numerical-range:generalizations:higher-rank-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], [6] and [7].

Given a matrix $M$ of dimension $d$ and $k\geq1$, Choi, Kribs, and Życzkowski (see [8])
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 $M_d$. 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 $\mathbb{C}^d$ such that $(Mu,u)=\lambda$ for **all**
unit vectors in $S$. In particular, we see that
\[
W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad.
\]

Note that, this numerical range is different from the $k$-numerical range as for a Hermitian matrix $A$, we get \[ \Lambda_k(A) = [\lambda_k, \lambda_{N-k+1}], \] where $\lambda_k$ are the eigenvalues of $A$ in an increasing order. On the other hand, the $k$-numerical range is given by \[ W_k = \left[\frac{1}{k}\sum_{i=1}^k\lambda_i, \frac{1}{k}\sum_{i=0}^{k-1} \lambda_{d-i} \right]. \] Hence, we get \[ \Lambda_k(A) \subset W_k(A). \]

A comparison between the $k$-numerical range and higher-rank numerical range in the case $k=2$. Note that $\Lambda_2 \subset W_2$. The matrix used in this example is $A = \mathrm{diag}(1, 2, 4, 8)$.

Numerical range (blue) and real numerical shadow of the matrix $U_5 = \mathrm{diag}(\left\{\ee^{2\pi \ii k/5}\right\}_{k=1}^5)$. The red polygon inside is $\Lambda_2(U_5)$.

Numerical range (blue) and real numerical shadow of the matrix $U_7 = \mathrm{diag}(\left\{\ee^{2\pi \ii k/5}\right\}_{k=1}^7)$. The red polygon inside is $\Lambda_2(U_7)$ and the green polygon is $\Lambda_3(U_7)$.

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.
John Holbrook, Nishan Mudalige, Mike Newman, Rajesh Pereira, 2015. Bounds on polygons of higher rank numerical ranges. *Linear Algebra and its Applications*, 474, Elsevier, pp.230–242.

8.
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-rank-numerical-range.txt · Last modified: 2018/11/03 22:54 by lpawela