Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 492154, 11 pages
doi:10.1155/2009/492154
Research Article

Numerical Radius and Operator Norm Inequalities

Khalid Shebrawi1 and Hussien Albadawi2

1Department of Applied Sciences, Al-Balqa' Applied University, 19117 Al-Salt, Jordan
2Department of Basic Sciences and Mathematics, Philadelphia University, 19392 Amman, Jordan

Received 4 November 2008; Accepted 2 March 2009

Academic Editor: Sever Dragomir

Copyright © 2009 Khalid Shebrawi and Hussien Albadawi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A general inequality involving powers of the numerical radius for sums and products of Hilbert space operators is given. This inequality generalizes several recent inequalities for the numerical radius, and includes that if A and B are operators on a complex Hilbert space H, then wr(A∗B)≤(1/2)‖|A|2r+|B|2r‖ for r≥1. It is also shown that if Xi is normal (i=1,2,…,n), then ‖∑i=1nXi‖r≤nr−1‖∑i=1n|Xi|r‖. Related numerical radius and usual operator norm inequalities for sums and products of operators are also presented.