Extremal case in Marcus-Oliveira conjecture and beyond

摘要

For A, C ∈ M_n the C-determinantal range of A is the following set on the complex plane {up triangle, open}_C(A) = {det(A - UCU*): UU* = I_n}. For normal matrices A and C with eigenvalues α_1,..., α_n and γ_1,..., 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 {up triangle, open}_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 {up triangle, open}C(A) and the radius of the convex hull of the points zσ, 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.