Shorted Operators Relative to a Partial Order in a Regular Ring

Blackwood, Brian; Jain, S. K.; Prasad, Kishore Chandra; Srivastava, Ashish K.

doi:10.1080/00927870902828629

Cited by 19 publications

(12 citation statements)

References 14 publications

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“…Decomposition of a Ring Induced by Minus Partial Order 1045 order as an intermediate step, it is proved in [3] that the equivalence still holds without the additional hypothesis. We show that a stronger result is in fact true.…”

Section: Elamentioning

confidence: 99%

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Decomposition of a ring induced by minus partial order

Rakić¹

2012

ELA

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Abstract. The minus partial order linear algebraic methods have proven to be useful in the study of complex matrices. This paper extends study of minus partial orders to general rings. It is shown that the condition a < − b, where < − is the minus partial order, defines two triples of orthogonal idempotents, and thus, a decomposition of a ring into a direct sum of abelian groups. Hence, several well-known results concerning minus partial order on real and complex matrices are generalized. Analogous decompositions for rectangular matrices over a ring and for Banach space operators are obtained. The equivalent conditions for the invariance of ab (1) a under the choices of b (1) are also obtained. An original inspiration for this work came from the study of minus partial order on complex matrices and from linear algebra methods.

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Section: Elamentioning

confidence: 99%

“…The equivalence of (3.12) and (3.15) was proved in [3] Lemma 3, and equivalence of (3.15)-(3.17) was proved in [9] Theorem 1. Thus, (3.12)-(3.17) are equivalent.…”

Section: Elamentioning

confidence: 99%

Decomposition of a ring induced by minus partial order

Rakić¹

2012

ELA

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“…In [21] it is proved that for complex matrices M and N of the same order, M ≤ − N if and only if G 1 (N ) ⊆ G 1 (M ). This result was latter extended to the setting of regular rings in [2]. In the next proposition we show that the relation "≤ − " is equivalent to the inclusion of the set of {1}-inverses for regular elements on a unital semiprime ring.…”

Section: Proposition 21 Let a Be A Unital Ring The Following Assermentioning

confidence: 66%

“…It is known that u has rank-one if and only if uau = Cu = 0, and that this is equivalent to the condition u = 0 and |σ(xu)| \ {0} ≤ 1, for all x ∈ A (also equivalent to |σ(ux)| \ {0} ≤ 1, for all x ∈ A). For every rank-one element u in A, there exists τ(u) ∈ C, such that u 2…”

Section: Remark 214 Note That the Condition Of Primality Cannot Be mentioning

confidence: 99%

Minus partial order and linear preservers

Burgos

Márquez-García

Campoy

2015

Linear and Multilinear Algebra

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In this paper we investigate the properties of minus partial order in unital rings. We generalize several results well known for matrices and bounded linear operators on Banach spaces. We also study linear maps preserving the minus partial order in unital semisimple Banach algebras with essential socle. A ≤ − Bif and only if rank(B − A) = rank(B ) − rank(A). He proved thatwhere A − denotes a {1}-inverse of A. This partial order is usually named the minus partial order.

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“…In particular, various characterizations of minus partial orderings were given in the literature. Some wellknown characterizations of the minus partial ordering are given below (see, e.g., [4]). …”

Section: Introductionmentioning

confidence: 99%