Numerical radius of Hadamard product of matrices

It is known that the numerical radius of the Hadamard product A∘B of two n-by-n matrices A and B is related to those of A and B by (a) w(A∘B)≤2w(A)w(B), (b) w(A∘B)≤w(A)w(B) if one of A and B is normal, and (c) w(A∘B)≤(maxi⁡aii)w(B) if A=[aij]i,j=1n is positive semidefinite. 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=[00a0] and B=[00b0] for (a), A=[a100a2] (|a1|≥|a2|) and B=[w(B)⁎⁎⁎] for (b), and A=[a1a3a2a4]≥0 (a1≥a4) and B=[w(B)⁎⁎⁎] for (c).