Matricial Radius: A Relation of Numerical radius with Matricial Range

Abstract

It has been shown that if $T$ is a complex matrix, then {\small\begin{align*} \omega(T)&=\frac{1}{n}\sup\left\{|\mathrm{Tr}\ X|;\ X\in W^n(T)\right\}\\ &=\frac{1}{n}\sup\left\{\|X\|_1;\ X\in W^n(T)\right\}\\ &= \sup\left\{ \omega(X);\ X\in W^n(T)\right\} \end{align*} } where $n$ is a positive integer, $\omega(T)$ is the numerical radius and $W^n(T)$ is the $n$'th matricial range of $T$.