On smoothing the operator coefficient of a first-order differential operator in a Banach space
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Winter Mathematical School "Modern Methods of Function Theory and Related Problems", Voronezh, January 28 - February 2, 2021, Part 1, Tome 206 (2022), pp. 3-14.

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In this paper, we consider a first-order differential operator acting in Lebesgue spaces. The method of similar operators allows one to reduce the operator considered to an operator with a more convenient potential.
Keywords: method of similar operators, first-order differential operator
Mots-clés : Fourier coefficients, Lebesgue space.
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A. G. Baskakov; I. A. Krishtal; N. B. Uskova. On smoothing the operator coefficient of a first-order differential operator in a Banach space. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Winter Mathematical School "Modern Methods of Function Theory and Related Problems", Voronezh, January 28 - February 2, 2021, Part 1, Tome 206 (2022), pp. 3-14. http://geodesic.mathdoc.fr/item/INTO_2022_206_a0/

[1] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965. | MR

[2] Baskakov A. G., Zamena Krylova—Bogolyubova v teorii nelineinykh vozmuschenii lineinykh operatorov, In-t mat. AN USSR, Kiev, 1980 | MR

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

[4] Baskakov A. G., “Teorema o privodimosti lineinykh differentsialnykh uravnenii s kvaziperiodicheskimi koeffitsientami”, Ukr. mat. zh., 35:4 (1983), 416–421 | MR | Zbl

[5] Baskakov A. G., “Zamena Krylova—Bogolyubova v teorii vozmuschenii lineinykh operatorov”, Ukr. mat. zh., 36:5 (1984), 606–611 | MR | Zbl

[6] Baskakov A. G., “Metod usredneniya v teorii vozmuschenii lineinykh differentsialnykh operatorov”, Differ. uravn., 21:4 (1985), 555–562 | MR | Zbl

[7] Baskakov A. G. Teorema o rasscheplenii operatora i nekotorye smezhnye voprosy analiticheskoi teorii vozmuschenii, Izv. AN SSSR. Ser. mat., 50:3 (1986), 435–457 | MR

[8] Baskakov A. G., Garmonicheskii analiz lineinykh operatorov, Izd-vo VGU, Voronezh, 1987 | MR

[9] Baskakov A. G., “Spektralnyi analiz vozmuschennykh nekvazianaliticheskikh i spektralnykh operatorov”, Izv. RAN. Ser. mat., 58:4 (1994), 3–32 | Zbl

[10] Baskakov A. G., “Ob abstraktnom analoge preobrazovaniya Krylova—Bogolyubova v teorii vozmuschennykh lineinykh operatorov”, Funkts. anal. prilozh., 33:2 (1999), 76–80 | MR | Zbl

[11] Baskakov A. G., Krishtal I. A., “Garmonicheskii analiz kauzalnykh operatorov i ikh spektralnye svoistva”, Izv. RAN. Ser. mat., 69:3 (2005), 3–54 | MR | Zbl

[12] Baskakov A. G., Sintyaeva K. A., “O neravenstvakh Bora—Favara dlya operatorov”, Izv. vuzov. Mat., 2009 No 12, 14–21 | Zbl

[13] Baskakov A. G., Derbushev A. V., Scherbakov A. O., “Metod podobnykh operatorov v spektralnom analize nesamosopryazhennogo operatora Diraka s negladkim potentsialom”, Izv. RAN. Ser. mat., 75:3 (2011), 3–28 | MR | Zbl

[14] Burlutskaya M. Sh., “O smeshannoi zadache dlya uravneniya s chastnymi proizvodnymi pervogo poryadka s involyutsiei i s periodicheskimi kraevymi usloviyami”, Zh. vychisl. mat. mat. fiz., 54:1 (2014), 3–12 | MR | Zbl

[15] 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

[16] Baskakov A. G., Uskova N. B., “Metod Fure dlya differentsialnykh uravnenii pervogo poryadka s involyutsiei i gruppy operatorov”, Ufim. mat. zh., 10:3 (2018), 11–34 | MR | Zbl

[17] Baskakov A. G., Krishtal I. A., Uskova N. B., “Metod podobnykh operatorov v issledovanii spektralnykh svoistv vozmuschennykh differentsialnykh operatorov pervogo poryadka”, Itogi nauki i tekhn. Ser. Sovr. mat. prilozh. Temat. obz., 171 (2019), 3–18.

