Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type

Yu. I. Karlovich ; I. M. Spitkovsky

Izvestiya. Mathematics , Tome 34 (1990) no. 2, pp. 281-316.

Résumé

The authors study $n$- and $d$-normality and compute the index of systems of singular integral equations with a semi-almost periodic matrix-valued coefficient $G$, as well as the index of operators of convolution type on the half-line and a finite interval converging to it. At the base of the investigation lies factorization with almost periodic factors of matrix-valued functions describing the asymptotics of $G$ at $\pm\infty$. Bibliography: 38 titles.

Bibliographie

[1] Litvinchuk G. S., Spitkovskii I. M., Faktorizatsiya matrits-funktsii, Dep. v VINITI 17.04.84, No 2410-84, Odessa, 1984, 460 pp.; Litvinchuk G. S., Spitkovskii I. M., Factorization of measurable matrix functions, Akademie Verlag Basel, Berlin ; Birkhäuser Verlag, Boston, Stuttgart, 1987 | MR | Zbl

[2] Clancey K., F., Gohberg I. Z., Factorization of matrix functions and singular integral operators, Birkhauser Verlag, Basel, Boston, Stuttgart, 1981 | MR | Zbl

[3] Duduchava R. V., “Integralnye uravneniya svertki s razryvnymi predsimvolami, singulyarnye integralnye uravneniya s nepodvizhnymi osobennostyami i ikh prilozheniya k zadacham mekhaniki”, Tr. Tbilissk. Matem. in-ta AN GruzSSR, 60, 1979, 5–135 | MR

[4] Sarason D. E., “Toeplitz operators with semi-almost-periodic symbols”, Duke Math. J., 44:2 (1977), 357–363 | DOI | MR

[5] Saginashvili A. I., “Singulyarnye integralnye operatory s koeffitsientami, imeyuschimi razryvy polu-pochti-periodicheskogo tipa”, Soobsch. AN GruzSSR, 94:2 (1979), 289–291 | MR | Zbl

[6] Saginashvili A. I., “Singulyarnye integralnye operatory s polu-pochti-periodicheskimi razryvami u koeffitsientov”, Soobsch. AN GruzSSR, 95:3 (1979), 541–543 | MR | Zbl

[7] Saginashvili A. I., “Singulyarnye integralnye uravneniya s koeffitsientami, imeyuschimi razryvy polu-pochti-periodicheskogo tipa”, Tr. Tbilissk. in-ta AN GruzSSR, 66, 1980, 84–95 | MR | Zbl

[8] Duduchava R. V., Saginashvili A. I., “Integralnye uravneniya svertki na poluosi s polu-pochti-periodicheskimi predsimvolami”, Diff. uravn., 17:2 (1981), 301–312 | MR | Zbl

[9] Tribel X., Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR

[10] Ganin M. P., “Ob integralnom uravnenii Fredgolma s yadrom, zavisyaschim ot raznosti argumentov”, Izv. vuzov. Matematika, 1963, no. 2, 31–43 | MR | Zbl

[11] Novokshenov V. Yu., “Uravneniya v svertkakh na konechnom otrezke i faktorizatsiya ellipticheskikh matrits”, Matem. zametki, 27:6 (1980), 935–946 | MR | Zbl

[12] Paltsev B. V., “O zadache Dirikhle dlya odnogo differentsialnogo operatora, vstrechayuschegosya v teorii sluchainykh protsessov”, Izv. AN SSSR. Ser. matem., 41:6 (1977), 1348–1387 | MR

[13] Paltsev B. V., “Uravneniya svertki na konechnom intervale dlya odnogo klassa simvolov, imeyuschikh stepennuyu asimptotiku na beskonechnosti”, Izv. AN SSSR. Ser. matem., 44:2 (1980), 322–394 | MR

[14] Paltsev B. V., “Obobschenie metoda Vinera–Khopfa dlya uravnenii svertki na konechnom intervale s simvolami, imeyuschimi stepennuyu asimptotiku na beskonechnosti”, Matem. sb., 113:3 (1980), 355–399 | MR

[15] Karlovich Yu. I., Spitkovskii I. M., “O neterovosti, $n$- i $d$-normalnosti singulyarnykh integralnykh operatorov s matrichnymi koeffitsientami, dopuskayuschimi razryvy polu-pochti-periodicheskogo tipa”, Shkola po teorii operatorov v funktsionalnykh prostranstvakh, Tezisy dokladov, Minsk, 1982, 81–82

