How to generalize $D$-stability

Abstract

In this paper, we introduce the following concept which generalizes known definitions of multiplicative and additive $D$-stability, Schur $D$-stability, $H$-stability, $D$-hyperbolicity and many others. Given a subset ${\mathfrak D} \subset {\mathbb C}$, a matrix class ${\mathcal G} \subset {\mathcal M}^{n \times n}$ and a binary operation $\circ$ on ${\mathcal M}^{n \times n}$, an $n \times n$ matrix $\mathbf A$ is called $({\mathfrak D}, {\mathcal G}, \circ)$-stable if $\sigma({\mathbf G}\circ {\mathbf A}) \subset {\mathfrak D}$ for any ${\mathbf G} \in {\mathcal G}$. Such an approach allows us to unite several well-known matrix problems and to consider common ways of their analysis. Here, we make a survey of existing results and open problems on different types of stability, study basic properties of $({\mathfrak D}, {\mathcal G}, \circ)$-stable matrices and relations between different $({\mathfrak D}, {\mathcal G}, \circ)$-stability classes.