Tightening Bounds on the Numerical Radius for Hilbert Space Operators

Abstract

Let $S$ be a bounded linear operator on a Hilbert space. We show that if $S$ is accretive (resp. dissipative the sense that $\frac{S-S^{\ast }}{2i}$ is positive) in the sense that $\frac{S+S^{\ast }}{2}$ is positive, then \[\frac{\sqrt{3}}{3}\left\| S \right\|\le ω\left( S \right),\] where $\Vert \cdot \Vert$ and $\omega \left(\cdot \right)$ denote the operator norm and the numerical radius, respectively.