Mathematics Faculty Articles

Title

An Operator Inequality and Matrix Normality

Document Type

Article

Publication Date

6-1-1996

Publication Title

Linear Algebra and its Applications

ISSN

0024-3795

Volume

240

First Page

105

Last Page

110

Abstract

Let A be a bounded linear operator on a Hilbert space H; denote |A| = (A∗A)12and 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)12(|A∗|v,v)12 ∀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 ≠ 12.

Comments

Under an Elsevier user license

DOI

10.1016/0024-3795(94)00189-8

Peer Reviewed

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