Operator Inequalities and Related Norm Inequalities

算子不等式和相关的范数不等式

• 批准号: 10640183
• 负责人: ANDO Tsuyoshi
• 金额: $ 1.6万
• 依托单位: Hokusei Gakuen University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640183/
• 关键词: matrix inequality; norm inequality; trace; eigenvalue inequality; operator monotone function; 凸集合; 端点; 同時縮小化可能性; 束構造

We set the following three actual research objectives :(1) study of the lattice structure of the cone of positive semi-definite operators,(2) study of super- and lower-subadditivity related to operator concave/convex functions,(3) study of convexity properties related to the trace.Concerning the first objective, we have succeeded in determining the condition for a pair of operators to admit the infimum in the cone of positive semi-definite operators [Reference 2].Concerning the second objective, we have succeeded in establishing the following inequality|||f (A+B)|||【less than or equal】 |||f (A)+f (B)|||,which is valid for any non-negative operator concave function f (t) on the half-line [0, ∞), any pair of positive semi-definite operators A,B and any unitarily invariant norm |||・|||[Reference 3].Concerning the third objective, we have obtained various estimates of the traces of multiple products of two matrices [Refernece 5]. Also we have succeeded in generalizing some known determinant and trace inequalities to majorization type inequalities for eigenvalues [Reference 7].

我们设定了以下三个实际的研究目标：（1）研究半正定算子锥的格结构，（2）研究与算子凹/凸函数有关的超次可加性和下次可加性，（3）研究与迹有关的凸性。关于第一个目标，我们成功地确定了一对算子在半正定算子锥中允许下确界的条件[参考文献2]。关于第二个目标，我们成功地建立了下列不等式||| f（A+B）||| [less大于或等于] ||| f（A）+f（B）|||，它对半直线[0，∞）上的任何非负算子凹函数f（t），任何半正定算子对A，B和任何酉不变范数成立 |||·|||关于第三个目标，我们得到了两个矩阵的重积迹的各种估计。我们还成功地将一些已知的行列式和迹不等式推广到特征值的控制型不等式[参考文献7]。

项目成果

T.Ando and X.Zhan: "Norm inequalties related to operator monotone functions"Mathematische Annalen. 315. 771-780 (1999)

T.Ando 和 X.Zhan：“与算子单调函数相关的范数不等式”Mathematische Annalen。

T.Ando and X.Zhan: "Norm inequalities related to operator monotone functions"Mathematische Annalen. 315. 771-780 (1999)

T.Ando 和 X.Zhan：“与算子单调函数相关的范数不等式”Mathematische Annalen。

T.Ando: "Extreme points of a positive operator ball"Operator Theory : Advances and Applications. 118. 53-66 (2000)

T.Ando：“正算子球的极值点”算子理论：进展和应用。

T.Ando: "Bloomfield-Watson-Knott type inequalities for eigenvalues"Taiwanese Journal of Mathematics. (in press).

T.Ando：“特征值的布卢姆菲尔德-沃森-诺特型不等式”台湾数学杂志。

T.Ando: "Problem of infimum in the positive cone"Analytic and Geometric Inequalities and Applications, Kluwer Academic Publisher, Amsterdam. 1-12 (1999)

T.Ando：“正圆锥中的下确界问题”《分析和几何不等式及其应用》，Kluwer 学术出版社，阿姆斯特丹。