Extremality of numerical radii of matrix products.

For two n -by- n matrices A and B , it was known before that their numerical radii satisfy the inequality w ( A B ) ≤ 4 w ( A ) w ( B ) , and the equality is attained by the 2-by-2 matrices A = [ 0 1 0 0 ] and B = [ 0 0 1 0 ] . 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 ⊗ [ 0 1 0 0 ] and B = [ 0 1 0 0 ] ⊗ 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 ( A B ) = 4 w ( A ) w ( B ) (resp., w ( A B ) = 2 w ( 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 [ 0 a 0 0 ] ⊕ A ′ and [ 0 0 b 0 ] ⊕ B ′ (resp., ⊕ A ′ and ⊕ B ′ ) with w ( A ′ ) ≤ | a | / 2 and w ( B ′ ) ≤ | b | / 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 ( A B ) = 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 [ a 1 0 0 a 2 ] and [ b 1 0 0 b 2 ] with | a 1 | ≥ | a 2 | and | b 1 | ≥ | b 2 | . [ABSTRACT FROM AUTHOR]