发表机构
Institute of Mathematics, University of the Philippines Diliman; Graduate School of Science and Technology, Sophia University（菲律宾大学迪利曼数学研究所; 上智大学理工研究生院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一类完全PPT映射族，给出了其$k$-正、$k$-余正和$k$-PPT的充要条件，并应用于已知映射，同时探讨相关奇异值不等式。
AI 中文摘要
给定$\alpha\in \mathbb R$和$B_1,\ldots,B_q\in M_{m,n}$，考虑映射$\Phi^{\pm}_{\alpha,\mathfrak{B}}(X)=\alpha \mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*$对所有$X\in M_n$成立。本研究给出了保证$\Phi_{\alpha,\mathfrak{B}}^\pm$分别为$k$-正、$k$-余正和$k$-PPT的关于$\alpha$和$B_1,\ldots,B_q$的充分必要条件。作为应用，恢复了关于映射$\alpha \mbox{tr}(X)I_n\pm X$的已知结果。此外，还考虑了与这些正性约束相关的奇异值不等式。
英文摘要
Given $α\in \mathbb R$ and $ B_1,\ldots,B_q\in M_{m,n}$, consider the map \[Φ^{\pm}_{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on $α$ and $B_1,\ldots,B_q$ are given that guarantee $Φ_{α,\mathfrak{B}}^\pm$ is $k$-positive, $k$-copositive, and $k$-PPT, respectively. As application, known results about the map $α\mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.
