/bjma/09-3-16

Primary: 47C10
Secondary: 06A06 , ‎15A09 , 16U99

Keywords: annihilator , core partial order , linear bounded operator , Rickart ring , sharp partial order

Rights: Copyright © 2015 Tusi Mathematical Research Group

Vol.9 • No. 3 • 2015
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