On Location of the Matrix Spectrum with Respect to a Parabola

Abstract

Abstract In the present article, we consider the problem on location of the matrix spectrum with respect to a parabola. In terms of solvability of a matrix Lyapunov type equation, we prove theorems 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 of the matrix spectrum with respect to a parabola.