Abstract
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.
Original language | English |
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Pages (from-to) | 365-368 |
Number of pages | 4 |
Journal | Linear and Multilinear Algebra |
Volume | 57 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1 Jan 2009 |
Externally published | Yes |
Keywords
- Eigenvalue inequalities
- Helly's theorem
- Higher rank numerical range
ASJC Scopus subject areas
- Algebra and Number Theory