Abstract
We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F *, thenT is an additive injective operator preserving rank-additivity onS n(F) if and only if there exists an invertible matrixU∈M n(F) and an injective field homomorphism ϕ ofF to itself such thatT(X)=cUX ϕUT, ∀X=(xij)∈Sn(F) wherec∈F *,X ϕ=(ϕ(x ij)). As applications, we determine the additive operators preserving minus-order onS n(F) over the fieldF.
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Xiao-min Tang received his MS from Heilongjang University. Since 1999 he has been at the Heilongjiang University. In 2002, he received a lecturer form Heilongjiang Education Committee. His research interests center on the theory of matrix algebra and its applications.
Chong-guang Cao received his MS from Northeastern Normal University. Since 1981 he has been at the Heilongjiang University. In 1992, he received a Professor form Heilongjiang Education Committee. His research intrests center on the theory of classical group and matrix algebra.
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Tang, XM., Cao, CG. Additive operators preserving rank-additivity on symmetry matrix spaces. JAMC 14, 115–122 (2004). https://doi.org/10.1007/BF02936102
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DOI: https://doi.org/10.1007/BF02936102