Three observations on the determinantal range

Let A, C ∈ Mn, the algebra of n × n complex matrices. The set of complex numbersΔC(A)={det(A-UCU*):U*U=In}is the C-determinantal range of A. In this note, it is proved that ΔC(A) is an elliptical disc for A, C ∈ M2. A necessary and sufficient condition for ΔC(A) to be a line segment is given when A and C are normal matrices with pairwise distinct eigenvalues. The linear operators L that satisfy the linear preserver property ΔC(A) = ΔC(L(A)), for all A, C ∈ Mn, are characterized.