On the convexity and circularity of the numerical range for quaternionic matrices - Mendes, S.; Carvalho, L.; Diogo, C. (New Trends in Quaternions and Octonions, 2019)
Let $A\in\mathcal{M}_n(\mathbb{H})$ be a $n× n$ matrix over the quaternions $\mathbb{H}$. The quaternionic numerical range of $A$ is the subset $W_{\mathbb{H}}(A)\subset\mathbb{H}$ defined by $$W_{\mathbb{H}}(A)=\{x^*Ax:x\in\mathbb{D}_{\mathbb{H}^n}(0,1)\}$$ where $\mathbb{D}_{\mathbb{H}^n}$ denotes the unit ball with centre in the origin of $\mathbb{H}^n$. Contrary to the case of complex matrices where the numerical range is always convex (Toeplitz-Hausdorff Theorem), convexity is no longer a property of every quaternionic numerical range. We study the convexity of the numerical range of quaternionic matrices. Quite specific, we prove that a certain class of quaternionic matrices always has convex numerical range and we give necessary and sufficient conditions for a $3× 3$ nilpotent quaternionic matrix to have convex numerical range. Another property that has been studied for complex and quaternionic matrices is the circularity of the numerical range. We establish the circularity of the numerical range for a class of quaternionic matrices. Moreover, we give necessary and sufficient conditions for a $3× 3$ nilpotent quaternionic matrix to have circular numerical range. Joint work with Cristina Diogo and Luís Carvalho from ISCTE-IUL.