Extremality of numerical radii of matrix commutators and Jordan products

Abstract

It is known that the numerical radius of the commutator (resp., Jordan product) AB - BA (resp., AB + BA) of two n-by-n matrices A and B is related to those of A and B by w(AB +/- BA) <= 4w(A)w(B). In this paper, we give complete characterizations of A and B for which the equality is attained. The matrices involved can be considered as elaborate generalizations of the equality-attaining A = [0 a (0 0)] and B = [0 0 (b) (0)]. (C) 2017 Elsevier Inc. All rights reserved.