Generalized \(M\)-matrices and ordered Banach algebras
A matrix \(A\in \mathbb{R}^{n\times n}\) is an \(M\)-matrix if and only if the mapping \(x\mapsto-Ax\) is quasimonotone increasing (qmi) and if the right spectral bound of \(-A\) is negative. Here qmi is meant with respect to the natural cone \(K=\{x\in \mathbb{R}^n: x_k\geq 0\}\). One possibility of generalizing \(M\)-matrices is to consider qmi linear mappings on \(\mathbb{R}^n\) with respect to other cones \(K\subseteq \mathbb{R}^n\). The paper presents results on such mappings in the Banach algebra setting and discusses some special cones. Moreover, by means of one-sided estimates it is possible to get informations on the right spectral bound of qmi mappings.
scientific article; zbMATH DE number 1903563
scientific article