Numerical Radii for Tensor Products of Matrices (1306.2423v1)

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 $n_B=\sup{\ell\ge 1 : B^{\ell}\neq 0}$.