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 $\frac{S-S^{\ast }}{2i}$ is positive) in the sense that $\frac{S+S^{\ast }}{2}$ is positive, then $$\frac{\sqrt{3}}{3}\parallel S\parallel \le \omega \left(S\right),$$ where $\parallel \cdot \parallel$ and $\omega (\cdot )$ denote the operator norm and the numerical radius, respectively.