The \(k\)-rank numerical radii (Q455456)

The \(k\)-rank numerical range \(\Lambda_k(A)\) of \(A\in M_n\) is defined as the subset \(\Lambda_k(A) =\{\lambda\in {\mathbb C}: X^*AX=\lambda I_k \text{ for some } X \in M_{n,k}, X^*X=I_k \}\subset \mathbb C\). When \(k=1\), it reduces to the classical numerical range. The study of the higher rank numerical range has close connections with problems of quantum computing. In the present paper, the \(k\)-rank numerical range is expressed via an intersection of any countable family of the classical numerical ranges. Reviewer's remarks. Theorem 2.1 is essentially Corollary 4.9 in [\textit{C.-K. Li, Y.-T. Poon} and \textit{N.-S. Sze}, J. Math. Anal. Appl. 348, No. 2, 843--855 (2008; Zbl 1151.47008)]. The comment after Theorem 2.1 (p.\,102) is incorrect; one needs to know convexity or the characterization of \(\Lambda_k(A)\) in order to represent \(\Lambda_k(A)\) as the intersection of compressions.