A Fiedler-type theorem for the determinant of \(J\)-positive matrices (Q2805348)

Let \(J\in\mathbb{C}^{n\times n}\) satisfy \(J^{\ast}=J\), \(J^2=I\) and have inertia \((r,n-r)\). A matrix \(A\in\mathbb{C}^{n\times n}\) is \(J\)-Hermitian if \(A=JA^{\ast}J\). It is \(J\)-positive (definite) if, in addition, \(A=JP\) for some positive (definite) matrix \(P\). All its eigenvalues are then real.NEWLINENEWLINELet \(A,C\in\mathbb{C}^{n\times n}\) be \(J\)-positive with eigenvalues \(a_1\geq\dots\geq a_n\) and \(c_1\geq\dots\geq c_n\), respectively. Assuming \(a_r>0>a_{r+1}\) and \(c_r>0>c_{r+1}\), the authors prove that NEWLINE\[NEWLINE \det{(A+C)}\geq\prod_{j=1}^n(a_j+c_j) NEWLINE\]NEWLINE if \(n-r\) is even, and that the inequality reverses if \(n-r\) is odd.NEWLINENEWLINEThe motivation of the paper rises from the following: \textit{M. Fiedler} [Proc. Am. Math. Soc. 30, 27--31 (1971; Zbl 0277.15010)] proved that if \(A\) and \(C\) are Hermitian, then NEWLINE\[NEWLINE \min_{\sigma\in S_n}\prod_{j=1}^n(a_j+c_{\sigma(j)})\leq\det{(A+C)}\leq \max_{\sigma\in S_n}\prod_{j=1}^n(a_j+c_{\sigma(j)}). NEWLINE\]NEWLINE \textit{M. Marcus} [Indiana Univ. Math. J. 22, 1137--1149 (1973; Zbl 0243.15025)] and \textit{G. N. de Oliveira} [``Research problem: normal matrices'', Linear Multilinear Algebra 12, 153--154 (1982)] conjectured that if \(A\) and \(C\) are normal with eigenvalues \(a_1,\dots,a_n\) and \(c_1,\dots,c_n\), respectively, then NEWLINE\[NEWLINE \det{(A+C)}\in\mathrm{co}\Big\{\prod_{j=1}^n(a_j+c_{\sigma(j)}):\sigma\in S_n\Big\}. NEWLINE\]NEWLINE Here \(\mathrm{co}\) denotes the convex hull.