Quaternionic numerical range of complex matrices (Q2020687)

The numerical range of an \(n\)-square matrix \(A\) over a quaternionic ring (called quaternionic numerical range) is defined as the set \[W_{\mathbb{H}}(A)=\{x^*Ax: x^*x=1, x\in \mathbb{H}^n\},\] where \(\mathbb{H}\) is the ring of Hamiltonian quaternions. The noncommutativity of the quaternionic multiplication makes the studying object very different from the complex field. A known way to study the quaternionic numerical range is through the bild defined as the intersection of \(W_{\mathbb{H}}(A)\) and \(\mathbb{C}\). Even the bild itself is in general very difficult to compute. In the present work, the authors consider the case when \(A\) is a complex matrix. The bild in this case is studied in depth. Conditions under which the bild of a complex matrix coincides with its complex numerical range and for which the quaternionic numerical range is convex are presented.