Choi, Man-Duen; Giesinger, Michael; Holbrook, John A.; Kribs, David W. Geometry of higher-rank numerical ranges. (English) Zbl 1137.15016 Linear Multilinear Algebra 56, No. 1-2, 53-64 (2008). Let \(T\) be a square complex matrix, \(k\) be a positive integer and \(I_k\) denotes the \(k \times k\) identity matrix. The rank-\(k\) numerical range of \(T\) is defined to be \[ \Lambda_k(T)=\{\lambda \in {\mathbb C}: PTP=\lambda P \text{ for some rank-\(k\) projection } P\}. \]Throughout some partial results and suggestive computational experiments, the authors investigate the issue of convexity and a possible extension of the Toeplitz-Hausdorff theorem. In particular, they prove that \(\Lambda_k(T)\) is convex for all square matrices \(T\) if and only if \(0 \in \Lambda_k\left( \left[\begin{smallmatrix} I_k & X \\ Y& -I_k \end{smallmatrix}\right]\right)\), and this if and only if for all \(M, R \in M_k({\mathbb C})\) such that \(R\) is positive definite and there exists a Hermitian \(H \in M_k({\mathbb C})\) such that \(I_k+MH+HM^*-HRH=H\). Recently, H. J. Woerdeman [ibid. 56, No. 1–2, 65–67 (2008; Zbl 1137.15018)] and C.-K. Li and N.-S. Sze [Canonical forms, higher rank numerical ranges, totally isotropic subspaces and matrix equations (preprint)] established the convexity of the higher-rank numerical range. Reviewer: Mohammad Sal Moslehian (Mashhad) Cited in 2 ReviewsCited in 26 Documents MSC: 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory Keywords:higher rank numerical range; convexity; Toeplitz-Hausdorff theorem Citations:Zbl 1137.15018 PDFBibTeX XMLCite \textit{M.-D. Choi} et al., Linear Multilinear Algebra 56, No. 1--2, 53--64 (2008; Zbl 1137.15016) Full Text: DOI References: [1] Choi M-D, Higher-rank numerical ranges of unitary and normal matrices (2006) [2] DOI: 10.1016/S0034-4877(06)80041-8 · Zbl 1120.81011 · doi:10.1016/S0034-4877(06)80041-8 [3] DOI: 10.1016/j.laa.2006.03.019 · Zbl 1106.15019 · doi:10.1016/j.laa.2006.03.019 [4] DOI: 10.1080/03081089308818222 · Zbl 0814.15027 · doi:10.1080/03081089308818222 [5] DOI: 10.1103/PhysRevA.55.900 · doi:10.1103/PhysRevA.55.900 [6] DOI: 10.1080/03081089108818047 · Zbl 0717.15018 · doi:10.1080/03081089108818047 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.