Automorphisms of Mn, partially ordered by the star order

Legiša, Peter

doi:10.1080/03081080500209646

Cited by 13 publications

(10 citation statements)

References 17 publications

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“…Since then A * A = A * B if and only if (A * A) * = (A * B) * if and only if A 2 = BA which is equivalent to AA * = BA * , we may conclude that the star, the left-star, and the right-star partial orders are the same partial order on H + n (F). Maps on M n (F) preserving these orders have already been studied (see [10,19]). It would be interesting to describe (surjective) maps that preserve the star order (in both directions) on the set H + n (F) of all real or complex positive semidefinite matrices.…”

Section: Discussionmentioning

confidence: 99%

Preservers of partial orders on the set of all variance-covariance matrices

Golubić

Marovt

2020

Filomat

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Let H+n(R) be the cone of all positive semidefinite n x n real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L?wner and the minus partial orders. Motivated by applications in statistics we study these partial orders on H+n(R). We describe the form of all surjective maps on H+ n (R), n > 1, that preserve the L?wner partial order in both directions. We present an equivalent definition of the minus partial order on H+n(R) and also characterize all surjective, additive maps on H+ n (R), n ? 3, that preserve the minus partial order in both directions.

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Section: Discussionmentioning

confidence: 99%

Preservers of partial orders on the set of all variance-covariance matrices

Golubić

Marovt

2020

Filomat

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“…Since then A * A = A * B if and only if (A * A) * = (A * B) * if and only if A 2 = BA which is equivalent to AA * = BA * , we may conclude that the star, the leftstar, and the right-star partial orders are the same partial order on H + n (F). Maps on M n (F) preserving these orders have already been studied (see [10,19]). It would be interesting to describe (surjective) maps that preserve the star order (in both directions) on the set H + n (F) of all real or complex positive semidefinite matrices.…”

Section: Discussionmentioning

confidence: 99%

Preservers of partial orders on the set of all variance-covariance matrices

Golubić¹,

Marovt²

2019

Preprint

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Let H + n (R) be the cone of all positive semidefinite n×n real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the Löwner and the minus partial orders. Motivated by applications in statistics we study these partial orders on H + n (R). We describe the form of all surjective maps on H + n (R), n > 1, that preserve the Löwner partial order in both directions. We present an equivalent definition of the minus partial order on H + n (R) and also characterize all surjective, additive maps on H + n (R), n ≥ 3, that preserve the minus partial order in both directions.

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“…Among other things, characterizations of isomorphisms on certain partial orders are very interesting topics. Guterman in [9] characterized linear bijective maps on M n preserving the star order and Legiša in [11] considered automorphisms of M n with respect to the star order. Recently, several authors consider star order preserving maps on certain subsets of B(H) or a general von Neumann algebra with respect to the star order when H is infinite dimensional.…”

Section: Introductionmentioning

confidence: 99%

Star order automorphisms on the poset of type 1 operators

Wang

2020

Preprint

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Let H be a complex infinite dimensional Hilbert space and B(H) the algebra of all bounded linear operators on H. The star partial order is defined by A * ≤ B if and only if A * A = A * B and AA * = AB * for any A and B in B(H). We give a type decomposition of operators with respect to star order. For any A ∈ B(H), there are unique type 1 operator A 1 which is 0 or the supremum of those rank 1 operators less than A 1 and type 2 operator A 2 which is not greater than any rank 1 operator in star order such that A i * ≤ A(i = 1, 2) and A = A 1 + A 2 . Moreover, we determine all automorphisms on the poset of type 1 operators. As a consequence, we characterize continuous automorphisms on B(H). keywords star order; type decomposition; automorphism 2010 MSC 06A06, 47B49

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Automorphisms of Mn, partially ordered by the star order

Cited by 13 publications

References 17 publications

Preservers of partial orders on the set of all variance-covariance matrices

Preservers of partial orders on the set of all variance-covariance matrices

Preservers of partial orders on the set of all variance-covariance matrices

Star order automorphisms on the poset of type 1 operators

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