A Study on Equality Case of Some Numerical Radius Inequalities

Let A=U|A| be the polar decomposition of an n-by-n matrix A. The generalized Aluthge transform is then given by D_t(A)=|A|^t U|A|^(1-t) for t in [0,1]. We investigate the behavior of the numerical radius w(D_t(A)). In particular, for the following numerical radius inequalities:(1) 2w(A) - ∥A∥ ≦ w(D_t(A)) ≦ w(A),(2) w(A) ≦ (∥A∥ + ∥A^2∥^(1/2))/2, and(3) w(A)^2 ≦ (∥A∥^2 + w(A^2))/2.We consider when these inequalities become equalities. We want to give complete characterizations of A for which the equality is attained.