[16] Karlovich Yu. I., Spitkovskii I. M., “O neterovosti nekotorykh singulyarnykh integralnykh operatorov s matrichnymi koeffitsientami klassa SAP i svyazannykh s nimi sistem uravnenii svertki na konechnom promezhutke”, Dokl. AN SSSR, 269:3 (1983), 531–535 | MR | Zbl

[17] Karlovich Yu. I., Spitkovskii I. M., “Factorization problem for almost periodic matrix functions and Fredholm theory of Toeplitz operators with semi-almost-periodic matrix symbols”, Lect. Notes in Math., 1043, 1984, 279–282

[18] Levitan B. M., Pochti periodicheskie funktsii, GITTL, M., 1953

[19] Levin B. Ya., Raspredelenie kornei tselykh funktsii, GITTL, M., 1956

[20] Gokhberg I. Ts., Feldman I. L., Uravneniya v svertkakh i proektsionnye metody ikh resheniya, Nauka, M., 1971 | MR

[21] Boyarskii B. V., “Ob ustoichivosti zadachi Gilberta dlya golomorfnogo vektora”, Soobsch. AN GruzSSR, 21:4 (1958), 391–398 | Zbl

[22] Gokhberg I. Ts., Krein M. G., “Sistemy integralnykh uravnenii na polupryamoi s yadrami, zavisyaschimi ot raznosti argumentov”, UMN, 13:2 (1958), 3–72 | MR

[23] Vekua N. P., Sistemy singulyarnykh integralnykh uravnenii, Nauka, M., 1970 | MR | Zbl

[24] Gantmakher F. R., Teoriya matrits, Nauka, M., 1970 | MR

[25] Simonenko I. B., “Nekotorye obschie voprosy teorii kraevoi zadachi Rimana”, Izv. AN SSSR. Seriya matem., 32:5 (1968), 1138–1146 | MR | Zbl

[26] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[27] Sarason D. E., “Approximation of piecewise continuous functions by quotients of bounded analytic functions”, Canadian J. Math., 24 (1972), 642–65 | MR

[28] Spitkovskii I. M., “Nekotorye otsenki dlya chastnykh indeksov izmerimykh matrits-funktsii”, Matem. sb., 111:2 (1980), 227–248 | MR

[29] Krein S. G., Lineinye uravneniya v banakhovom prostranstve, Nauka, M., 1971 | MR

[30] Spitkovskii I. M., “O blochnykh operatorakh i svyazannykh s nimi voprosakh teorii faktorizatsii matrits-funktsii”, Dokl. AN SSSR, 254:4 (1980), 816–820 | MR | Zbl

[31] Spitkovskii I. M., Kriterii neterovosti blochno-treugolnykh operatorov i svyazannye s nim voprosy teorii faktorizatsii matrits-funktsii, Dep. v VINITI 28.05.81, No 2410-84, Odessa, 1981, 23 pp.

[32] Krupnik N. Ya., Banakhovy algebry s simvolom i singulyarnye integralnye operatory, Shtiintsa, Kishinev, 1984 | MR | Zbl

[33] Gokhberg I. Ts., Krupnik N. Ya., Vvedenie v teoriyu odnomernykh singulyarnykh integralnykh operatorov, Shtiintsa, Kishinev, 1973 | MR

[34] Karlovich Yu. I., Kravchenko V. G., “Algebra singulyarnykh integralnykh operatorov s kusochno nepreryvnymi koeffitsientami i kusochno gladkim sdvigom na slozhnom konture”, Izv. AN SSSR. Ser. matem., 47:5 (1983), 1030–1077 | MR

[35] Karlovich Yu. I., Kravchenko V. G., “O singulyarnom integralnom operatore s nekarlemanovskimi sdvigami na razomknutom konture”, Dokl. AN SSSR, 236:4 (1977), 792–795 | MR | Zbl

[36] Riss. F., Sekefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979

[37] Spitkovskii I. M., “Faktorizatsiya nekotorykh klassov polu-pochti-periodicheskikh matrits-funktsii i ee prilozheniya k sistemam uravnenii tipa svertki”, Izv. vuzov. Matematika, 1983, no. 4, 88–94 | MR | Zbl

[38] Karlovich Yu. I., Spitkovskii I. M., “K teorii sistem uravnenii tipa svertki s polu-pochti-periodicheskimi simvolami v prostranstvakh besselevykh potentsialov”, Dokl. AN SSSR, 86:4 (1986), 799–803 | MR