Geodesic
On spectral properties of one difference operator with involution
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 1, Tome 208 (2022), pp. 15-23.

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

We consider a difference operator with involution acting in the complex Hilbert space $l_2(\mathbb{Z})$. Using the method of similar operators, we reduce it to the diagonal (block diagonal) form, which allows one to obtain various spectral characteristics of the original operator and to construct biinvariant subspaces for it.
Keywords: method of similar operators, difference operator, spectrum, spectral projector. 
@article{INTO_2022_208_a2,
     author = {G. V. Garkavenko and N. B. Uskova},
     title = {On spectral properties of one difference operator with involution},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {15--23},
     publisher = {mathdoc},
     volume = {208},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_208_a2/}
}
TY  - JOUR
AU  - G. V. Garkavenko
AU  - N. B. Uskova
TI  - On spectral properties of one difference operator with involution
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 15
EP  - 23
VL  - 208
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_208_a2/
LA  - ru
ID  - INTO_2022_208_a2
ER  - 
%0 Journal Article
%A G. V. Garkavenko
%A N. B. Uskova
%T On spectral properties of one difference operator with involution
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 15-23
%V 208
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2022_208_a2/
%G ru
%F INTO_2022_208_a2
G. V. Garkavenko; N. B. Uskova. On spectral properties of one difference operator with involution. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 1, Tome 208 (2022), pp. 15-23. http://geodesic.mathdoc.fr/item/INTO_2022_208_a2/

[1] Baskakov A. G., “Metody abstraktnogo garmonicheskogo analiza v teorii vozmuschenii lineinykh operatorov”, Sib. mat. zh., 24:1 (1983), 21–39 | MR

[2] Baskakov A. G., Krishtal I. A., Uskova N. B., “Metod podobnykh operatorov v spektralnom analize operatornykh beskonechnykh matrits”, Prikl. mat. fiz., 52:2 (2020), 71–85 | MR

[3] Baskakov A. G., Krishtal I. A., Uskova N. B., “Metod podobnykh operatorov v spektralnom analize operatornykh beskonechnykh matrits. Primery. I”, Prikl. mat. fiz., 52:3 (2020), 185–194 | MR

[4] Baskakov A. G., Polyakov D. M., “Metod podobnykh operatorov v spektralnom analize operatora Khilla s negladkim potentsialom”, Mat. sb., 208:1 (2017), 3–47 | MR | Zbl

[5] Baskakov A. G., Uskova N. B., “Obobschennyi metod Fure dlya sistemy differentsialnykh uravnenii pervogo poryadka s involyutsiei i gruppy operatorov”, Differ. uravn., 54:2 (2018), 276–280 | Zbl

[6] Baskakov A. G., Uskova N. B., “Spektralnyi analiz differentsialnykh operatorov s involyutsiei i gruppy operatorov”, Differ. uravn., 54:9 (2018), 1281–1291

[7] Broitigam I. N., Polyakov D. M., “Asimptotika sobstvennykh znachenii beskonechnykh blochnykh matrits”, Ufim. mat. zh., 11:3 (2019), 10–29 | Zbl

[8] Burlutskaya M. Sh., “Smeshannaya zadacha dlya sistemy differentsialnykh uravnenii pervogo poryadka s nepreryvnym potentsialom”, Izv. Saratov. un-ta. Nov. ser. Mat. Mekh. Inform., 16:2 (2016), 145–151 | MR | Zbl

[9] Burlutskaya M. Sh., “Klassicheskoe i obobschennoe reshenie smeshannoi zadachi dlya sistemy uravnenii pervogo poryadka s nepreryvnym potentsialom”, Zh. vychisl. mat. mat. fiz., 59:3 (2019), 380–390 | MR | Zbl

[10] Burlutskaya M. Sh., Khromov A. P., “Funktsionalno-differentsialnye operatory s involyutsiei i operatory Diraka s periodicheskimi kraevymi usloviyami”, Dokl. RAN., 454:1 (2014), 15–17 | MR | Zbl

[11] Vladykina V. E., Shkalikov A. A., “Spektralnye svoistva obyknovennykh differentsialnykh operatorov s involyutsiei”, Dokl. RAN., 484:1 (2019), 12–17 | MR | Zbl

