Additive mappings on symmetric matrices

Kuzma, Bojan; Orel, Marko

doi:10.1016/j.laa.2006.02.010

Cited by 11 publications

(8 citation statements)

References 18 publications

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“…By the structural results of rank-one linear preservers and rank-one non-increasing linear mappings on symmetric matrices (see a complete result under a more general setting in [8], [12]), the structure of ψ can be established immediately.…”

Section: It Is Easily Verified Thatmentioning

confidence: 99%

Linear spaces and preservers of bounded rank-two per-symmetric triangular matrices

Chooi¹,

Kwa²,

Lim³

et al. 2014

ELA

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Abstract. Let F be a field and m, n be integers m, n 3. Let SMn(F) and STn(F) denote the linear space of n × n per-symmetric matrices over F and the linear space of n × n per-symmetric triangular matrices over F, respectively. In this note, the structure of spaces of bounded rank-two matrices of STn(F) is determined. Using this structural result, a classification of bounded rank-two linear preservers ψ : STn(F) → SMm(F), with F of characteristic not two, is obtained. As a corollary, a complete description of bounded rank-two linear preservers between per-symmetric triangular matrix spaces over a field of characteristic not two is addressed.

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Section: It Is Easily Verified Thatmentioning

confidence: 99%

Linear spaces and preservers of bounded rank-two per-symmetric triangular matrices

Chooi¹,

Kwa²,

Lim³

et al. 2014

ELA

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“…We need the following result proved in [7,10] for the spaces of symmetric matrices and in [16] for the spaces of Hermitian matrices where K = F . (i) η(X) = ζ(X)B for some additive functional ζ : H m → F and some rank-one matrix B ∈ H n , (ii) η(X) = λP X σ P * for some n × m matrix P over K, some nonzero endomorphism σ of K commuting with the automorphism −, and some nonzero λ ∈ F .…”

Section: Mh Limmentioning

confidence: 99%

“…Many authors have studied the structures of linear and additive rank-one preservers and rank-one non-increasing maps between spaces of Hermitian matrices [2], [3], [7]- [11], [16]- [18]. This paper is concerned with linear and additive maps on tensor products of spaces of Hermitian matrices that carry the set of tensor product of rank-one matrices into itself.…”

Section: Introductionmentioning

confidence: 99%

Additive preservers of tensor product of rank one hermitian matrices

Lim

2012

ELA

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Abstract. Let K be a field of characteristic not two or three with an involution and F be its fixed field. Let Hm be the F -vector space of all m-square Hermitian matrices over K. Let ρm denote the set of all rank-one matrices in Hm. In the tensor product spaceIn this paper, additive maps T from Hm ⊗ Hn to Hs ⊗ Ht such that T (ρm ⊗ ρn) ⊆ (ρs ⊗ ρt) ∪ {0} are characterized. From this, a characterization of linear maps is found between tensor products of two real vector spaces of complex Hermitian matrices that send separable pure states to separable pure states. Also classified in this paper are almost surjective additive maps L fromWhen K is algebraically closed and K = F , it is shown that every linear map on k i=1 Hm i that preserves k i=1 ρm i is induced by k bijective linear rank-one preservers on Hm i , i = 1, . . . , k.

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“…We show, in Lemma 2.5, that every nonzero compound-commuting additive map is rank-one nonincreasing, i.e., the map sends rank one matrices to matrices of rank less than or equal to one. By using the structural results of rank-one nonincreasing additive maps on the space of block triangular matrices in [3], symmetric matrices in [8,7], and Hermitian matrices in [9,10,11], respectively, we characterize compoundcommuting additive maps on the spaces of square matrices, triangular matrices and symmetric matrices over arbitrary fields, and on the space of Hermitian matrices over a field with proper involution. We will see in Theorem 2.8 that the classification of compound-commuting additive maps on triangular matrices is quite different and essentially more complicated than the corresponding theorems on spaces of square matrices, symmetric matrices, and Hermitian matrices.…”

Section: Is φ-Commuting If (ψ • φ)(A) = (φ • ψ)(A) For Allmentioning

confidence: 99%

Compound-Commuting Additive Maps on Matrix Spaces

Chooi¹

2011

Journal of the Korean Mathematical Society

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Abstract. In this note, compound-commuting additive maps on matrix spaces are studied. We show that compound-commuting additive maps send rank one matrices to matrices of rank less than or equal to one. By using the structural results of rank-one nonincreasing additive maps, we characterize compound-commuting additive maps on four types of matrices: triangular matrices, square matrices, symmetric matrices and Hermitian matrices.

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Additive mappings on symmetric matrices

Cited by 11 publications

References 18 publications

Linear spaces and preservers of bounded rank-two per-symmetric triangular matrices

Linear spaces and preservers of bounded rank-two per-symmetric triangular matrices

Additive preservers of tensor product of rank one hermitian matrices

Compound-Commuting Additive Maps on Matrix Spaces

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