Advanced Refinements of Numerical Radius Inequalities

Document Type : Full Length Article

Authors

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

Abstract

By taking into account that the computation of the numerical radius is an optimization problem, we prove, in this paper, several refinements of the numerical radius inequalities for Hilbert space operators. It is shown, among other inequalities, that if A is a bounded linear operator on a complex Hilbert space, then
ω(A)≤½√(|| |A|2+|A*|2||+|| |A| |A*|+|A*| |A| ||),
where ω(A), ||A||, and |A| are the numerical radius, the usual operator norm, and the absolute value of A, respectively. This inequality provides a refinement of an earlier numerical radius inequality due to Kittaneh, namely,
ω(A)≤½(||A||+||A2||)½.
Some related inequalities are also discussed.

Keywords