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Geometry of higher-rank numerical ranges. (English) Zbl 1137.15016

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.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory

Citations:

Zbl 1137.15018
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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
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