Numerical ranges of nilpotent operators

For any operator A on a Hilbert space, let W(A), w(A) and w₀(A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if Aⁿ=0, then w(A)?(n-1)w₀(A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w(A)=(n-1)w₀(A), (2) A is unitarily equivalent to an operator of the form aA_n⊕A^′, where a is a scalar satisfying |a|=2w₀(A), A_n is the n-by-n matrix