Some equality conditions with respect to the dual norm of the numerical radius*†
K. Okubo · Linear and Multilinear Algebra · 1993
There are several inequalities comparing the numerical radius w(·)(or it's dual norm w *(⋅)) with other norms on Mn , the space of all n×n complex matrices. We give conditions for the case of equality in these inequalities. In particular, we show that w *(A) =∥A∥1 (the trace norm of A) if and only if A is normal and 1=w *(A)/n=∥A∥∞ (the spectral norm of A) if and only if A is unitarily similar to a block matrix on where A 13 A 22 are unitary and ∥A 31∥∞≤1. Moreover we characterize matrices A that satisfy the equality w *(A) =2∥A∥1.