Research of Norm Inequalities on Matices Algebra

Matices代数范数不等式的研究

• 批准号: 11640146
• 负责人: OKUBO Kazuyoshi
• 金额: $ 2.3万
• 依托单位: HOKKAIDO UNIVERSITY OF EDUCATION
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640146/
• 关键词: Trace; Numerical range; Numerical radius; Spectrum; Square root of matrix; rank reducing; Approximant; Operator radius; Spectral norm; positive semidefinite; Majorization; Golden-Thompson inequality; numerical range

Let M_n be the algebra of all n×n complex matrices. The spectral norm, which is the operator norm of linear operator on C^n, is the one of the important norms on C^n. Another important norm is the numerical radius in M_n, which is of measurement of magnitude of a particle in quantum mechanics. The operator radius ω_ρ ( )(ρ>0) were defined by J.A.R.Holbrook. These radius interpolate among the spectral norm, the numerical radius and the spectral radius. We give an explicit description of all matrices A∈M_2 such that ω_ρ(A)【less than or equal】1. This description leads to the formulas for ρ-radii when the eigenvalues of such matrices either have equal absolute values or (mod π) argument.Trace inequalities for multiple products of powers of two matrices are discussed via the method of log majorization. For instance, the trace inequality|Tr (A^<p1>B^<q1>A^<p2>B^<q2>…A^<pK>B^<qK>|【less than or equal】Tr (AB)is obtained for positive semidefinite matrices A, B and p_i, q_i【greater than or equal】0 with p1+…+pK=q1+…+qK=1 under some additional condition.For A∈M_n (C), let W (A) denote the numerical range of A.It is shown that if W (A)∩(-∞, 0)=φ, then A has a unique square root B∈M_n (C) with W (B) in the closed right half plane.

设M_n是所有n×n复矩阵的代数。谱范数是C^n上线性算子的算符范数，是C^n上的重要范数之一，另一个重要范数是M_n中的数值半径，它是量子力学中对粒子量级的度量。算子半径ω_ρ()(ρ和0)是由J.A.R.Holbrook定义的。这些半径在谱范数、数值半径和谱半径之间进行内插。给出了所有矩阵A∈M_2的显式刻画，使得ω_ρ(A)[小于或等于]1.当这类矩阵的特征值具有相等的绝对值或(modρ)自变量时，这种刻画导出了π半径的公式.利用对数优选法讨论了两个矩阵的幂的乘积的迹不等式.例如，迹不等式|Tr(A^&lt；p1&gt；B^&lt；q1&gt；A^&lt；p2&gt；B^&lt；q2&gt；…对于具有p1+…的半正定矩阵A，B和p_i，q_i[大于或等于]0，得到了A^&lt；pk&gt；B^&lt；qk&gt；|[小于等于]Tr(AB+PK=Q1+…对A∈M_n(C)，设W(A)表示A的数值范围，证明了如果W(A)∩(-∞，0)=φ，则A有唯一的平方根B∈M_n(C)且W(B)在右半闭平面内.

项目成果

N.Komuro: "Properties on the set of upper bounds in partially ordered linear space"Journal of Hokkaido University of Education. 5. 15-20 (2001)

N.Komoro：“偏序线性空间上界集合的性质”北海道教育大学学报。

大久保和義(共著C.R.Johnson,R.Reams): "Uniquness of matrix square roots and applications"Linear Algebra and its Applications. 323. 51-60 (2001)

Kazuyoshi Okubo（合著者 C.R.Johnson、R.Reams）：“矩阵平方根的唯一性及其应用”线性代数及其应用 323. 51-60 (2001)。

大久保和義(共著I.Spitkovsky): "On the characterization of 2×2 ρ-contraction matrices"Linear Algebra and its Applications. 325. 177-189 (2001)

Kazuyoshi Okubo（合著者 I. Spitkovsky）：“关于 2×2 ρ 收缩矩阵的表征”线性代数及其应用 325. 177-189 (2001)。

長谷川和泉: "Infinitesimal projective transformations on tangent bundle with the horigontal lift connection"Journal of Hokkaido University of Education. (To appear). (2001)

长谷川泉：“具有水平升力连接的切丛上的无穷小射影变换”北海道教育大学学报（待发表）。

大久保和義(共著安藤毅,日合文雄): "Trace inequalities for multiple products of two matrices"Mathematical Inequalities & Applications. 3. 307-318 (2000)

Kazuyoshi Okubo（合著者 Takeshi Ando 和 Fumio Higo）：“跟踪两个矩阵的多重乘积的不等式”数学不等式与应用 3. 307-318 (2000)。