On the C-determinantal range for special classes of matrices

Abstract

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 − U CU * ) : U U * = 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].

More on geometry of Krein space C-numerical range

“…Next, assuming C is J-unitarily diagonalizable with real eigenvalues, we present a correct characterization of the matrices A, C for which W C J (A) is a subset of the real line. The statement presented in a previous attempt made in [1,Theorem 5.2] for the (J, C)-tracial range, assuming that the matrix JC has pairwise distinct main diagonal entries, is incomplete (as well as the statement in [11], corresponding to the case J = I n , as can be seen in [5]). Now, we correct the above result.…”