New Estimates for the Numerical Radius

In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and refinements of some well known inequalities, including multiplicative behavior of the numerical radius and norm bounds. Among many other applications, it is shown that if $T$ is accretive-dissipative, then $$\frac{1}{\sqrt{2}}\left\| T \right\|\le \omega \left( T \right),$$ where $\omega \left( \cdot \right)$ and $\left\| \cdot \right\|$ denote the numerical radius and the usual operator norm, respectively. This inequality provides a considerable refinement of the well known inequality $\frac{1}{2}\|T\|\leq \omega(T).$