Condition for the higher rank numerical range to be non-empty

It is shown that the rank-k numerical range of every n-by-n complex matrix is non-empty if k < n/3 + 1. The proof is based on a recent characterization of the rank-k numerical range by Li and Sze, the Helly's theorem on compact convex sets, and some eigenvalue inequalities. In particular, the result implies that rank-2 numerical range is non-empty if n ≥ 4. This confirms a conjecture of Choi et al. If k ≥ n/3 + 1, an n-by-n complex matrix is given for which the rank-k numerical range is empty. An extension of the result of bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.