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

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.

4 pages; to appear in LAMA

References in corpus (2)

• Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations
• Quantum Error Correcting Codes From The Compression Formalism

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