On location of the matrix spectrum with respect to a parabola

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 $\mathcal{P}_i$ (bounded by a parabola) and $\mathcal{P}_e$ (lying outside the closure of $\mathcal{P}_i$). 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.