Numerical Radius and Operator Norm Inequalities

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.