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(AB)≤4w(A)w(B), and the equality is attained by the 2-by-2 matrices A=[_{0 0}^{0 1}] and B = [_{1 0}^{0 0}]. Moreover, the constant "4" here can be reduced to "2" if A and B commute, and the corresponding equality is attained by A=I₂ ⊗ [_{0 0}^{0 1}] and B=[_{0 0}^{0 1}] ⊗ I₂. 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) 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 [_{0 0} ^{0 a}] ⊗ A′ and [_{b 0}^{0 0}] × B′ (resp., (Formula presented.) ⊗ A′ and (Formula presented.) ⊗ 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(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 (Formula presented.) and (Formula presented.) with |a₁|≥|a₂| and |b₁|≥|b₂|.