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

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