Extremality of numerical radii of tensor products of matrices

Abstract

For n-by-n and m-by-m complex matrices A and B, respectively, it is known that the inequality w(A⊗B)≤‖A‖w(B) holds, where w(⋅) and ‖⋅‖ 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 the equality w(A⊗B)=‖A‖w(B) holds if and only if A and B have k-by-k compressions A1 and B1, respectively, such that rank(‖A‖2Ik−A1⁎A1)≤minθ∈R⁡dim⁡ker⁡(w(B)Ik−Re(eiθB1)). We also give some consequences of this result. In particular, we show that if rankB≤sup⁡{k∈N:‖Ak‖=‖A‖k}, then w(A⊗B)=‖A‖w(B).