On Location of the Matrix Spectrum with Respect to a Parabola

Abstract

In the present article, we consider the problem on location of the matrix spectrum withrespect to a parabola. In terms of solvability of a matrix Lyapunov type equation, we provetheorems on location of the matrix spectrum in certain domains {cal P}_i (bounded by a parabola) and {cal P}_e (lying outside the closure of {cal P}_i). A solution to the matrix equation is constructed.We use this equation and prove an analog of the Lyapunov–Krein theorem on dichotomy ofthe matrix spectrum with respect to a parabola.