Numerical Radii for Tensor Products of Matrices

TL;DR

This paper characterizes when the numerical radius of tensor products of matrices equals the product of their norms, providing conditions involving unitary parts, Jordan blocks, and permutationally irreducible matrices.

Contribution

It offers new necessary and sufficient conditions for equality in the numerical radius inequality for tensor products, extending understanding of matrix behavior.

Findings

Equality holds if $A$ has a unitary part or $W(B)$ is a centered circular disc.

For certain $A$, $w(A)$ is bounded below by a cosine function, with equality characterizing Jordan blocks.

For nonnegative $B$, the equality condition relates to block-shift structure and the index $p_A$.

Abstract

For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\otimes B)\le\|A\|w(B)$ holds, where $w(\cdot)$ and $\|\cdot\|$ denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if $\|A\|=1$ and $w(A\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and the numerical range $W(B)$ of $B$ is a circular disc centered at the origin, (2) if $\|A\|=\|A^k\|=1$ for some $k$, $1\le k<\infty$, then $w(A)\ge\cos(\pi/(k+2))$, and, moreover, the equality holds if and only if $A$ is unitarily similar to the direct sum of the $(k+1)$-by-$(k+1)$ Jordan block $J_{k+1}$ and a matrix $B$ with $w(B)\le\cos(\pi/(k+2))$, and (3) if $B$ is a nonnegative matrix with its real part (permutationally) irreducible, then $w(A\otimes B)=\|A\|w(B)$ if and only if either $p_A=\infty$ or $n_B\le p_A<\infty$ and $B$ is permutationally similar to a block-shift matrix \[[ {array}{cccc} 0 & B_1 & & & 0 & \ddots & & & \ddots & B_k & & & 0 {array} ]\] with $k=n_B$, where $p_A=\sup\{\ell\ge 1: \|A^{\ell}\|=\|A\|^{\ell}\}$ and $n_B=\sup\{\ell\ge 1 : B^{\ell}\neq 0\}$.