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arXiv · 0706.1540

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

Chi-Kwong Li·Yiu-Tung Poon·Nung-Sing Sze
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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 $Λ_2(A)$ 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.

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Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze. 2007-11-04. Condition for the higher rank numerical range to be non-empty. https://doi.org/10.1080/03081080701786384

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