• Journal of applied mathematics & informatics
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• 제14권1_2호
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• pp.115-122
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• 2004

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. Let $S_{n}(F)$ be the space of all $n\;\times\;n$ symmetry matrices over a field F with 2, $3\;\in\;F^{*}$, then T is an additive injective operator preserving rank-additivity on $S_{n}(F)$ if and only if there exists an invertible matrix $U\;\in\;M_n(F)$ and an injective field homomorphism $\phi$ of F to itself such that $T(X)\;=\;cUX{\phi}U^{T},\;\forallX\;=\;(x_{ij)\;\in\;S_n(F)$ where $c\;\in;F^{*},\;X^{\phi}\;=\;(\phi(x_{ij}))$. As applications, we determine the additive operators preserving minus-order on $S_{n}(F)$ over the field F.

• 호남수학학술지
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• 제23권1호
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• pp.51-57
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• 2001

Let $\mathcal{B}(H)$ and $\mathcal{B}(K)$ denote the algebras of all bounded linear operators on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. We show that if ${\phi}:\mathcal{B}(H){\rightarrow}\mathcal{B}(K)$ is an additive mapping satisfying ${\phi}({\mid}A{\mid}^2)={\mid}{\phi}(A){\mid}^2$ for every $A{\in}\mathcal{B}(H)$, then there exists a mapping ${\psi}$ defined by ${\psi}(A)={\phi}(I){\phi}(A)$, ${\forall}A{\in}\mathcal{B}(H)$ such that ${\psi}$ is the sum of $two^*$-homomorphisms one of which C-linear and the othere C-antilinear. We will also study some conditions implying the injective and rank-preserving of ${\psi}$.