Upper and Lower Bounds for Numerical Radii of Block Shifts

TL;DR

This paper establishes bounds for the numerical radii of block shift matrices using operator norm and minimum modulus, providing conditions for when these bounds are tight.

Contribution

It introduces bounds for the numerical radii of block shift matrices based on operator norm and minimum modulus, with precise equality conditions.

Findings

Numerical radius of A'' is less than or equal to that of A.

Numerical radius of A is less than or equal to that of A'.

Exact conditions for equality in the bounds are characterized.

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.