[12] Garkavenko G. V., “O diagonalizatsii nekotorykh klassov lineinykh operatorov”, Izv. vuzov. Mat., 1994, no. 11, 14–19 | MR | Zbl

[13] Garkavenko G. V., Uskova N. B., “Spektralnyi analiz raznostnykh operatorov vtorogo poryadka s rastuschim potentsialom”, Tavrich. vestn. inform. mat., 28:3 (2015), 40–48

[14] Garkavenko G. V., Uskova N. B., “Metod podobnykh operatorov v issledovanii spektralnykh svoistv odnogo klassa raznostnykh operatorov”, Vestn. Voronezh. un-ta. Ser. Fiz. Mat., 2016, no. 3, 101–111 | Zbl

[15] Garkavenko G. V., Uskova N. B., “Spektralnyi analiz odnogo klassa raznostnykh operatorov s rastuschim potentsialom”, Izv. Saratov. un-ta. Nov. ser. Mat. Mekh. Inform., 16:4 (2016), 395–402 | MR | Zbl

[16] Garkavenko G. V., Uskova N. B., “Asimptotika sobstvennykh znachenii raznostnogo operatora s rastuschim potentsialom i polugruppy operatorov”, Mat. fiz. komp. model., 20:4 (2017), 6–17 | MR

[17] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965

[18] Danford N., Shvarts Dzh. T., Lineinye operatory. Spektralnye operatory. T. 3, Mir, M., 1974

[19] Krishtal I. A., Uskova N. B., “Spektralnye svoistva differentsialnykh operatorov pervogo poryadka s involyutsiei i gruppy operatorov”, Sib. elektron. mat. izv., 16 (2019), 1091–1132 | Zbl

[20] Rudin U., Funktsionalnyi analiz, Mir, M., 1975 | MR

[21] Skrynnikov A. V., “O kvazinilpotentnom variante metoda Fridrikhsa v teorii podobiya lineinykh operatorov”, Funkts. anal. prilozh., 17:3 (1983), 89–90 | MR | Zbl

[22] Baskakov A. G., Krishtal I. A., Romanova E. Yu., “Spectral analysis of a differential operator with an involution”, J. Evol. Equations., 17 (2017), 669–684 | DOI | MR | Zbl

[23] Baskakov A. G., Krishtal I. A., Uskova N. B., “Similarity techniques in the spectral analysis of perturbed operator matrices”, J. Math. Anal. Appl., 477:2 (2019), 930–960 | DOI | MR | Zbl

[24] Baskakov A. G., Krishtal I. A., Uskova N. B., “On the spectral analysis of a differential operator with an involution and general boundary conditions”, Eurasian Math. J., 11:2 (2020), 30–39 | DOI | MR | Zbl

[25] Boutet de Monvel A., Zielinski L., “Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices”, Centr. Eur. J. Math., 12:3 (2014), 445–463 | MR | Zbl

[26] Hinkkannen A., “On the diagonalization of a certain class of operators”, Michigan Math. J., 32:3 (1985), 349–359 | DOI | MR

[27] Ikebe Y., Asai N., Miyazaki Y., Cai D., “The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application”, Lin. Alg. Appl., 241–243 (1996), 599–618 | DOI | MR | Zbl

[28] Janas J., Malejki M., “Alternative approaches to asymptotic behavior of eigenvalues of some unbounded Jacobi matrices”, J. Comput. Appl. Math., 200 (2007), 342–356 | DOI | MR | Zbl

[29] Janas J., Naboko S., “Infinite Jacobi matrices with unbounded entries: Asymptotics of eigenvalues and the transformation operator approach”, SIAM J. Math. Anal., 36:2 (2004), 643–658 | DOI | MR | Zbl

[30] Kopzhassarova A. A., Lukashov A. L., Sarsenbi A. M., “Spectral properties of non-self-adjoint perturbations for a spectral problem with involution”, Abstr. Appl. Anal., 2012 (2012), 590781 | MR | Zbl

[31] Malejki M., “Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices”, Lin. Alg. Appl., 431 (2009), 1952–1970 | DOI | MR | Zbl

[32] Malejki M., “Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices”, Opuscula Math., 30:3 (2010), 311–330 | DOI | MR | Zbl

[33] Malejki M., “Eigenvalues for some complex infinite tridiagonal matrices”, J. Adv. Math. Comp. Sci., 26:5 (2018), 1–9 | DOI