Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Preservers of Gershgorin Set of Jordan Product of Matrices

  • Received : 2017.07.24
  • Accepted : 2018.03.09
  • Published : 2018.12.23
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Abstract

For $A,B{\in}M_2(\mathbb{C})$, let the Jordan product be AB + BA and G(A) the eigenvalue inclusion set, the Gershgorin set of A. Characterization is obtained for maps ${\phi}:M_2(\mathbb{C}){\rightarrow}M_2(\mathbb{C})$ satisfying $$G[{\phi}(A){\phi}(B)+{\phi}(B){\phi}(A)]=G(AB+BA)$$ for all matrices A and B. In fact, it is shown that such a map has the form ${\phi}(A)={\pm}(PD)A(PD)^{-1}$, where P is a permutation matrix and D is a unitary diagonal matrix in $M_2(\mathbb{C})$.

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References

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