Tightening Bounds on the Numerical Radius for Hilbert Space Operators

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