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

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

2007-06-11

arxiv.org/abs/0706.1540v2

Linear and Multilinear Algebra, 57:365-368, 2009

Mathematics

Functional Analysis

4 pages; to appear in LAMA

Scientific paper

10.1080/03081080701786384

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.

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