Upper and Lower Bounds for Numerical Radii of Block Shifts

Abstract: For any $n$-by-$n$ matrix $A$ of the form [[\begin{array}{cccc} 0 & A_1 & & \ & 0 & \ddots & \ & & \ddots & A_{k-1} \ & & & 0\end{array}],] we consider two $k$-by-$k$ matrices [A'=[\begin{array}{cccc} 0 & |A_1| & & \ & 0 & \ddots & \ & & \ddots & |A_{k-1}| \ & & & 0\end{array}] \ {and} \ A''=[\begin{array}{cccc} 0 & m(A_1) & & \ & 0 & \ddots & \ & & \ddots & m(A_{k-1}) \ & & & 0\end{array}],] where $|\cdot|$ and $m(\cdot)$ denote the operator norm and minimum modulus of a matrix, respectively. It is shown that the numerical radii $w(\cdot)$ of $A$, $A'$ and $A''$ are related by the inequalities $w(A'')\le w(A)\le w(A')$. We also determine exactly when either of the inequalities becomes an equality.