Research of Norm Inequalities on Matices Algebra

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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.