Geodesic
On location of the matrix spectrum with respect to a parabola
Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 26-40

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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. 
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     author = {G. V. Demidenko and V. S. Prokhorov},
     title = {On location of the matrix spectrum with respect to a parabola},
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G. V. Demidenko; V. S. Prokhorov. On location of the matrix spectrum with respect to a parabola. Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 26-40. http://geodesic.mathdoc.fr/item/MT_2023_26_1_a1/