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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{MT_2023_26_1_a1,
     author = {G. V. Demidenko and V. S. Prokhorov},
     title = {On location of the matrix spectrum with respect to a parabola},
     journal = {Matemati\v{c}eskie trudy},
     pages = {26--40},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2023_26_1_a1/}
}
TY  - JOUR
AU  - G. V. Demidenko
AU  - V. S. Prokhorov
TI  - On location of the matrix spectrum with respect to a parabola
JO  - Matematičeskie trudy
PY  - 2023
SP  - 26
EP  - 40
VL  - 26
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2023_26_1_a1/
LA  - ru
ID  - MT_2023_26_1_a1
ER  - 
%0 Journal Article
%A G. V. Demidenko
%A V. S. Prokhorov
%T On location of the matrix spectrum with respect to a parabola
%J Matematičeskie trudy
%D 2023
%P 26-40
%V 26
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2023_26_1_a1/
%G ru
%F MT_2023_26_1_a1
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/

[1] Godunov S. K., Sovremennye aspekty lineinoi algebry, Nauchnaya kniga, Novosibirsk, 1997

[2] Yu. L. Daletskni, M. G. Krein, Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970

[3] A. G. Mazko, “Lokalizatsiya spektra i ustoichivost dinamicheskikh sistem”, Trudy Instituta matematiki NAN Ukrainy, 28, In-t matematiki NAN Ukrainy, Kiev, 1999

[4] E. A. Biberdorf, M. A. Blinova, N. I. Popova, “Modifications of the dichotomy method of a matrix spectrum and their application to stability tasks”, Numer. Anal. Appl., 11:2 (2018), 108–120

[5] E. A. Biberdorf, “Development of the matrix spectrum dichotomy method”, Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy A Liber Amicorum to Professor Godunov, eds. Demidenko G.V., Romenski E., Toro E., Dumbser M., Springer Nature, Cham, 2020, 37–43

[6] A. Bulgak, H. Bulgak, Linear algebra, Selcuk University, Konya, 2001

[7] A. Bulgak, G. Demidenko, I. Matveeva, “On location of the matrix spectrum inside an ellipse”, Selcuk J. Appl. Math., 4:1 (2003), 25–41

[8] G. Demidenko, “On a functional approach to spectral problems of linear algebra”, Selcuk J. Appl. Math., 2:2 (2001), 39–52

[9] A. G. Mazko, Matrix equations, spectral problems and stability of dynamic-systems, Cambridge Scientific Publishers, Cambridge, 2008

[10] M. Sadkane, A. Touhami, “On modifications to the spectral dichotomy algorithm”, Numer. Fund. Anal. Optim., 34:7 (2013), 817–791

[11] S. Traore, M. Dosso, “Spectral dichotomy methods of a matrix with respect to the general equation of the parabola”, Eur. J. Pure Appl. Math., 15:2 (2022), 681–725

[12] L. N. Trefethen, “Computation of pseudospectra”, Acta Numer., 8 (1999), 247–295