Advanced Refinements of Numerical Radius Inequalities

By taking into account that the computation of the numerical radius is an op- timization problem, we prove, in this paper, several refinements of the numerical radius in- equalities 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) ≤ 1 2 r |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) ≤ 1 2 ∥A∥ + A 2 1 2 . Some related inequalities are also discussed. Received: 23 April 2021, Revised: 20 August 2021, Accepted: 08 September 2021.