Star, left-star, and right-star partial orders in Rickart ∗-rings
Citations
17 citations
Cites background from "Star, left-star, and right-star par..."
...In [15], authors also generalized the left-star and the right-star partial orders from B(H) to Rickart ∗-rings....
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...In [15], equivalent definitions of the star partial order ≤ ∗ on A...
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References
199 citations
"Star, left-star, and right-star par..." refers background in this paper
...It is known (see [4] or [12]) that in proper involutory rings, an element a is a ∗-regular element if and only if aa∗ and a∗a are both regular....
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...For example, Drazin [4] defined the star partial order in the following way: A ≤ ∗ B when A∗ A = A∗ B and AA∗ = B A∗, (1) A, B ∈ B(H)....
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...Since any Rickart ∗-ring is a proper involutory ring and since Drazin introduced the star partial order (1) in a broader sense of proper ∗-semigroups, the next theorem is a direct consequence of Theorem 1 ((i) ⇔ (iii))....
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...For example, Drazin [4] defined the star partial order in the following way:...
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...Drazin introduced in [4] the star partial order in a broader sense of proper ∗-semigroups....
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156 citations
"Star, left-star, and right-star par..." refers background in this paper
...Since bb†a = ab†b = ab†a = a, we may conclude from Lemma 1 in [17] (compare items (iv) and (v)) that there exists x ∈ A such that axa = a, xax = x , xa = xb and ax = bx ....
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78 citations
"Star, left-star, and right-star par..." refers result in this paper
...Many of these results are motivated by the corresponding results for complex matrices.[7,8] Even in the case of matrices, the proofs are not elementary....
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76 citations
"Star, left-star, and right-star par..." refers result in this paper
...Many of these results are motivated by the corresponding results for complex matrices.[7,8] Even in the case of matrices, the proofs are not elementary....
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...The result is, in some way, analogous to Theorem 1 in [7] where the case of complex matrices is considered....
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75 citations
"Star, left-star, and right-star par..." refers methods in this paper
...Inspired by a paper of Baksalary and Mitra [9], Liu, Benítez and Zhong generalized in [13] the left-star and the right-star partial orders from the set of all complex m × n matrices to a C† in the following way....
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...Baksalary and Mitra proved that for A, B ∈ Mm,n , A ≤∗ B implies A ∗≤ B and A ≤∗ B, and A ∗≤ B or A ≤∗ B implies A −≤ B....
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...The left-star and the right-star partial orders were introduced by Baksalary and Mitra in [9] in the following way....
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