[18] 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

[19] 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

[20] Baskakov A. G., Krishtal I. A., Uskova N. B., “O spektralnykh svoistvakh operatora Diraka na pryamoi”, Differ. uravn., 57:2 (2021), 153–161 | Zbl

[21] Burlutskaya M. Sh., Khromov A. P., “Klassicheskoe reshenie dlya smeshannoi zadachi s involyutsiei”, Dokl. RAN., 435:2 (2010), 151–154 | Zbl

[22] Burlutskaya M. Sh., Khromov A. P. Smeshannye zadachi dlya giperbolicheskikh uravnenii pervogo poryadka, Dokl. RAN., 441:2 (2011), 156–159 | MR | Zbl

[23] Burlutskaya M. Sh., Khromov A. P. Metod Fure v smeshannoi zadache dlya uravneniya s chastnymi proizvodnymi pervogo poryadka s involyutsiei, Zh. vychisl. mat. mat. fiz., 51:12 (2011), 2233–2246 | MR | Zbl

[24] Broitigam I. N., Polyakov D. M., “Ob asimptotike sobstvennykh znachenii differentsialnogo operatora chetvertogo poryadka s matrichnymi koeffitsientami”, Differ. uravn., 54:4 (2018), 458–474 | MR | Zbl

[25] Broitigam I. N., Polyakov D. M., “Ob asimptotike sobstvennykh znachenii differentsialnogo operatora tretego poryadka”, Algebra i analiz., 31:4 (2019), 16–47 | MR

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

[27] Garkavenko G. V., Uskova N. B., “Spektralnyi analiz raznostnykh operatorov vtorogo poryadka s rastuschim potentsialom”, Tavrich. vestn. inform. mat., 2015, no. 3(28), 40–48

[28] 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

[29] Garkavenko G. V., Uskova N. B., “Metod podobnykh operatorov v issledovanii spektralnykh svoistv raznostnykh operatorov s rastuschim potentsialom”, Sib. elektron. mat. izv., 14 (2017), 673–689 | MR | Zbl

[30] Gantmakher F. R., Teoriya matrits, Fizmatlit, M., 2010 | MR

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

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

[33] Karpikova A. V., “Asimptotika sobstvennykh znachenii operatora Shturma—Liuvillya s periodicheskimi kraevymi usloviyami”, Ufim. mat. zh., 6:3 (2014), 28–34 | MR

[34] Karpikova A. V., “Asimptotika sobstvennykh znachenii integro-differentsialnogo operatora s periodicheskimi kraevymi usloviyami”, Vestn. Voronezh. un-ta. Ser. Fiz. Mat., 2015, no. 1, 153–156 | Zbl

[35] Katrakhov V. V., Sitnik S. M., “Metod operatorov preobrazovaniya i kraevye zadachi dlya singulyarnykh ellipticheskikh uravnenii”, Sovr. mat. Fundam. napr., 64:2 (2018), 211–426 | MR

[36] 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

[37] Levitan B. M., Pochti periodicheskie funktsii, Gostekhizdat, M., 1953 | MR

[38] Polyakov D. M., “Spektralnyi analiz differentsialnogo operatora chetvertogo poryadka s periodicheskimi i antiperiodicheskimi kraevymi usloviyami”, Algebra i analiz., 27:5 (2015), 117–152

[39] Polyakov D. M., “Spektralnyi analiz nesamosopryazhennogo operatora chetvertogo poryadka s negladkimi koeffitsientami”, Sib. mat. zh., 56:1 (2015), 165–184 | MR | Zbl

[40] Polyakov D. M., “Spektralnye svoistva odnomernogo operatora Shrëdingera”, Vestn. Voronezh. un-ta. Ser. Fiz. Mat., 2016, no. 2, 146–152 | Zbl

[41] Polyakov D. M., “Odnomernyi operator Shrëdingera s kvadratichno summiruemym potentsialom”, Sib. mat. zh., 59:3 (2018), 596–615 | MR | Zbl

[42] Polyakov D. M., “O spektralnykh kharakteristikakh nesamosopryazhennogo operatora chetvertogo poryadka s matrichnymi koeffitsientami”, Mat. zametki., 105:4 (2019), 637–642 | Zbl

[43] Polyakov D. M., “Otsenki dlin spektralnykh lakun operatorov Shrëdingera i Diraka”, Differ. uravn., 56:5 (2020), 595–604 | MR | Zbl

[44] Polyakov D. M., “Spektralnye otsenki dlya operatora chetvertogo poryadka s matrichnymi koeffitsientami”, Zh. vychisl. mat. mat. fiz., 60:7 (2020), 1201–1223 | Zbl

