论文标题
线性K-power的支架和功率产品效果
Linear k-power preservers and trace of power-product preservers
论文作者
论文摘要
令$ v $为$ n \ times n $复合物或真实的一般矩阵，遗传矩阵，对称矩阵，正定义（分别为半明确）矩阵，对角线矩阵或上三角矩阵。修复$ k \ in \ mathbb {z} \ setMinus \ {0，1 \} $。我们表征了$ψ（a^k）=ψ（a^k）=ψ（a）^k $的线性映射$ψ：v \ to v $ in $ v $中的开放型邻居$ s $ i_n $。 $ k $ - 动力保存器一定是$ k $ - 功率的保留剂，而案例$ k = 2 $对应于约旦同构。应用结果，我们将MAPS $ ϕ，ψ：v \到V $表征“ $ \ operatatorName” \ operatatorName {tr}（ϕ（a）ψ（b）^k）= \ operatatorName {tr}（ab^k）$ for ALL $ a，b \ in s $ in S $，$ ϕ $和$ ϕ $和$ψ$都是线性的。”这些特征系统地扩展了文献中的现有结果，并且在量子信息理论等领域中有许多应用。矩阵上的一些结构定理和功率序列被广泛用于我们的特征中。
Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus \{0, 1\}$. We characterize linear maps $ψ:V\to V$ that satisfy $ψ(A^k)=ψ(A)^k$ on an open neighborhood $S$ of $I_n$ in $V$. The $k$-power preservers are necessarily $k$-potent preservers, and the case $k=2$ corresponds to Jordan homomorphisms. Applying the results, we characterize maps $ϕ,ψ:V\to V$ that satisfy "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}(AB^k)$ for all $A\in V$, $B\in S$, and $ψ$ is linear" or "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}(AB^k)$ for all $A, B\in S$ and both $ϕ$ and $ψ$ are linear." The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.
