R. Lemos

On Generalized Numerical Ranges of Operators on an Indefinite Inner Product Space

In this article, numerical ranges associated with operators on an indefinite inner product space are investigated. Boundary generating curves, corners, shapes and computer generations of these sets are studied. In particular, the Murnaghan–Kippenhahn theorem for the classical numerical range is generalized.

Extremal case in Marcus–Oliveira conjecture and beyond

For A, C ∈ M n the C-determinantal range of A is the following set on the complex plane ▵ C (A) = {det(A − UCU*): UU* = I n }. For normal matrices A and C with eigenvalues α1, … , α n and γ1, … , γ n , respectively, Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973), pp. 1137–1149] and Oliveira [G.N. de Oliveira, Normal matrices (research problem), Linear Multilinear Algebra 12 (1982), pp. 153–154] conjectured that ▵ C (A) is a subset of the convex hull of the points , σ ∈ S n , where S n is the symmetric group of degree n. We investigate the extremal set of matrices for which the equality holds in the Marcus–Oliveira conjecture. We illustrate the use of the obtained results by two different applications. The first one deals with the equality case between the radius of ▵ C (A) and the radius of the convex hull of the points z σ, σ ∈ S n . The second one is the characterization of additive Frobenius endomorphisms for the determinantal range or radius on the space M n and on its real subspace of Hermitian matrices.

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 - U C U * ) : U U * = I n } . If A, C are both Hermitian matrices, then by a result of Fiedler (1971) 11 this set is a real line segment.In our 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 Li concerning the C-numerical range of a Hermitian matrix, see Condition 5.1?(a) in Li, (1994) 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 Li et?al. (2008) 21, Theorem 3.3).