[45] Polyakov D. M., “O nelokalnom vozmuschenii periodicheskoi zadachi dlya differentsialnogo operatora vtorogo poryadka”, Differ. uravn., 57:1 (2021), 14–21 | Zbl

[46] Romanova E. Yu., “Spektralnyi analiz differentsialnogo operatora s involyutsiei”, Vestn. NGU. Ser. mat. mekh. inform., 14:4 (2014), 64–78 | Zbl

[47] Romanova E. Yu., “Spektralnyi analiz operatora Diraka v lebegovykh prostranstvakh”, Vestn. Voronezh. un-ta. Ser. Fiz. Mat., 2015, no. 2, 142–149 | Zbl

[48] Sitnik S. M., Shishkina E. L., Metod operatorov preobrazovaniya dlya differentsialnykh uravnenii s operatorom Besselya, Fizmatlit, M., 2019

[49] Uskova N. B., “K odnomu rezultatu R. Ternera”, Mat. zametki., 76:6 (2004), 905–917 | MR

[50] Uskova N. B., “O spektralnykh svoistvakh operatora Shturma—Liuvillya s matrichnym potentsialom”, Ufim. mat. zh., 7:3 (2015), 88–99 | MR

[51] Uskova N. B., “O spektralnykh svoistvakh odnogo differentsialnogo operatora vtorogo poryadka s matrichnym potentsialom”, Differ. uravn., 52:5 (2016), 579–588 | Zbl

[52] Uskova N. B., “Matrichnyi analiz spektralnykh proektorov vozmuschennykh samosopryazhennykh operatorov”, Sib. elektron. mat. izv., 16 (2019), 369–405 | MR | Zbl

[53] Fridrikhs K. O., Vozmuschenie spektra v gilbertovom prostranstve, Mir, M., 1969

[54] Shelkovoi A. N., “Spektralnye svoistva differentsialnogo operatora vtorogo poryadka, opredelyaemogo nelokalnymi kraevymi usloviyami”, Mat. fiz. kompyut. model., 21:4 (2018), 18–33 | MR

[55] Scherbakov A. O., “Spektralnyi analiz nesamosopryazhennogo operatora Shturma—Liuvillya s singulyarnym potentsialom”, Nauch. ved. Belgorod. un-ta. Ser. Mat. Fiz., 12 (155):31 (2013), 102–108

[56] Scherbakov A. O., “Regulyarizovannyi sled operatora Diraka”, Mat. zametki., 98:1 (2015), 134–146 | MR

[57] Baskakov A. G., Garkavenko G. V., Glazkova M. Yu., Uskova N. B., “On spectral properties of one class difference operators”, J. Phys. Conf. Ser., 1479 (2020), 01 | DOI

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

[59] Baskakov A. G., Krishtal I. A., Uskova N. B., “Linear differential operator with an involution as a generator of an operator group”, Oper. Matrices., 12:3 (2018), 723–756 | DOI | MR | Zbl

[60] 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

[61] 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

[62] Baskakov A. G., Krishtal I. A., Uskova N. B., “Closed operator functional calculus in Banach modules and applications”, J. Math. Anal. Appl., 492:2 (2020), 124473 | DOI | MR | Zbl

[63] Garkavenko G. V., Zgolich A. R., Uskova N. B. Spectral analysis of one class of the integro-differential operators, J. Phys. Conf. Ser., 1203 (2019), 012102 | DOI

[64] Delsarte J., “Hypergroupes et operateurs de permutation et de transmutation”, Colloque C.N.R.S. Nancy., 1956, 29–45 | MR | Zbl

[65] Delsarte J., Lions J. L. Transmutations d'operateurs differentiels dans le domaine complexe, Commun. Math. Helv., 32 (1957), 113–128 | DOI | MR | Zbl

[66] Friedrichs K. O., Lectures on Advanced Ordinary Differential Equations, Gordon and Breach, New York, 1965 | MR | Zbl

[67] Kravchenko V. V., Forward and inverse Sturm–Liouville problems: A Method of Solution, Springer-Verlag, Basel, 2020 | MR

[68] Kravchenko V. V., Sitnik S. M., Transmutation Operators and Applications, Birkhäuser, Basel, 2020 | MR | Zbl

[69] Polyakov D. M., “Formula for regularized trace of a second-order differential operator with involution”, J. Math. Sci., 251:5 (2020), 748–759 | DOI | MR | Zbl

[70] Reiter H., Stegeman J. D., Classical harmonic analysis and locally compact groups, Oxford Univ. Press, Oxford, 2000 | MR | Zbl

[71] Shishkina E., Sitnik S., Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics, Academic Press, 2020 | MR | Zbl