Numerical Radius Inequalities for Sums and Products of Operators

Wasim Audeh

doi:10.4236/alamt.2019.93003

Abstract

A numerical radius inequality due to Shebrawi and Albadawi says that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then for all r≥1. We give sharper numerical radius inequality which states that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then where . Moreover, we give many numerical radius inequalities which are sharper than related inequalities proved recently, and several applications are given.

Highlights

Hirzallah, Kittaneh and Shebrawi have proved in [8] that: If X ∈ B ( H ), :X + Re X − Im X ≤ w( X ) (4)they proved that: Re X − Im X − X + 2 + 2 ≤ w(X ) (5)they showed that: if X,Y ∈ B ( H ), : w 0 Y
The second lemma follows from the spectral theorem for positive operators and Jensen’s inequality
The first result in this paper is numerical radius inequality which is sharper than the inequality (7)

Summary

Introduction

We give many numerical radius inequalities which are sharper than related inequalities proved recently, and several applications are given. An important property of the numerical radius norm is its weak unitary invariance, that is, for X ∈ B ( H ) , w(U ∗ XU ) = w( X )

Objectives

Results

Conclusion