Higher rank numerical range

The rank–$k$ numerical ranges, denoted below by ${\mathrm{\Lambda }}_k$, 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 ${\mathrm{\Lambda }}_k(M)$ 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 ${\mathrm{\Lambda }}_k(M)$, let us mention [5] and [6].

Given a matrix $M\in M_N$ and $k\ge 1$, Choi, Kribs, and Życzkowski (see [7]) defined the rank–$k$ numerical range of $M$ as

$${\mathrm{\Lambda }}_k(M)=\{\lambda \in \mathbb{C}:\mathrm{\exists }P\in P_k\mathrm{such}\mathrm{that}PMP=\lambda P\},$$

where $P_k$ denotes the set of rank–$k$ orthogonal projections in $M_N$. It is not hard to verify that ${\mathrm{\Lambda }}_K(M)$ 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 all unit vectors in $S$. In particular, we see that $$W(M){=}_1(M{)}_2(M{)}_3(M).$$

References

1. [1]M. D. Choi, J. A. Holbrook, D. W. Kribs, and K. Życzkowski, “Higher-rank numerical ranges of unitary and normal matrices,” Operators and Matrices, vol. 1, pp. 409–426, 2007, [Online]. Available at: https://www.semanticscholar.org/paper/HIGHER-RANK-NUMERICAL-RANGES-OF-UNITARY-AND-NORMAL-Choi-Holbrook/5b13b6b5a92ce54bbf1699375a3ba26cbceb90ae.
   https://www.semanticscholar.org/paper/HIGHER-RANK-NUMERICAL-RANGES-OF-UNITARY-AND-NORMAL-Choi-Holbrook/5b13b6b5a92ce54bbf1699375a3ba26cbceb90ae
2. [2]M. D. Choi, M. Giesinger, J. A. Holbrook, and D. W. Kribs, “Geometry of higher-rank numerical ranges,” Linear and Multilinear Algebra, vol. 56, no. 1-2, pp. 53–64, 2008, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081080701336545.
   https://www.tandfonline.com/doi/abs/10.1080/03081080701336545
3. [3]H. J. Woerdeman, “The higher rank numerical range is convex,” Linear and Multilinear Algebra, vol. 56, no. 1-2, pp. 65–67, 2008, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081080701352211.
   https://www.tandfonline.com/doi/abs/10.1080/03081080701352211
4. [4]C. K. Li and N. S. Sze, “Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations,” Proceedings of the American Mathematical Society, vol. 136, no. 9, pp. 3013–3023, 2008, [Online]. Available at: https://www.ams.org/journals/proc/2008-136-09/S0002-9939-08-09536-1/.
   https://www.ams.org/journals/proc/2008-136-09/S0002-9939-08-09536-1/
5. [5]C. K. Li, Y. T. Poon, and N. S. Sze, “Condition for the higher rank numerical range to be non-empty,” Linear and Multilinear Algebra, vol. 57, no. 4, pp. 365–368, 2009, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081080701786384.
   https://www.tandfonline.com/doi/abs/10.1080/03081080701786384
6. [6]H.-L. Gau, C.-K. Li, and P. Y. Wu, “Higher-rank numerical ranges and dilations,” Journal of Operator Theory, vol. 0, pp. 181–189, 2010, [Online]. Available at: https://www.jstor.org/stable/24715918.
   https://www.jstor.org/stable/24715918
7. [7]M. D. Choi, D. W. Kribs, and K. Życzkowski, “Higher-rank numerical ranges and compression problems,” Linear algebra and its applications, vol. 418, no. 2, pp. 828–839, 2006, [Online]. Available at: https://arxiv.org/abs/math/0511278.
   https://arxiv.org/abs/math/0511278