Refined upper bounds for the numerical radius via weighted operator means
Shankhadeep Mondal, Ram Narayan Mohapatra, Kasun Tharuka Dewage
TL;DR
This work introduces a family of refined upper bounds for the numerical radius $w(A)$ of bounded linear operators by employing weighted geometric means of $|A|$ and $|A^*|$, controlled by a parameter $\theta \in [0,1]$. It develops a weighted geometric mean inequality and strengthens it via spectral--numerical interpolation, producing sharper hybrid bounds that improve on classical results (e.g., those of Kittaneh and Bhunia--Paul) except in normal or degenerate cases. The authors extend these bounds to $2\times2$ operator matrices and fully characterize equality scenarios, revealing rigidity phenomena where equality occurs only in highly structured situations (often normal or scalar cases). They also provide explicit finite-dimensional examples illustrating the improvements and rigidity, highlighting the practical relevance for non-normal operators. The approach blends operator means with spectral information, offering a framework that could extend to broader operator settings and applications in matrix analysis and frame theory.
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.