Further inequalities for certain powers of positive definite matrices

Abstract

Let A i , P, i = 1, …, m, be n by n complex matrices such that each A i is positive definite. It is shown, among other inequalities, that If 0 < a j ≤ s n (A j ), and P is Hermitian, then for every unitarily invariant norm , where If s 1(A j ) ≤ b j , and , then for every unitarily invariant norm ,