An Inverse Problem for the $k$-Rank Numerical Range

Abstract

For an $n \times n$ matrix $A$ and an integer $k \in [1,n]$, the concept of the higher rank numerical range $\Lambda_k(A)=\left\{z \in \mathbb{C}:V^*AV=zI_k, \; V \in \mathbb{C}^{n \times k}, \; V^*V=I_k\right\}$ has been introduced in relation to the study of error correcting codes and has been extensively studied. In this paper, an inverse problem for $\Lambda_k(A)$ is considered, where, for a given point $z \in \Lambda_k(A)$, we investigate the construction of a generating isometry $V \in \mathbb{C}^{n \times k}$ for $z$, i.e., such that $V^*AV=zI_k$. Under this scope, the notion of a covering number for higher rank numerical range points is introduced, generalizing a corresponding definition for $k=1$.