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

TL;DR

This paper establishes a condition on the matrix size for the non-emptiness of the rank-$k$ numerical range, confirming a conjecture and extending the result to infinite-dimensional operators.

Contribution

It provides a new sufficient condition for the non-emptiness of the rank-$k$ numerical range for matrices and extends the analysis to infinite-dimensional operators.

Findings

The rank-$k$ numerical range is non-empty if $n \\ge 3k - 2$ for $n$-by-$n$ matrices.

Confirmed the conjecture that $\\Lambda_2(A)$ is non-empty if $n \\ge 4$.

Demonstrated the existence of matrices with empty rank-$k$ numerical range when $3k-2>n>0$.

Abstract

It is shown that the rank-$k$ numerical range of every $n$-by-$n$ complex matrix is non-empty if $n \ge 3k - 2$. 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 $\Lambda_2(A)$ is non-empty if $n \ge 4$. This confirms a conjecture of Choi et al. If $3k-2>n>0$, an $n$-by-$n$ complex matrix is given for which the rank-$k$ numerical range is empty. Extension of the result to bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.