Research of Norm Inequalities on Matices Algebra

设M_n是所有n×n复矩阵的代数。谱范数是C^n上线性算子的算子范数，是C^n上重要的范数之一。另一个重要的指标是M_n中的数值半径，它是量子力学中粒子大小的度量。算子半径ω_ρ（）（ρ>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插值。本文给出了所有矩阵A∈M_2使得ω_ρ（A）[小于或等于]1的显式描述.当这类矩阵的特征值具有相等的绝对值或（mod π）辐角时，给出了这类矩阵的ρ-半径的计算公式，并利用对数优化方法讨论了两个矩阵的幂的重积的迹不等式.例如，迹不等式|Tr（A^<p1>B^<q1>A^<p2>B^<q2>. A^<pK>B^<qK>|在一定条件下，得到了半正定矩阵A，B和pi，qi [大于等于]0且p1+.+pK=q1+.+qK=1的Tr（AB）.对于A∈ Mn（C），设W（A）表示A的数值值域.证明了若W（A）&lt;$（-∞，0）=φ，则A在闭右半平面上有唯一的平方根B∈M_n（C），其中W（B）。

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.