On the C-determinantal range for special classes of matrices

Let A and C be square complex matrices of size n, the C-determinantal range of A is the subset of the complex plane {det (A− UCU∗) : UU∗ = In}. If A,C are both Hermitian matrices, then by a result of M. Fiedler [11] this set is a real line segment. In this paper we study this set for the case when C is a Hermitian matrix. Our purpose is to revisit and improve two well-known results on this topic. The first result is due to C.K. Li concerning the C-numerical range of a Hermitian matrix, see Condition 5.1 (a) in [20]. The second one is due to C.-K. Li, Y.-T. Poon and N.-S. Sze about necessary and sufficient conditions for the C-determinantal range of A to be a subset of the line, see [21, Theorem 3.3].