ON THE C -DETERMINANTAL RANGE FOR SPECIAL CLASSES OF MATRICES ALEXANDER GUTERMAN, RUTE LEMOS, AND GRAC ¸ A SOARES Abstract. 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 * = I n }. 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]. 1. Introduction Let M n be the algebra of n × n complex matrices, U n be the group of n × n unitary matrices and S n be the symmetric group of degree n. Let A, C ∈ M n . Definition 1.1. The C -determinantal range of A is the subset of the complex plane denoted and defined by △ C (A)= {det (A − UCU ∗ ): U ∈ U n } and the C -determinantal radius of A is d C (A) = max{|z | : z ∈△ C (A)}. The set △ C (A) is compact and connected, but in general it is not convex (see for instance [2, Example 2]) and it may not be simply connected [1]. It is clear that △ A (C )=(−1) n △ C (A) and this set is unitarily invariant, that is, △ C (A)= △ V CV ∗ (UAU ∗ ) for any U, V ∈ U n . Definition 1.2. The σ-points of △ C (A) are defined by z σ = n  i=1 (α i − γ σ(i) ), σ ∈ S n , where α 1 ,...,α n and γ 1 ,...,γ n are the eigenvalues of A and C , respectively. It is easy to see that all the (not necessarily distinct) n! σ-points belong to △ C (A). The characterization of the C -determinantal range of A for Hermitian matrices A and C was obtained by M. Fiedler [11], who proved that △ C (A) is a real line segment, whose endpoints are the minimal and maximal σ-points of △ C (A). The C -determinantal range of A is intimately connected with a famous conjecture of M. Mar- cus [22] and G. N. de Oliveira [24], which can be reformulated as follows: for normal matrices A, C ∈ M n it holds that △ C (A) is a subset of the convex hull of the σ-points z σ , σ ∈ S n . This Key words and phrases. C -Determinantal range, C -numerical range, Marcus-Oliveira conjecture, σ-points, real sets 2010 Mathematics Subject Classification. 15A15, 15A60, 15A86 . 1