Generalization of sharp and core partial order using annihilators

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 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 I R . Following the idea we also extend and discuss the recently introduced concept of core partial order.

On some partial orders on a certain subset of the power set of rings

“…Rakić introduced another generalization of the left-sharp and the rightsharp orders in [22]. Namely, for a, b ∈ R, we say that a♯ 1 ≤ b if a ∈ I R and a = p {a} b = bq for some idempotent q ∈ R with r R (a) = r R (q).…”

“…It turns out (see [22]) that when R = B(H) with dim H < ∞, I R is exactly the set G(R) of all group invertible operators in B(H). Independently to [15] where definition (1.1) was introduced, Rakić presented in [22] another generalization of the sharp partial order to rings using annihilators. Namely, for a, b ∈ R, we say that a ≤ ♯ b if a ∈ I R and a = pb = bp for some idempotent p ∈ R with r R (a) = r R (p) and l R (a) = l R (p).…”

“…The first goal of this paper is to extend the concept of the sharp partial order ≤ ♯ (introduced on I R by Rakić in [22]) to J R and study its properties. This is done in Section 2.…”

“…In [3], Baksalary and Trenkler introduced the core partial order on the set of all n × n complex matrices that have the group inverse (see also [13,14]). This order was generalized by Rakić in [22] to I R where R is a Rickart * -ring. Namely, for a, b ∈ R we write…”

On partial orders in proper $*$-rings

“…Mitra introduced in [9] a partial order on the set of all n × n matrices over a field F which have the group inverse. This order, known as the sharp partial order, was generalized in [6] and independently in [13] to rings. The definition from [6] follows.…”