Three observations on the determinantal range (Q1779394)

Let \(A,C\) be \(n\times n\) complex matrices. The set \(\Delta _C(A)=\{\det (A-UCU^*)\,:\,U^*U=I_n\}\) is the \(C\)-\textit{determinantal range} of~\(A\). The authors prove that \(\Delta_C(A)\) is an elliptical disc when \(A,C\in {\mathbb C}^{2\times 2}\). Necessary conditions and sufficient conditions for \(\Delta_C(A)\) to be a line segment are given when \(A\) and \(C\) are diagonal matrices with pairwise disjoint entries. Further, linear operators \(L\) that satisfy the linear preserver property \(\Delta_C(A)=\Delta_C(L(A))\) for all \(A,C\in {\mathbb C}^{n\times n}\) are characterized.