Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices

We present new upper and lower bounds for the numerical radius of a bounded linear operator defined on complex Hilbert space, which improve existing bounds. Among many other inequalities proved in this article, we show that non-zero $T$ space $H,$ $w(T)\geq \frac{\|T\|}{2}+\frac{m(T^2)}{2\|T\|}, $ where $w(T)$ is $m(T^2)$ Crawford number $T^2$. This substantially improves inequality \frac{\|T\|}{2} .$ also obtain some matrices illustrate with examples these are better than