数值域可加性研究

On the Additive Property of Numerical Ranges

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。

Abstract: 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.