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# Higher order numerical range

The rank–$k$ numerical ranges, denoted below by $\Lambda_k$, were introduced c. 2006 by Choi, Kribs, and Życzkowski as a tool to handle compression problems in quantum information theory. Since then their theory and applications have been advanced with remarkable enthusiasm. The sequence of papers [1], [2], [3], [4], for example, led to a striking extension of the classical Toeplitz–Hausdorff theorem (convexity of $W(M)$): all the $\Lambda_k(M)$ are convex (though some may be empty), and they are intersections of conveniently computable half–planes in $\mathbb{C}$. Among the many more recent papers concerning the $\Lambda_k(M)$, let us mention CITE!! [LPS2009,GLW2010].

Given a matrix $M\in M_N$ and $k\geq1$, Choi, Kribs, and \.Zyczkowski (see [CK\.Z2006a,CK\.Z2006b]) defined the rank–$k$ numerical range of $M$ as $\Lambda_k(M)=\{\lambda\in\mathbb{C}:\exists P\in P_k\mbox{ such that }PMP=\lambda P\},$ where $P_k$ denotes the set of rank–$k$ orthogonal projections in $M_N$. It is not hard to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that there is some $k$–dimensional subspace $S$ of $\mathbb{C}^N$ such that $(Mu,u)=\lambda$ for \textbf{all} unit vectors in $S$. In particular, we see that $W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad.$

1. M. D. Choi, J. A. Holbrook, D. W, Kribs, K. Życzkowski, 2007. Higher-rank numerical ranges of unitary and normal matrices. Operators and Matrices, 1, pp.409–426.
2. M. D. Choi, M. Giesinger, J. A. Holbrook, D. W. Kribs, 2008. Geometry of higher-rank numerical ranges. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.53–64.
3. H. J. Woerdeman, 2008. The higher rank numerical range is convex. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.65–67.
4. C. K, Li, N. S. Sze, 2008. Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations. Proceedings of the American Mathematical Society, 136, pp.3013–3023.