Geodesic
Lyapunov Transformation of Differential Operators with Unbounded Operator Coefficients
Matematičeskie zametki, Tome 99 (2016) no. 1, pp. 11-25

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

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator of multiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established.
Keywords: Lyapunov transformation, evolution operator, perturbed differential operator, Cauchy problem, Lyapunov kinematic similarity, exponential dichotomy, splitting pair of functions, Bohl spectrum. 
M. S. Bichegkuev. Lyapunov Transformation of Differential Operators with Unbounded Operator Coefficients. Matematičeskie zametki, Tome 99 (2016) no. 1, pp. 11-25. http://geodesic.mathdoc.fr/item/MZM_2016_99_1_a1/
@article{MZM_2016_99_1_a1,
     author = {M. S. Bichegkuev},
     title = {Lyapunov {Transformation} of {Differential} {Operators} with {Unbounded} {Operator} {Coefficients}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {11--25},
     year = {2016},
     volume = {99},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_99_1_a1/}
}
TY  - JOUR
AU  - M. S. Bichegkuev
TI  - Lyapunov Transformation of Differential Operators with Unbounded Operator Coefficients
JO  - Matematičeskie zametki
PY  - 2016
SP  - 11
EP  - 25
VL  - 99
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_2016_99_1_a1/
LA  - ru
ID  - MZM_2016_99_1_a1
ER  - 
%0 Journal Article
%A M. S. Bichegkuev
%T Lyapunov Transformation of Differential Operators with Unbounded Operator Coefficients
%J Matematičeskie zametki
%D 2016
%P 11-25
%V 99
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_2016_99_1_a1/
%G ru
%F MZM_2016_99_1_a1

[1] S. G. Krein, Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967 | MR | Zbl

[2] Yu. L. Daletskii, M. G. Krein, Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nelineinyi analiz i ego prilozheniya, Nauka, M., 1970 | MR | Zbl

[3] E. Khille, R. Fillips, Funktsionalnyi analiz i polugruppy, IL, M., 1962 | MR | Zbl

[4] A. G. Baskakov, “Polugruppy raznostnykh operatorov v spektralnom analize lineinykh differentsialnykh operatorov”, Funkts. analiz i ego pril., 30:3 (1996), 1–11 | DOI | MR | Zbl

[5] A. G. Baskakov, “Issledovanie lineinykh differentsialnykh uravnenii metodami spektralnoi teorii raznostnykh operatorov i lineinykh otnoshenii”, UMN, 68:1 (2013), 77–128 | DOI | MR | Zbl

[6] M. S. Bichegkuev, “K spektralnomu analizu raznostnykh i differentsialnykh operatorov v vesovykh prostranstvakh”, Matem. sb., 204:11 (2013), 3–20 | DOI | MR | Zbl

[7] M. S. Bichegkuev, “O spektre raznostnykh i differentsialnykh operatorov v vesovykh prostranstvakh”, Funkts. analiz i ego pril., 44:1 (2010), 80–83 | DOI | MR | Zbl

[8] A. G. Baskakov, “Lineinye differentsialnye operatory s neogranichennymi operatornymi koeffitsientami i polugruppy raznostnykh operatorov”, Matem. zametki, 59:6 (1996), 811–820 | DOI | MR | Zbl

[9] A. G. Baskakov, “O korrektnosti lineinykh differentsialnykh operatorov”, Matem. sb., 190:3 (1999), 3–28 | DOI | MR | Zbl

[10] M. S. Bichegkuev, “Ob usloviyakh razreshimosti raznostnykh uravnenii s nachalnym usloviem iz podprostranstva”, Sib. matem. zhurn., 51:4 (2010), 751–768 | MR | Zbl

[11] M. S. Bichegkuev, “K spektralnomu analizu differentsialnykh operatorov s neogranichennymi operatornymi koeffitsientami”, Differents. uravneniya, 51:4 (2015), 427–435 | DOI | Zbl

[12] A. Ben-Artzi, I. Gohberg, M. A. Kaashoek, “Invertibility and dichotomy of differential operators on a half-line”, J. Dynam. Differential Equations, 5:1 (1993), 1–36 | DOI | MR | Zbl

[13] B. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR | Zbl

[14] N. P. Erugin, Lineinye sistemy obyknovennykh differentsialnykh uravnenii, Izd-vo AN BSSR, Minsk, 1963 | MR | Zbl

[15] A. G. Baskakov, “Spektralnyi analiz differentsialnykh operatorov s neogranichennymi operatornymi koeffitsientami, raznostnye otnosheniya i polugruppy raznostnykh otnoshenii”, Izv. RAN. Ser. matem., 73:2 (2009), 3–68 | DOI | MR | Zbl

[16] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., 194, Springer-Verlag, New York, 2000 | MR | Zbl

[17] A. G. Baskakov, “Garmonicheskii i spektralnyi analiz operatorov s ogranichennymi stepenyami i ogranichennykh polugrupp operatorov na banakhovom prostranstve”, Matem. zametki, 97:2 (2015), 174–190 | DOI | MR | Zbl

[18] A. G. Baskakov, N. S. Kaluzhina, “Teorema Berlinga dlya funktsii s suschestvennym spektrom iz odnorodnykh prostranstv i stabilizatsiya reshenii parabolicheskikh uravnenii”, Matem. zametki, 92:5 (2012), 643–661 | DOI | MR | Zbl

[19] A. G. Baskakov, N. S. Kaluzhina, D. M. Polyakov, “Medlenno menyayuschiesya na beskonechnosti polugruppy operatorov”, Izv. vuzov. Matem., 2014, no. 7, 3–14 | Zbl

[20] A. G. Baskakov, “Spektralnye kriterii pochti periodichnosti reshenii funktsionalnykh uravnenii”, Matem. zametki, 24:2 (1978), 195–206 | MR | Zbl

[21] A. G. Baskakov, A. Yu. Duplischeva, “Raznostnye operatory i operatornye matritsy vtorogo poryadka”, Izv. RAN. Ser. matem., 79:2 (2015), 3–20 | DOI | MR | Zbl