Geometry of higher-rank numerical ranges

Abstract

We consider geometric aspects of higher-rank numerical ranges for arbitrary N × N matrices. Of particular interest is the issue of convexity and a possible extension of the Toeplitz–Hausdorff Theorem. We derive a number of reductions and obtain partial results for the general problem. We also conduct graphical and computational experiments. Added in proof: Following acceptance of this paper, our subject has developed rapidly. First, Hugo Woerdeman established convexity of the higher-rank numerical ranges by combining Proposition 2.4 and Theorem 2.12 with the theory of algebraic Riccati equations. See Woerdeman, H., 2007, The higher rank numerical range is convex, Linear and Multilinear Algebra, to appear. Subsequently Chi-Kwong Li and Nung-Sing Sze followed a different approach that not only yields convexity but also provides important additional insights. See Li, C.-K. and Sze, N.-S., 2007, Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations, preprint. See also Li, C.-K., Poon, Y.-T., and Sze, N.-S., 2007, Condition for the higher rank numerical range to be non-empty, preprint.