An operator inequality and matrix normality

Abstract

Let A be a bounded linear operator on a Hilbert space H; denote |A| = (A ∗A) 1 2 and the norm of x ϵ H by ‖ x‖. It is proved that |(Au, v)|≤⦀A| au‖ ⦀A ∗| 1−a‖ ∀u, v ϵ H for any 0 < α < 1. In particular, |(Au, v)|≤(|A|u, u) 1 2 (|A ∗|v,v) 1 2 ∀u, v ϵ H. When H is of finite dimension, it is shown that A must be a normal operator if it satisfies |(Au, u)|≤(|A|u, u) a(|A ∗|u, u) 1−a ∀u ϵ H for some real number α ≠ 1 2 .