Extremality of numerical radii of tensor products of matrices

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‖²I_k−A₁ ^⁎A₁)≤min_θ∈R⁡dim⁡ker⁡(w(B)I_k−Re(e^iθB₁)). We also give some consequences of this result. In particular, we show that if rankB≤sup⁡{k∈N:‖A^k‖=‖A‖^k}, then w(A⊗B)=‖A‖w(B).