1 Numerical ranges of nilpotent operators

By Hwa-Long Gau and Pei Yuan Wu.

For any operator A on a Hilbert space, let w(A) and w_{0}(A) denote its numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if A is nilpotent with nilpotency n, then w(A) is at most the product of n − 1 and w_{0}(A). When A attains its numerical radius, we also determine a necessary and sufficient condition for the equality to hold.