Revisiting the Upper Bild of Quaternionic Numerical Ranges

Abstract

The upper bild of the quaternionic numerical range provides a convex representative of the generally nonconvex quaternionic numerical range. Although its convexity is known, the geometry of the upper bild is difficult to describe explicitly, even for 2-by-2 quaternionic matrices. In this paper, we revisit the upper bild of quaternionic numerical ranges from both algebraic and geometric viewpoints. We first point out gaps in Kumar's proof. We then investigate several special classes of 2-by-2 quaternionic matrices and describe their shapes and boundary structures in the closed upper complex half-plane directly in terms of the standard eigenvalues and the off-diagonal entry of the Schur form. For the general 2-by-2 case, we use a different canonical form. Following So and Thompson, we prove the concavity of the upper-boundary function through the associated complex representation. For the lower boundary, we obtain a distance-to-a-disk representation and use convex optimization to prove its convexity. We also treat the degenerate cases. Together with the connectedness of the vertical sections, these results give an alternative proof of the convexity of the upper bild for arbitrary quaternionic matrices.