Refined upper bounds for the numerical radius via weighted operator means
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 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.
Cite
@article{arxiv.2602.04134,
title = {Refined upper bounds for the numerical radius via weighted operator means},
author = {Shankhadeep Mondal and Ram Narayan Mohapatra and Kasun Tharuka Dewage},
journal= {arXiv preprint arXiv:2602.04134},
year = {2026}
}