Extremality of numerical radii of matrix products

Abstract

For two n-by-n matrices A and B, it was known before that their numerical radii satisfy the inequality w(AB) <= 4w(A)w(B), and the equality is attained by the 2-by-2 matrices A = [GRAPHICS] and B = [GRAPHICS] . Moreover, the constant "4" here can be reduced to "2" if A and B commute, and the corresponding equality is attained by A = I-2 circle times [GRAPHICS] and B = [GRAPHICS] circle times I-2. In this paper, we give a complete characterization of A and B for which the equality holds in each case. More precisely, it is shown that w(AB) = 4w(A)w(B) (resp., w(AB) = 2w(A)w(B) for commuting A and B) if and only if either A or B is the zero matrix, or A and B are simultaneously unitarily similar to matrices of the form [GRAPHICS] circle plus A\' and [GRAPHICS] circle plus B\' (resp., [GRAPHICS] circle plus A\' and [GRAPHICS] circle plus B\') with w(A\') <= vertical bar a vertical bar/2 and w(B\') <= vertical bar b vertical bar/2. An analogous characterization for the extremal equality for tensor products is also proven. For doubly commuting matrices, we use their unitary similarity model to obtain the corresponding result. For commuting 2-by-2 matrices A and B, we show that w(AB) = w(A)w(B) if and only if either A or B is a scalar matrix, or A and B are simultaneously unitarily similar to [GRAPHICS] and [GRAPHICS] with vertical bar a(1)vertical bar >= vertical bar a(2)vertical bar and vertical bar b(1)vertical bar >= vertical bar b(2)vertical bar. (C) 2016 Elsevier Inc. All rights reserved.