On the defect index of quadratic self-adjoint operator pencils (Q1810191)

Consider the matrix pencil of the form \(L(s)=sA+\frac{1}{s}B-C\) where \(A,B,C\) are Hermitian matrices acting in the space \(H=\mathbb{C}^{n}\) and satisfying the following conditions: \[ A=(a_{ij}) _{i,j=1,2,\dots,n},\quad A>0,\;a_{ii}>\sum_{\underset{j\neq i}{j=1}}^{n}(a_{ij}) +e_{i}^{A},e_{i}^{A}\geq 0, \] \[ B=\left\| b_{ij}\right\| _{i,j=1,2,\dots,n},\quad B>0,\;b_{ii}>\sum_{\underset{j\neq i}{j=1}}^{n}\left| b_{ij}\right| +e_{i}^{B},e_{i}^{B}\geq 0,\;C=J_{k}=\text{diag}( I_{n-k},-I_{k}) . \] This paper obtains simple sufficient conditions ensuring the equality \( k\_\left( L\right) =2k\) and the e-dichotomy of pencils of the form of \(L(s).\)