数值域可加性研究  

On the Additive Property of Numerical Ranges

作　　者：许然 郭亚明 孙星睿 曾志坚 吴德玉[1] 

机构地区：[1]内蒙古大学数学科学学院,内蒙古 呼和浩特

出　　处：《理论数学》2020年第3期221-225,共5页Pure Mathematics

基　　金：内蒙古大学校级大学生创新创业训练计划资助项目(项目编号:201911225);内蒙自然科学基金(2019MS01019)资助项目。

摘　　要：在数值域理论中,线性算子数值域的次可加性是平凡的,但可加性成立的条件却是苛刻的。对有限维复空间上的数值域可加性成立条件进行研究,利用数值域的酉不变性和正规矩阵数值域恰为其特征值的凸组合等性质,先探讨两个Hermite矩阵的数值域可加性成立的充要条件,其次探讨Hermite矩阵和反Hermite矩阵数值域可加性成立的充要条件,最后给出了二维正规矩阵数值域可加性成立的充要条件,即两个矩阵的特征值对应的特征空间有非空交集且第一个矩阵的特征值连线平行于第二个矩阵的特征值连线。In the numerical range theory, the subadditivity of the numerical range of linear operators is trivial, but the conditions for the additive property to be established are harsh. This paper studies the conditions for the additive property to the numerical range on the finite dimensional complex space. By using the unitary invariance of the numerical range and the properties that the numerical range of normal matrix is just the convex combination of its eigenvalues, the necessary and sufficient conditions for the additive property to the numerical range of two Hermite matrices are discussed first. Secondly, the necessary and sufficient conditions for the additive property to the numerical range of Hermite matrix and skew Hermite matrix are discussed. Finally, the necessary and sufficient conditions for the additive property to the numerical range of two-dimensional normal matrix are given, that is, the feature spaces corresponding to the eigenvalues of the two matrices have nonzero intersections and the eigenvalue connection of the first matrix is parallel to the eigenvalue connection of the second matrix.

关 键 词：数值域可加性 酉不变性 HERMITE矩阵 正规矩阵 

分 类 号：O15[理学—数学]