Numerical Radii for Tensor Products of Matrices

Abstract

For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\otimes B)\le \Vert A\Vert w(B)$ holds, where $w(\cdot )$ and $\Vert \cdot \Vert$ 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 $\Vert A\Vert =1$ 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 $\Vert A\Vert =\Vert A^k\Vert =1$ 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 $w(A\otimes B)=\Vert A\Vert w(B)$ 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 $p_A=\sup \{\ell \ge 1:\Vert A^{\ell }\Vert =\Vert A{\Vert }^{\ell }\}$ and $n_B=\sup \{\ell \ge 1:B^{\ell }\ne 0\}$.

Cited by in corpus (2)

• Numerical radius inequalities for tensor product of operators
• Numerical radius and ${\ell }_p$ operator norm of Kronecker products and Schur powers: inequalities and equalities