A research problem

We present a conjecture which when true would generalize T. Ando's characterization of the numerical radius of (bounded linear) operators on a Hilbert space (see [A]). Some evidence for the validity of the conjecture is given. In the finite dimensional case we shall restate the conjecture in terms of convex matrix sets and norms on matrices that are invariant under unitary similarities (u.s.i. norms).