Refined upper bounds for the numerical radius via weighted operator means

TL;DR

This paper introduces refined upper bounds for the numerical radius of operators on Hilbert spaces using weighted geometric means, improving existing bounds and providing sharper inequalities especially for non-normal operators.

Contribution

The paper develops a new family of inequalities for the numerical radius based on weighted operator means, extending and refining prior bounds with applications to operator matrices.

Findings

New upper bounds for numerical radius using weighted geometric means

Hierarchy of hybrid inequalities combining spectral and numerical radius estimates

Complete characterization of equality cases and examples illustrating strict bounds

Abstract

We establish new upper bounds for the numerical radius of bounded linear operators on a complex Hilbert space by introducing weighted geometric means of the modulus of an operator and its adjoint. This approach yields a family of inequalities that extend and strictly refine several well-known bounds due to Kittaneh and Bhunia--Paul, except in normal or degenerate cases. Further improvements are obtained by interpolating numerical radius estimates with spectral radius bounds, leading to a hierarchy of hybrid inequalities that provide sharper control for non-normal operators. Applications to $2\times2$ operator matrices are presented, and the equality cases are completely characterized, revealing strong rigidity phenomena. Explicit examples are included to illustrate the strictness of the new bounds.