An Alternative Estimate for the Numerical Radius of Hilbert Space Operators

TL;DR

This paper introduces a new lower bound for the numerical radius of operators on Hilbert spaces and identifies specific conditions where the numerical radius of a block operator matrix equals the average of the norms of its components.

Contribution

It provides an alternative estimate for the numerical radius and characterizes when the numerical radius of a certain block operator matrix equals the average of the norms of its entries.

Findings

New lower bound for numerical radius of Hilbert space operators

Conditions for equality of numerical radius and average of norms in block matrices

Enhanced understanding of numerical radius behavior in operator matrices

Abstract

We give an alternative lower bound for the numerical radii of Hilbert space operators. As a by-product, we find conditions such that \begin{equation*} \omega\left(\left[\begin{array}{cc} 0 & R \\ S & 0 \end{array}\right]\right)=\frac{\Vert R \Vert +\Vert S\Vert }{2} \end{equation*} where $R, S \in \mathbb{B}(\mathcal{H})$.