An Operator Inequality and Matrix Normality

Authors

Charles R. Johnson, College of William and Mary
Fuzhen Zhang, University of California, Santa BarbaraFollow

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

NSUWorks Citation

Johnson, Charles R. and Zhang, Fuzhen, "An Operator Inequality and Matrix Normality" (1996). Mathematics Faculty Articles. 28.
https://nsuworks.nova.edu/math_facarticles/28

DOI

10.1016/0024-3795(94)00189-8