Maps preserving trace of products of matrices

• URL: http://arxiv.org/abs/2103.12552v2
• Date: Sun, 9 Jan 2022 07:20:28 GMT
• Title: Maps preserving trace of products of matrices
• Authors: Huajun Huang and Ming-Cheng Tsai
• Abstract summary: We prove the linearity and injectivity of two maps $phi_1$ and $phi$ on certain subsets of $M_n$. We apply it to characterize maps $phi_i:mathcalSto mathcalS$ ($i=1, ldots, m$) satisfyingoperatornametr (phi_m(A_m))=operatornametr (A_m)$$ in which $mathcalS$ is the set of $n$-by-$n
• Score: 1.4620086904601473
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove the linearity and injectivity of two maps $\phi_1$ and $\phi_2$ on certain subsets of $M_n$ that satisfy $\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $\phi_i:\mathcal{S}\to \mathcal{S}$ ($i=1, \ldots, m$) satisfying $$\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general, Hermitian, or symmetric matrices for $m\ge 3$, or positive definite or diagonal matrices for $m\ge 2$. The real versions are also given.

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