numerical-range:generalizations: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 [5] and [6].

Given a matrix $M\in M_N$ and $k\geq1$, Choi, Kribs, and Życzkowski (see [7])
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 **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.

5.
C. K. Li, Y. T. Poon, N. S. Sze, 2009. Condition for the higher rank numerical range to be non-empty. *Linear and Multilinear Algebra*, 57, Taylor & Francis, pp.365–368.

6.
H. L. Gau, C. K. Li, P. Y. Wu, 2010. Higher-rank numerical ranges and dilations. *Journal of Operator Theory*, 63, pp.181.

7.
M. D. Choi, D. W. Kribs, K. Życzkowski, 2006. Higher-rank numerical ranges and compression problems. *Linear algebra and its applications*, 418, pp.828–839.

numerical-range/generalizations/higher-order-numerical-range.txt · Last modified: 2013/05/08 12:56 by lpawela